vérification symbolique de modèles à l’aide de systèmes de
TRANSCRIPT
HAL Id: tel-00579490https://tel.archives-ouvertes.fr/tel-00579490v2
Submitted on 12 Apr 2011
HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, estdestinée au dépôt et à la diffusion de documentsscientifiques de niveau recherche, publiés ou non,émanant des établissements d’enseignement et derecherche français ou étrangers, des laboratoirespublics ou privés.
Vérification symbolique de modèles à l’aide de systèmesde ré-écriture dédiés
Duy-Tùng Nguyên
To cite this version:Duy-Tùng Nguyên. Vérification symbolique de modèles à l’aide de systèmes de ré-écriture dédiés.Ordinateur et société [cs.CY]. Université d’Orléans, 2010. Français. �NNT : 2010ORLE2030�. �tel-00579490v2�
!"#$%&"'( )*+%,(-!&
!"#$ %"!&"'(#$
)!*$+!$) $& &$!,+"#"-*$)
!"#$!%#&$' ( &)#
'./&$ !"#$%&"$ '! (
)012'345 !6 78!
*+,-./,. 0. ( 9: ;<=;>?@ 9A:A
!"# !$%&'(# )& *#+,& ,& - !"#$%& '$ ()%*+,$&-+#. ')/&(.0*-
.(/0( )('&1 2 30(+)(%3 - 1*2!&30#+4%$
#B?CD<E=C;4 F1G>;HCI0@ J@ G;JKH@F L H*ECJ@ J@
F1F=KG@F J@ ?B2B<?C=0?@F JBJCBF
56789 '+&+:.$ ;0& <
=$0*>?+"@$( A/BCD9BD 4#!5&//&"#6 7'(8&#/(%3 ,9:#)3+'/
DEFF/D59BD8 <
+'+$& GBA68 4#!5&//&"#6 7'(8&#/(%3 ,& ;&'<8&
H0I&+"$ J/D /K 4#!5&//&"#6 7'(8&#/(%3 ,& 4+#(/ =>
=BDL <
L!@0* G/1A6B5 ?+@%#& ,& 0!'53#&'0&/6 7'(8&#/(%3 ,9:#)3+'/
E@3$' G/BE==EK1 4#!5&//&"#6 7'(8&#/(%3 ,& 4+#(/ =>>
+'+$& GBA68 4#!5&//&"#6 7'(8&#/(%3 ,& ;&'<8&
=$0*>?+"@$( A/BCD9BD 4#!5&//&"#6 7'(8&#/(%3 ,9:#)3+'/
8$&:$ 6E E 4#!5&//&"#6 AB2 C+0D+'
H0I&+"$ J/D /K 4#!5&//&"#6 7'(8&#/(%3 ,& 4+#(/ =>
!"!#$%!"!&'(
! "#!$% !$ "&'" ()!*#!) +#!' , )!*!)-#!) ./ 0#1#!) 2'-3% !" ./ 456)#-! 7&)1&$8
(&') *95:&#) ;5#" +93&$$!') 1! )5((&)"!) -!""! "3<%!8 (&') +9#$"=)>" ?'9#+% &$" (&)"=
, *&$ ")5:5#+8 !" +!')% ()=-#!'@ -&**!$"5#)!% !" )!*5)?'!%/ .5 A)5"#"'1! %951)!%%!
5'%%# , ./ B&35$ 2&#-3'"8 ./ C3*!1 2&'5DD5$#8 ./ E!)A! F51151 !" ./ G)=A&#)!
E'")! (&') 5:&#) 5--!("= 1! ;5#)! (5)"#! 1' D')H 1! -!""! "3<%!/ .5+3!')!'%!*!$"8
./ G)=A&#)! E'")! $! (!'@ (5% (5)"#-#(!) 1! +5 %&'"!$5$-! , -5'%! 1' ()&6+<*! 1!
")5$%(&)"/
! )!*!)-#! =A5+!*!$" ./ !5$I.#-3!+ J&':)!')8 1956&)1 !$-51)5$" 1! %"5A! 1!
0KC 5' L56&)5"&#)! 2&)1!+5#% 1! M!-3!)-3! !$ N$;&)*5"#?'! OL52MNP8 +9'$#:!)%#"=
1! 2&)1!5'@ N8 ('#% 1#)!-"!') 1! "3<%! , +9'$#:!)%#"= 19Q)+=5$%8 (&') 5:&#) %'#:# !"
%'((&)"= *&$ ")5:5#+ 5' -&')% 1! -!% 5$$=!%/
! "#!$% =A5+!*!$" , )!*!)-#!) ./ B&35$ 2&#-3'"/ E&$ !@(=)#!$-! 15$% +! 1&I
*5#$! 1' %H%"<*!% 1! )=I=-)#"')! 5 ="= +!% (+'% ()=-#!'@/
J!""! "3<%! $95')5#" %5$% 1&'"! (5% ="= '$! !@(=)#!$-! 5'%%# !$)#-3#%%5$"! !"
5A)=56+! %5$% +5 -&$:#:#5+#"= 1!% *!*6)!% 1' L56&)5"&#)! 19N$;&)*5"#?'! 4&$15*!$I
"5+! 19Q)+=5$% OLN4QP8 +9'$#:!)%#"= 19Q)+=5$%/ N+% %&$" ")&( $&*6)!'@ (&') ?'! D! +!%
-#"! "&'%8 *5#% D! +!% )!*!)-#! -35+!')!'%!*!$"/
.!)-# , *!% (5)!$"%8 , *5 %&!') 5#$%# !" , *5 -&(#$! (&') +!') 5(('# *&)5+/
K" *!)-# !$-&)! , "&'% *!% 5*#% ?'! D! $95# (5% -#"= #-#8 *5#% ?'# *=)#"!)5#!$" 1!
+9>")!/
!"#$%&' (
!"#$%&'"($! )*!*#+,-
!"#$%&'(&()
!" #!$%&'()!" *! !"#$%&'#() +&# *,-. /0)" *! #1!&#! 2&" 340215! 677789 :00!"
+&# ,#, *,;!0+//,!" *2&" )& /1!<'!1 #!</" /21 *!" ,()'/!" *! $%!1$%!)1" !# "+&# *!
/0)" !& /0)" )#'0'",!" *2&" 0! <'0'!) '&*)"#1'!0 /+)1 0=2&20>"! *=)&! ?12&*! ;21',#, *!
">"#@<!" A">"#@<!" <2#,1'!0"B 0+?'$'!0"B ">"#@<!" 1,2$#'C"B ">"#@<!" #!</" 1,!0D9 E0 !"#
<2'&#!&2&# /1+);, ()! $!" #!$%&'()!" "+&# !F$2$!" !# "+&# C1,()!<<!&# )#'0'",!"
/+)1 *,#!$#!1 *!" G+?)!" *2&" *!" $2" '&*)"#1'!0"9 H! &+<G1!)"!" ,#)*!" "+&# !&
$+)1" /+)1 ,021?'1 0!)1" $%2</" *=2//0'$2#'+&" !# 2<,0'+1!1 0!)1 !F$2$'#,9 4!$' &+)"
$+&*)'# . /!&"!1 ()! 0!" 2//0'$2#'+&" '&*)"#1'!00!" $+&#'&)!&# . "! <)0#'/0'!1 *!
<2&'@1! "'?&'I$2#';! *2&" 0!" /1+$%2'&!" 2&&,!"9
H2&" 0!" 2&&,!" J7B 0!" '&*)"#1'!" *!" $+</+"2&#" ,0!$#1+&'()!"B *2&" 0!)1
1!$%!1$%! *=+)#'0" /+)1 2<,0'+1!1 0! &';!2) *! $+&I2&$! *! 0!)1" /1+*)'#"B +&# 2*+/#,
0!" *#&+"&,,-. /- *!%#.#(). 0#)&#"-. AKHHD 3K1>2&# LJMNB K1>2&# LJJ68B )&!
"#1)$#)1! *! 02 !"#$%&'#() .1,2(3#45-B /+)1 #12'#!1 *!" $+</+"2&#" *! /0)"
!& /0)" $+</0!O!"9 !" KHH "+&# *!" "#1)$#)1!" $+*2&# *!" C+&$#'+&" G++0,!&&!"9
E0" /!);!&# P#1! ;)" $+<<! *!" 21G1!" +Q 0!" ,#2#" 1!/1,"!&#!&# *!" $%+'O *!
;20!)1" *! ;21'2G0!" G++0,!&&!" R )& +1*1! #+#20 ")1 0!" ;21'2G0!" ?212&#'# 0=)&'$S
'#, *) $+*2?! *=)&! C+&$#'+&9 !" #!$%&'()!" *! /21#2?! *! "#1)$#)1!"B $+<G'&,!"
. *!" <,#%+*!" *! 1,*)$#'+&"B $+&*)'"!&# . *!" '</0,<!#'+&" !O#1P<!<!&# !CS
I$2$!" !& /12#'()! 3T'&2#+ LJJ7B U)0?221* LJJJ89 V'&"'B *!" ;,1'I$2#'+&" !O%2)"S
#';!" +&# /) P#1! 1,20'",!" ")1 *!" ">"#@<!" $+</1!&2&# *!" <'00'21*" *=,#2#"
3T'&2#+ LJJ7B U)0?221* LJJJ8 AW+'1 $C9 0=21#'$0! !"#$%&'( )%*+& ,-+(.'/0 1 1020
2343+2 4/* 5+"%/* 3K)1$% LJJ68D9 ! /+);+'1 *=!O/1!""'+& *!" KHH !"# ")F"2&#
/+)1 <2&'/)0!1 )&! ?12&*! $02""! *! ">"#@<!" I&'" 3K)1$% LJJ689 4+<<! 0! &+<G1!
*! ;21'2G0!" *!" ">"#@<!" ,#)*'," !"# )& C2$#!)1 $1'#'()!B *! &+<G1!)"!" "#1)$#)1!"
6 &4 577 +&# ,#, /1+/+",!" 3T'&!1 LJJJB 4'21*+ 677789 H=2)#1!" "#1)$#)1!" +&#
/0)#X# $%!1$%, . ,#!&*1! 0! *+<2'&! *=2//0'$2#'+& *! $!" #!$%&'()!" 3K2%21 LJJYB
Z)/#2 LJJ[B Z)/#2 LJJ\B T2)G+1?&! LJJJB ]!^!0 LJJJB T2)G+1?&! 677789
H2&" 0! $2*1! *! /1+-!#" '&*)"#1'!0"B *2&" 34+);1!)1 67768B '0" +&# $+&_) )&!
&+);!00! "#1)$#)1! 6 &4 577B 0!"*#&+"&,,-. /- *!%#.#(). /- *())!-. AHHHD9
=+G-!$#'C ,#2'# *! C+)1&'1 )& +)#'0 `!O'G0! ()' /!)# P#1! 2)#2&# ()! /+""'G0! 2*2/#,
/+)1 02 ;,1'I$2#'+& *! #+)# #>/! *! <+*@0!" !# ()' +^1! *!" $2/2$'#," *! #12'#!<!&#
"'<'02'1!" 2)O KHH9 V 02 *'^,1!&$! *!" KHHB 0!" +/,12#'+&" ")1 $!" "#1)$#)1!" &!
"+&# /2" /1,S*,I&'!"B <2'" )&! $02""! *=+/,12#!)1"B 2//!0,! %+<+<+1/%'"<!"B !"#
'+*)'#! /+)1 /!1<!##1! . )& )#'0'"2#!)1 *! $+&$!;+'1 "!" /1+/1!" +/,12#'+&"9 H2&"
! "#$% &'( )* +,&'-./0& -, 12,2'$3(
! "#$%&!' &!( )*+,*-&!( .! (#./ 0*( -##&1!..!( 2 !&&!( 0+!..!./ &!3+( )*&!3+( $*.(
$!( $#"*,.!( .#. .1 !((*,+!"!./ -#+.1(4 5.! *3/+! *+* /1+,(/,63! ,./1+!((*./! !(/
637*3 3. #+$+! (3+ &!( )*+,*-&!( !(/ 0+1(300#(1 $*.( &* $18.,/,#.4 9! 0&3(' 3.!
)*+,*-&! 0!3/ *00*+*:/+! 0&3(,!3+( ;#,( $*.( 3. "<"! =!",.4 >!//! 0+#0+,1/1 !(/
/+%( 3/,&! 63*.$ #. "*.,03&! $!( (/+3 /3+!( $?.*",63!( #""! &!( 8&!(4 @+A ! B
&* C+*.$! D!E,-,&,/1 $! &* (/+3 /3+!' &!( 999 (#./ =#,(,( F$*.( &! *$+! $3 0+#G!/
HIJKL' 3. 0+#G!/ JMNO PQ+1*./ RSSTUV #""! &!( (/+3 /3+!( 0#3+ &* )1+,8 */,#.
$! (?(/%"!( $1 +,/( !. O;W P@,&&,!+( RSSTU' 3. &*.C*C! $! 0+#/#/?0*C! $! =*3/ .,)!*34
M#3( *)#.( "#./+1 &7*0/,/3$! $!( 999 B /+*,/!+ $!( 0+#C+*""!( $3 O;X' 3. &*.C*C!
0#3+ &* )1+,8 */,#. PMC3?!. RSSY' MC3?!. RSSZU4
O!( /! =.,63!( $! 42520' &/'( .(6 &('7(6 P9!+(=#[,/\ ]^^S' Q**$!+ ]^^_'
I=&!-3( = RSSR' `*./!"* RSSaU' *)! 3.! -*(! "*/=1"*/,63! (#&,$! !/ $! .#"b
-+!3(!( +! =!+ =!( (#./ *00&, *-&!( *3/#3+ $!( 0+#-&%"!( ,.=1+!./( B &* +1b1 +,/3+!
c.B.d. /!+",.*,(#.' *& 3& $!( * !((,-&!(' !/ 4 W*+", &!( /! =.,63!( $! )1+,8 *b
/,#.' .#3( +!/+#3)#.( !. +1b1 +,/3+! &7163,)*&!. ! $3 !"#$%&'#&()*+ 0#3+ &!( (?(b
/%"!( 8.,(4 W#3+ &!( (?(/%"!( ,.8.,(' &!( /! =.,63!( $! !"#$%& '()!$ *+!,-./"
#./ 1/1 ",(! !. 0&* ! Pc!(/!. ]^^Z' d#&0!+ ]^^_' Q#3*GG*., RSSS' e-$3&&* RSSY'
@!.!/ RSS]' f!3,&&*$! RSST' Q#, =3/ RSS_U4 9*.( &!( $!3E' 3. *& 3& $!( * !((,-&!(
#3 $73.! (3+b*00+#E,"*/,#. !(/ &! #!3+ $! &* /! =.,63!4
>! , .#3( #.$3,/ B #.(/+3,+! 3.! .#3)!&&! (/+3 /3+! $! &* )1+,8 */,#. (?"-#&b
,63! -*(1! (3+ &!( (?(/%"!( $! +1b1 +,/3+!(4
!"#$%&'(
O7#-G! /,; $! !//! /=%(! !(/ $! #.(/+3,+! 3. .#3)!*3 /?0! $! (?(/%"!( $! +1b
1 +,/3+! 0#3+ &* )1+,8 */,#. (?"-#&,63! $! "#$%&!(4 M#3( *00!&#.( &!( ,-,./ #, "#
01%1&0).20# 3!*&.)!**#$,4 M#3( "#./+#.( 63! .#/+! "#$%&! * &* 03,((*. ! $7!E0+!((,#.
$!( (?(/%"!( $! +1b1 +,/3+! !/ 637,& !(/ -,!. *$*0/1 B &71/3$! $! 0+#0+,1/1( $! (g+!/1
!/ $! 0+#0+,1/1( $! &#C,63! /!"0#+!&&! $! "#$%&!(4
M#3( *&&#.( "!//+! !. 1),$!. ! 3.! (#3( &*((! $! (?(/%"!( ;#. /,#..!&(' &!(
1$1 #*.4)0#, !/ &!( 1$1 #*.4)0#, 5 "0!).#' 0+1(!+)*./ &* 03,((*. ! $7!E0+!((,#. $!(
(?(/%"!( ;#. /,#..!&( !/ $!( /! =.,63!( $7* 1&1+*/,#. $!( *& 3&( *-#3/,((*./ B 3.
#3/,& $! )1+,8 */,#. (?"-#&,63! !h * !4
)*+%,&!-%&*+(
M#( #./+,-3/,#.( (#./ $,),(1!( !. $!3E *(0! /( i e(0! / /=1#+,63! !/ e(0! /
!E01+,"!./*&4
j 86%(0& -' 9/(* M#3( *)#.( 0+#0#(1 3. #3/,& ;#+"!& B -*(! $! (?(b
/%"!( $! +1b1 +,/3+! 0#3+ &* )1+,8 */,#. (?"-#&,63! $! "#$%&!( $1 ,$*-&!(4
P>#3)+!3+ RSS_' Q#, =3/ RS]SU4
]4 >#. !0/,#. $!( (?(/%"!( $! +1b1 +,/3+! ;#. /,#..!&( !/ 1&1"!./*,+!(4
!"#$%&!'"()&# *#(!* +*$ $,$(-'*$ +* !./.0!)(1!* 02"$$)31*$4 +*$ %�/
()&##*2$ *( +*$ .2.'*#(")!*$5
65 7�*8()&# *( &8()')$"()&# +*$ "29&!)(:'*$ +;.<"21"()&# +*$ $,$(-'*$
%�()&##*2$ .2.'*#(")!*$5
=5 >&+.2)$"()&# +*$ !.$*"1? +* @*(!) &!+)#")!*$ A@B C *( :).!"!0:)31*$ 8"!
$,$(-'*$ %�()&##*2$ .2.'*#(")!*$5
D5 7�*8()&# +*$ $,$(-'*$ +* !./.0!)(1!* 8&1! +.0!)!* *( <.!)E*! +*$
8!&8!).(.$ +* 2&9)31*$ (*'8&!*22*$ (*22*$ 31* F F *( 7 F ")#$) 31* +*$
8!&8!).(.$ +* $G!*(.5
H !"#$%& $'#()*+$,&-./ I&1$ "<&#$ )'82.'*#(. 1# *#$*'J2* +;&1()2$ J"$.$
$1! +*$ $,$(-'*$ %�()&##*2$ .2.'*#(")!*$ 2)$(. 0)/+*$$&1$ K
L5 M'82.'*#("()&# +;1# <.!)E0"(*1! +* $,$(-'*$ %�()&##*2$ .2.'*#(")!*$5
65 7�*8()&# +;1# &1()2 +* (!"#$%&!'"()&# "1(&'"()31* +*$ '&+-2*$ +*
!.$*"1? +* @*(!) &!+)#")!*$ *( :).!"!0:)31*$ <*!$ +*$ $,$(-'*$ %�()&##*2$
.2.'*#(")!*$5
=5 N."2)$"()&# +;1# &1()2 8*!'*(("#( +* +.0!)!* *( <.!)E*! +*$ 8!&8!).(.$
+*$ 2&9)31*$ (*'8&!*22*$ (*22*$ 31* F F *( 7 F5
I&1$ "<&#$ 0&'8"!. #&(!* <.!)E0"(*1!4 +;1#* 8"!( "<*0 +*$ &1()2$ +* !./.0!)(1!*
(*2$ 31* )'J1O4 >"1+* *( P> $1! +*$ '&+-2*$ "!J&!*$0*#($4 +;"1(!* 8"!(
"<*0 +*$ &1()2$ +* <.!)E0"()&# (*2$ 31* Q@MI4 I1Q>R4 Q>SN *( TQUU $1!
+*$ '&+-2*$ +* !.$*"1? +* @*(!) "E# +* '&#(!*! #&$ 8*!%&!'"#0*$ 8&1! +*$
$,$(-'*$ +.0)+"J2*$5
!"#$%&#'%($ )* +# ',-&*
S8!-$ 2* 0:"8)(!* +;)#(!&+10() 2" (:-$* $*!" 8!.$*#(.* *# D 9!"#+*$ 8"!()*$
$&1$/+)<)$.*$ *# LV 0:"8)(!*$5
U"#$ 2" 8!*')-!* 8"!()* A0)(.*+*,-*)$"C4 #&1$ !"88*2&#$ 2*$ 8!)#0)8*$ +* 2"
'&+.2)$"()&# *( +* 2" <.!)E0"() *# 8"!()012)*! 2*$ 0�*8($ W J"$* +* $,$(-'*$
+* !./.0!)(1!*5 7*((* 8"!()* 0&#()*#( .9"2*'*#( 1#* .(1+* +* 0"$ 8&1! '&#(!*! 2*
+.'"!0:* +* 2" '&+.2)$"()&# *( +* 2" <.!)E0"()&# W 2;")+* +* $,$(-'*$ +* !./.0!)(1!*5
F" +*1?)-'* 8"!()* A12"&3+$ 4$ 5(6(%)*&7)$ 89,%&*9,,$.C 8!.$*#(* +*$
8!)#0)8*$ +* $,$(-'* !./.0!)(1!* %�()&##*2 *( +*$ "29&!)(:'*$ +;.<"21"()&# 8&1!
0*$ $,$(-'*$ 8"! 1#* "#"2,$* +;"00*$$)J)2)(.5 @&1! *?82&!*! (&1$ 2*$ .("($ 8&$$)J2*$
+;1# $,$(-'*4 &# 2* 8"!0&1!( *# 8"!("#( +;1# .("( )#)()"25 F* 8!&J2-'* +* 2;*?82&/
!"()&# *?:"1$()<* *$( 31;)2 , " J*"10&18 +* 8&$$)J)2)(.$4 'X'* 8&1! 1# &!+)#"(*1!5
@21$)*1!$ $&21()&#$ $&#( +)$8&#)J2*$4 #&1$ #;*# 8!.$*#(&#$ 31* 31*231*$ 1#*$5
F" (!&)$)-'* 8"!()* A!##.*%-&*9, : .- ()*;%-&*9,C '&#(!* 2" 0"8"0)(. +;/
*?8!*$$)&# +*$ 8!&J2-'*$ 0&##1$ *( 2" 8*!%&!'"#0* +*$ $,$(-'*$ %�()&##*2$ .2./
'*#(")!*$ 8"! +*$ '&+-2*$ +* !.$*"1? +* @*(!) &!+)#")!*$ *( :).!"!0:)31*$ ")#$) 31*
! "#$%!&'( )* +,&'-./0&!-, 12,2'$3(
!" #$%#$&'('" !" )%*&+,!" (!-#%$!))!" ./.0 1/.0 !( !" #$%#$&'('" ! "2$!('3 .!"
$'",)(4(" "(4(&"(&+,!" 5%," &"!5( )4 )&-&(! !" %,(&)" "45" ,(&)&"!$ )!" (!675&+,!" 8469
6')'$4(&%5 4&5"& +,! )4 64#46&(' ! 6! +,& ,(&)&"! 6!" (!675&+,!" #%,$ ),((!$ 6%5($!
)8!:#)%"&%5 6%-;&54(%&$! 45" )8454)<"! 8466!""&;&)&('3
=5>50 45" )4 !$5&?$! #4$(&! @4('5%(0&! (5 (& "-,03/5!-,5A0 5%," #$%#%"%5"
,5! !:(!5"&%5 ! 5%" "<"(?-!" ! $'9'6$&(,$! B )!" "<"(?-!" $'9'6$&(,$!" C%56(&%55!)"
#4$4-'($'"3 D%," 'E!)%##%5" 4,""& ,5 -% ?)! ! !"6$&#(&%5 ! "<"(?-!"0 )! )45*4*!
.CF0 !( -%5($%5" 6%--!5( )!" -% ?)!" '6$&(" 45" 6! )45*4*! #!,E!5( 454)<"!$ G
)84& ! ! 6!" 5%,E!4,: "<"(?-!" ! $'9'6$&(,$!3
!"#$ %$& '!()*+$&
!"#$%&'(#)%" *+"+$,-. )))
!"#$%& ' !#()%*%+"%#&, -
/ 0-+1."#2 2'$ -, 3+$)4(,#)%" &. 1%&5-.2 6
!" #$%&'()*+($, %-) ).)+/0-) 1&*2+(3) ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! 4
!"!" 5+162+61- %- 71(89- ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! 4
!"! :&)-*6; %- <-+1( ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! =
! >$?(@6-) +-08$1-''-) ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! A
! !" >$?(@6- +-08$1-''- '(,&*(1- >B> ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! A
! ! >$?(@6- +-08$1-''- *1C$1-)2-,+- DB> ! ! ! ! ! ! ! ! ! ! ! ! ! ""
! !E D$08*1*()$, %-) '$?(@6-) +-08$1-''-) ! ! ! ! ! ! ! ! ! ! ! ! ! "
!E F&1(G2*+($, ).0C$'(@6- ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! "
!E!" H(*?1*00-) %- %&2()($,) C(,*(1-) IJHHK ! ! ! ! ! ! ! ! ! ! ! "E
!E! F*1(*,+-) %- JHH ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! "L
!L M6-'@6-) $6+(') %- N&1(G2*+($, ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! "L
6 7$)"()8.2 &. -, 3+$)4(,#)%" 9 :,2. &. ;<=2 >
E!" <1(,2(8-) ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! "O
E!"!" F&1(G2*+($, P C*)- %- ).)+/0-) %- 1&Q&21(+61- ! ! ! ! ! ! ! ! ! "O
E!"! F&1(G2*+($, P 'R*(%- %- *6+$0*+-) %R*1C1-) ! ! ! ! ! ! ! ! ! ! !
E! <1$+$2$'- %RS1C(+1*?- S1C$1-)2-,+ IBS<K ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! 4
E! !" H-)21(8+($, %6 81$+$2$'- BS< ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! 4
E! ! F&1(G2*+($, %6 BS< P C*)- %- B:5) ! ! ! ! ! ! ! ! ! ! ! ! ! ! =
E! !E F&1(G2*+($, %6 BS< P 'R*(%- %- *6+$0*+-) %R*1C1-) ! ! ! ! ! ! A
E!E F&1(G2*+-61) C*)&) )61 %-) B:5) -+ %-) *6+$0*+-) %R*1C1-) ! ! ! ! ! ET
!"#$%& ' ./,$0*& 1& 2(3(4#%$5#& 67+4$%7++&) 8692.: ;;
? =@2#51.2 &. <+A+($)#'$. B%"(#)%"".-2 6C
L!" 5.)+/0-) %- :&Q&21(+61- U$,2+($,,-') IUB:5)K ! ! ! ! ! ! ! ! ! ! ! ! EA
L!"!" <1(,2(8-) ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! EA
L!"! B1*,)3$10*+($, %- B:5) N-1) UB:5) ! ! ! ! ! ! ! ! ! ! ! ! ! ! L"
L!"!E B1*,)3$10*+($, %- UB:5) N-1) B:5) ! ! ! ! ! ! ! ! ! ! ! ! ! LE
L! UB:5) V'&0-,+*(1-) IVUB:5)K ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! LL
L! !" <1(,2(8-) ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! LL
L! ! B1*,)3$10*+($, %- B:5) N-1) VUB:5) ! ! ! ! ! ! ! ! ! ! ! ! ! LW
L! !E B1*,)3$10*+($, %- UB:5) N-1) VUB:5) ! ! ! ! ! ! ! ! ! ! ! ! 4=
!!! "#$%& '&( )#*!+,&(
!" #$%&'( ) *+,-./ 0&#$%&'(1 ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! 23
!"!4 5+-67-8/( ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! 23
!"!9 %+:6(;,+<:.-,6 =/ #$%&'( >/+( &#$%&'( ! ! ! ! ! ! ! ! ! ! ?@
! A,6>/+(-,6 =/( (B(.C</( ;,67.-,66/D( >/+( %&'( ! ! ! ! ! ! ! ! ! ! ! ?4
- . #%/#*!01 '&( (2(*+)&( 3%3)&1*#!,&( 4-
2!4 EDF,+-.G</ =HI>:DJ:.-,6 7D:((-KJ/ ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ?3
2!9 L8.-<-(:.-,6 ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! 34
2!9!4 M>:DJ:.-,6 :>/7 D: (.+:.IF-/ =/ (:.J+:.-,6 ! ! ! ! ! ! ! ! ! ! ! 34
2!9!9 EDF,+-.G</ =HI>:DJ:.-,6 ,8.-<-(I/ ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! 32
!"#$%& ' ())*%+"$%,- . *" /0#%1+"$%,- 23
4 50'3%!(#*!01 6#, 78"9: ;<
?!4 N,=CD/( :+O,+/(7/6.( ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! P"
?!4!4 5+,.,7,D/ =/ 5/+7,D:./ ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! P"
?!4!9 5+,.,7,D/ =H#D/7.-,6 E+O,+/(7/6. ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! P2
?!9 &I(/:JQ =/ 5/.+- 5R% 05D:7/R%+:6(-.-,61 ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! PP
?!" &I(/:JQ =/ 5/.+- G-I+:+7G-KJ/( ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! S?
= >3,!?@#*!01 6#, 78"9: <A<
3!4 E6:DB(/ =H:77/((-O-D-.I 8,J+ DH-6./+TOD,7:F/ ! ! ! ! ! ! ! ! ! ! ! ! ! ! 4@"
3!9 E6:DB(/ =H:77/((-O-D-.I 8,J+ D/( -6>:+-:6.( ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! 4@
3!9!4 UJ/DKJ/( <,=CD/( :+O,+/(7/6.( ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! 4@S
3!9!9 UJ/DKJ/( <,=CD/( =/ +I(/:JQ =/ 5/.+- ! ! ! ! ! ! ! ! ! ! ! ! ! 444
3!" V,F-KJ/( ./<8,+/DD/( ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! 44"
3!"!4 V,F-KJ/ ./<8,+/DD/ D-6I:-+/ V%V ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! 44"
3!"!9 V,F-KJ/ ./<8,+/DD/ :+O,+/(7/6./ A%V ! ! ! ! ! ! ! ! ! ! ! ! ! 44?
; B&1@C)#,D( <E<
P!4 E88D-7:.-,6 (J+ =/( <,=CD/( :+O,+/(7/6.( ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! 49
P!4!4 &I(JD.:.( /Q8I+-</6.:JQ /6 FI6I+:D ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! 49
P!4!9 %&'(W $%&'( /. #$%&'( (J+ =/( <,=CD/( :+O,+/(7/6.( ! ! ! 492
P!9 E88D-7:.-,6 (J+ =/( +I(/:JQ =/ 5/.+- ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! 4"4
P!9!4 '-<JD:.-,6( =/( +I(/:JQ =/ 5/.+- ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! 4"4
P!9!9 '-<JD:.-,6 =/( +I(/:JQ =/ 5/.+- G-I+:+7G-KJ/( ! ! ! ! ! ! ! ! ! 4"9
P!" E88D-7:.-,6 (J+ DH-6./+TOD,7:F/ /. D: >I+-X7:.-,6 =/( D,F-KJ/( ./<T
8,+/DD/( ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! 4"?
P!"!4 E88D-7:.-,6 (J+ DH-6./+TOD,7:F/ ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! 4"?
P!"!9 E88D-7:.-,6 (J+ D: D,F-KJ/ V%V ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! 4"3
P!"!" E88D-7:.-,6 (J+ D: D,F-KJ/ A%V ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! 4"P
!"#$ %$& '!()*+$& ),
! "#$%&' ! ( !'$) *+ ,#+-#-' ./*$ ,# !0$&12#%&/+ .#$ 34567898
- .!/0!0$ 123+ #! 45+)67!()2/ 89:
!" #$%&'()*%+'$ ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ",-
!. /01*&+2%+'$1 34$4&5601 () 789 ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ",:
!; 789 2')& 65 <4&+=*5%+'$ 1>?@'6+A)0 ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ",B
8; 4$+& 3/ #!/0!0$ 123+ #! 45+)67!()2/ 1!+ <= >? 8@8
"C!" #$%&'()*%+'$ ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! "-;
"C!. D>1%E?01 464?0$%5+&01 1+?2601 ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! "-;
"C!; D>1%E?01 464?0$%5+&01 25&5?4%&41 FGHID1 ! ! ! ! ! ! ! ! ! ! ! ! ! ! "--
"C!;!" D>1%E?01 464?0$%5+&01 25&5?4%&41 FGHID1 J 6K5+(0 (01 +$<5&+L
5$%1 ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! "--
"C!;!. D>1%E?01 464?0$%5+&01 25&5?4%&41 MFGHID1N ! ! ! ! ! ! ! ! ! "-:
88 A2/7#3&)2/ 05/5+!#$ 8B8
C)"#)20+!1D)$ 8B:
E F+$3G$& 8B-
O!" PQ52+%&0 ,! D>1%E?01 (0 I4L4*&+%)&0 G'$*%+'$$061 ! ! ! ! ! ! ! ! ! ! ! ":
O!"!" D0*%+'$ ,!"! GHID1 ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ":
O!"!. D0*%+'$ ,!.! RGHID1 ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! "S.
O!"!; D0*%+'$ ,!;! IRGHID1 ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! "S;
O!. PQ52+%&0 -! T<56)5%+'$ (01 1>1%E?01 464?0$%5+&01 ! ! ! ! ! ! ! ! ! ! ! "S,
O!; PQ52+%&0 :! U'(46+15%+'$ 25& RGHID ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! "S-
O!, PQ52+%&0 S! 94&+=*5%+'$ 25& RGHID ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! "S-
C <(3%$ %$ 7!& %3 .H4 8II
V!" F&'*011)1 2&+$*+256 ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! "SS
V!. F&'*011)1 D0&<0)& ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! "SB
V!; F&'*011)1 P6+0$% ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! "S
V!, F&'*011)1 U4(+5 ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! "BC
A JJJ 123+ .H4 8K8
P!" P'(530 () ?'(E60 0$ /// ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! "B"
P!. W24&5%+'$1 () ?'(E60 0$ /// ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! "B"
!"#$%!" &'!($"
!"#$% & ' "()$*$+!$"%,
!"#$%&' (
!"#$%&' '() !* +"),-.*&,/% 0$
#/01!$'
!"#$! %# &'()*+ , -
!"# $% $&!'()*%+ ",-# *./-(#,"# 0% '*,1023% /% 3,/.0(#!)(," !-4 '*("$('!-4
$,"$%')# )%0# 5-% 0! #)*-$)-*% /% 6*('7% %) 0% *.#%!- /% 8%)*( ',-* 3,/.0(#%* /%# #9#:
)23%# *.!$)(;#< = '!*)(* /% 0! /%#$*(')(," /- 3,/20%+ '0-#(%-*# )%$&"(5-%# ',-* >.*(:
?%* $% 3,/20% '%->%") @)*% .)-/(.%# )%00%# 5-% 0%# !"#$%&' (&)*!+& &' A8"-%0( BCDD+
E3%*#," BCFG+ H(#)0! BCFI+ J!*/( BCFK+ L0!*7% BCFK+ =*",0/ BCFF+ M%*)& BCCNO+ 0!
,-+#./0(#!1 '2)3! #$%& AP*9!") BCFK+ P*9!") BCCI+ L0!*7% IGGGO+ 0%# (&/41#$%&'
5& +-5%/(#!1 30'-& '%+ 5&' !+5+&' *0+(#& ' AQ,0R3!"" BCCB+ Q,0R3!"" BCFK+
L0!*7% IGGGO %) 0%# (&/41#$%&' 5603'(+0/(#!1 AL0!*7% IGGGO< S,-# ",-# ;,$!0(#,"#
/!"# $% $&!'()*% #-* 0%# !"#$%&' (&)*!+& &' %) 0! ,-+#./0(#!1 '2)3! #$%& 5-( #%*,")
.)-/(.%# /!"# 0! #-()% /% $% 3.3,(*%<
='*2# !>,(* '*.#%"). 0! #)*-$)-*% /% 6*('7% %) 0% *.#%!- /% 8%)*(+ ",-# /(#,"# $,3:
3%") ",-# ',->,"# $,"#)*-(*% /%# 0,T(5-%# )%3',*%00%# U '!*)(* /V-"% )%00% #)*-$)-*%<
E"#-()%+ ",-# /.$*(>,"# $,33%") ",-# ',->,"# $,"#)*-(*% /%# #)*-$)-*%# #931,0:
(5-%# U '!*)(* /V-" )%0 3,/20%< E"?"+ ",-# 0(#),"# 5-%05-%# ,-)(0# /% >.*(?$!)(,"
$,""-# 1!#.# #-* $%# /%-4 )%$&"(5-%#<
".//0123
!" #$%&'()*+($, %-) ).)+/0-) 1&*2+(3) !
!"!" #$%&'$&%( )( *%+,-( ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! .
!"! /01(2&3 )( 4($%+ ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! 5
" " #$%&'()* +),-$.)//)* 0
! !" 678+9&( $(:,7%(;;( ;+<02+%( 6=6 ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! >
! ! 678+9&( $(:,7%(;;( 2%?7%(1'(<$( @=6 ! ! ! ! ! ! ! ! ! ! ! ! ! ""
! !A @7:,2%2+17< )(1 ;78+9&(1 $(:,7%(;;(1 ! ! ! ! ! ! ! ! ! ! ! ! ! "
" 1 23.&456+&$7 *8,9$/&'() :"
!A!" B+28%2::(1 )( )0'+1+7<1 ?+<2+%(1 CDBBE ! ! ! ! ! ! ! ! ! ! ! "A
!A! F2%+2<$(1 )( DBB ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! "G
" ; <()/'()* $(+&/* =) >3.&456+&$7 :;
!"! #$%&'()*+($, %-) ).)+/0-) 1&*2+(3) 4
!" #$%&'()*+($, %-) ).)+/0-) 1&*2+(3)
!"# $%&'!(# )*%+!&) ),# #-#./0,# &1%2.34# ,. ',"&# 2!0$!&.,0,(.#5 6( #-#./0,
&1%2.34 ,#. 2%$."&1 ,( #,# !"!#5 7' ,#. 18%',0,(. )12&3. $%& "( 4&%(293##,0,(. )*1.%.
), '*"( : '*%".&,5 6( .,' 2!"$', )*1.%.# ,#. %$$,'1 "(, !#"$%&!&'$5
6( #-#./0, &1%2.34 $,". ;.&, 0!)1'3#1 $%& "(, #.&"2."&, ), <&3$=, !" $%& "(
&1#,%" ), >,.&35 !"# $!"?!(# ".3'3#,& ',# .,29(3@",# ),# '!83@",# .,0$!&,'',# ,.
',# .,29(3@",# ), '% ?1&3A2%.3!( #-0+!'3@", $!"& ?1&3A,& 2,# 0!)/',#5 !"# %''!(#
$&1#,(.,& 2,# .,29(3@",# )%(# ',# #,2.3!(# B5B ,. B5C5
!"!" #$%&'$&%( )( *%+,-(
D% #.&"2."&, ), <&3$=, ,#. "( 8&%$9, ), .&%(#3.3!(#E1.%.# 2%$%+', ), 0!)1'3#,&
', 2!0$!&.,0,(. )*"( #-#./0, &1%2.345 F!&0,'',0,(.G "(, #.&"2."&, ), <&3$=, ,#.
"( @"%)&"$',.M = (S, S0, R, L) #"& "( ,(#,0+', A(3 ), $&!$!#3.3!(# %.!03@",# AP
.,' @", H
I S ,#. "( ,(#,0+', A(3 )*1.%.#5
I S0 ⊆ S ,#. '*,(#,0+', )*1.%.# 3(3.3%"J5
I R ⊆ S × S ,#. "(, &,'%.3!( ), .&%(#3.3!(#5 K,'',E23 )!3. ;.&, .!.%', (∀s ∈
S,∃s′ | (s, s′) ∈ R)5
I L : S → 2AP ,#. "(, 4!(2.3!( %##!23%(. : 29%@", 1.%. "( ,(#,0+', ), $&!$!E
#3.3!(# %.!03@",# ?&%3,# )%(# 2,. 1.%.5
D,# 2!0$!&.,0,(.# )*"( #-#./0, &1%2.34 #!(. )1A(3# $%& #,# .&%(#3.3!(#5 6(
2!0$!&.,0,(. ,#. "(, #1@",(2, 3(A(3, )*1.%.# .,' @", 29%@", 1.%. ,#. !+.,(" ),
'*1.%. $&121),(. $%& "(, .&%(#3.3!(5 6( 29,03( )%(# '% #.&"2."&, M : $%&.3& )*"(
1.%. s0 ,#. "(, #1@",(2, 3(A(3, )*1.%.# π = s0s1s2... .,' @", (si, si+1) ∈ R ∀i ≥ 05
!"# %''!(# 3''"#.&,& 2,.., #.&"2."&, $%& "( ,J,0$',5
56-07'- " ('&! )$* %!#)+!)#* ,* -#&./* M = (S, S0, R, L) +'00* 1" %)&2"$!* 3
4 S = {s0, s1}5
4 S0 = {s0}5
4 R = {(s0, s1), (s1, s0), (s1, s1)}5
4 L : s0 → {a, b} *! s1 → {a,¬b}5
611* *%! , +#&!* +'00* )$ 7#".8* ,* !#"$%&!&'$%9 !"!% ,"$% 1" :7)#* ;5<5 =&$%&> π0 =
!"# B5L I 6(, #.&"2."&, ), <&3$=,
s0s0s1 $?*%! ."% )$ +8*0&$ ,"$% M 5 @*.*$,"$!> π1 = s0s1s0s1s1...s1 *%! )$ +8*0&$
,"$% M 5
!"#$%&'( )* +,-.(/&0 01' ,# 2-'%34#&%5/ 6( .567,(0
!"#$%# &' ('$$' )$#*($*#' &' +#%!,'- ./*) !/*0/.) *$%1%)'# 1') 1/2%3*') $'45
!/#'11') '$ 1') $'(6.%3*') &' 1" 07#%8("$%/. )94:/1%3*' !/*# 07#%8'# (' 4/&;1'<
!"! #$%&'() *& +&,-.
=. #7)'"* &' >'$#% ')$ *. 4/&;1' 4"$674"$%3*' )'#0".$ ? #'!#7)'.$'# &%0'#)
)9)$;4') @%.A/#4"$%3*')- %.&*)$#%'1)- <<<B $#"0"%11".$ )*# &') 0"#%":1') &%)(#;$')< C1 ')$
("!":1' &' 4/&71%)'# 1' (/4!/#$'4'.$ &D*. )9)$;4' #7"($%A '$ )'#" 0* (/44' *.
4/&;1' &' &')(#%!$%/. (/.(%) &' )$#*($*#') &' +#%!,'<
E/#4'11'4'.$- *. #7)'"* &' >'$#% ')$ *. 3*%.$*!1' R =< S, T, Pre, Post,m0 >
$'1 3*' F
G S ')$ *. '.)'4:1' 8.% &' !1"(')<
G T ')$ *. '.)'4:1' 8.% &' $#".)%$%/.)<
G Pre(t) '$ Post(t) )/.$ &'*H "!!1%("$%/.) &' T &".) S 3*% ? $/*$' $#".)%$%/.
t &' T "))/(%'.$ 1') &'*H 7$"$) Pre(t) '$ Post(t) 3*% )/.$ #')!'($%0'4'.$
1D !"#"$% '$ 1' &'( &' 1" $#".)%$%/. t<
G m0 ')$ *. 4*1$%5'.)'4:1' &' S "!!'17 1' 4"#3*"2' %.%$%"1<
=. 4"#3*"2' m !'*$ "4'.'# ? *. "*$#' 4"#3*"2' m′@./$7 !"# m[t > m′
B
)'*1'4'.$ )D%1 'H%)$' *.' $#".)%$%/. t $'11' 3*' m > Pre(t) '$ m′ = m − Pre(t) +
Post(t)< I" #'1"$%/. &' A#".(6%))'4'.$ !'*$ J$#' 7$'.&*' %.&*($%0'4'.$ ? $/*$'
)73*'.(' &' $#".)%$%/.) !"# F m1[ε > m2 )% m1 = m2- '$ m1[σ.t > m2 )% ∃m :
m1[σ > m '$ m[t > m2< =. 4"#3*"2' m′')$ "(('))%:1' ? !"#$%# &D*. 4"#3*"2'
m )% ∃σ ∈ T ⋆ : m[σ > m′< K/*) ./$/.) Reach(R,m0) 1D'.)'4:1' &' 4"#3*"2')
"(('))%:1') ? !"#$%# &' m0<
I' #7)'"* &' >'$#% ')$ &%$ F
G 3/% )% ‖ S ‖< +∞ '$ ‖ T ‖< +∞<
G -,-.(/&#%'( )% ∀p ∈ S,∀t ∈ T - Pre(t)(p) ≤ 1 '$ Post(t)(p) ≤ 1<
G 85'/- )% ‖ Reach(R,m0) ‖< +∞<
G 0#%/ )% ∀m ∈ Reach(R,m0)- 4 ')$ )"%.- c.?.d. ∀p ∈ S,m(p) ≤ 1<
LM%.'# NOOOP !#7)'.$' 3*'13*') 'H'4!1') &' #7)'"*H &' >'$#%< K/*) "11/.) 1')
&7(#%#' (%5&'))/*)<
K/*) ./*) A/("1%)/.) )*# 1') #7)'"*H &' >'$#% !"#"47$#7) )/%$ !"# *. ./4:#' &'
Q'$/.) N - )/%$ !"# *. ./4:#' &' )/*) #7)'"*H %&'.$%3*')< R'*H #'!#7)'.$".$) &')
4/&;1') !"#"47$#7) !"# *. ./4:#' &' Q'$/.) N )/.$ 1' )9)$;4' &' +".:". '$ 1'
)9)$;4' &' EMS< T#/%) #'!#7)'.$".$) &') 4/&;1') !"#"47$#7) !"# *. ./4:#' &' )/*)
#7)'"*H %&'.$%3*') )/.$ 1' !#/:1;4' &') >6%1/)/!6')- 1' !#/$/(/1' &' S1/$$'&5U%.2 '$
1' !#/$/(/1' &' U/*.&5U/:%. M*$'H<
N< I' !#/:1;4' &') >6%1/)/!6') ')$ *. 'H'4!1' 'H!1%("$%A &D*. !#/:1;4' &' (/.5
(*##'.('< * &7:*$- (6"3*' !6%1/)/!6' ')$ '. 7$"$ V%&1'V '$ &7(%&' &' 4".2'# F
('1" 1' (/.&*%$ ? !#'.&#' 1') &'*H A/*#(6'$$') &' 2"*(6' '$ &' &#/%$'< >*%)- %1
1') 1%:;#' "!#;) "0/%# 4".27< I' 4/&;1' &') !6%1/)/!6') ')$ (/4!/)7 !"# N
)/*) #7)'"*H %&'.$%3*')< I' )/*) #7)'"* &' >'$#% !/*# 1' %5%;4' !6%1/)/!6' ')$
!"#* '. 82*#' W<W @"B< X' )/*) #7)'"* #'!#7)'.$' *. !6%1/)/!6' '$ 1" A/*#(6'$$'
!"! #$%&'()*+($, %-) ).)+/0-) 1&*2+(3) 4
! "#$%&'() *)+ ",-&$+$.,)+
pask ( i+1)modN
pwai t ( i+1)modN
pask
pwai t
R
bufi lde
bufful l
psend
ploadl
pok
Res
Task
Tbuf
T1load
T2loadT1send
T2send
Task
%! "#$/$0$&) *) 1$23*41$%-3 52/)6
0! 78+/'() *) 957
*! "#$/$0$&) *) 7&$//)*41-3:
)! 78+/'() *) ; 3% 3
!"# <=< > ?2)&@2)+ #A+) 26 *) ")/#-
!"#$%&'( )* +,-.(/&0 01' ,# 2-'%34#&%5/ 6( .567,(0
! "#$% &'($ ! )*(+',')*-. /0 1'!&"*-$$- ! "#$% 20!"*-3 &-)&%,-4$%- )0& +0
)+0"- )'(4$(++%- Fork(i+1)modN 3 -,$ !4- )0&$(- ! ,'!, &%,-0! ! )*(+',')*-
,!(504$3 "-+0 (++!,$&- "'66-4$ +-, ,'!, &%,-0!7 "'66!4(8!-4$ -4$&- -!7. !
"#$!%& !$' ()"*#+ #,#-!*'%."! !' $%.*.
9. :' ;+- ! <&'$'"'+- - &%,-0! =+'$$- >(42 ? /- &%,-0! - <-$&( )'!& !4
,-!+ 4'-! - "- )&'$'"'+- -,$ &-)&%,-4$% -4 @2!&- 9.9 A B. /- 6' ;+- 2+'C0+
-,$ "'6)',% )0& N $-+ ,'!, &%,-0!7 "'44-"$%, -4 )0&$02-04$ +-, $&04,(D
$('4, (Free(i + 1) mod N -$ Used(i + 1) mod N). E'66- +- )&'C+;6- -,
)*(+',')*-,3 "- &%,-0! -,$ ()"*#+ #,#-!*'%."! !' $%.*.
F. :' ;+- ! <&'$'"'+- - >'!4 D>'C(4 :!$-7 A)&%,-4$% )0& GH&01 IJJKLB ? M'(&
+0 @2!&- 9.9 ACB. E- )&'$'"'+- &%,'!$ !4 $N)- ,)%"(@8!- ! )&'C+;6- O-7"+!D
,('4 6!$!-++- )0&6( N )&'"-,,!, '&204(,%, 04, !4- 604(;&- "(&"!+%-3 -604D
04$ O0""% -& P !4- &-,,'!&"- )0&$02%-. E'66- +-, -!7 &%,-0!7 "(D -,,!,3
"- &%,-0! !$' ()"*#+ #,#-!*'%."! !' $%.*.
Q. =N,$;6- - R04C04 ? /0 @2!&- 9.9 A-B &-)&%,-4$- +- &%,-0! - <-$&( O!4
,N,$;6- - R04C04. E- 6' ;+- -,$ )0&06%$&% )0& +- 4'6C&- - S-$'4, N 0!
%C!$ -4 p1, p2, p3, -$ p4. E- &%,-0! -,$ ()"*#+ #,#-!*'%."! 60(, )0, $%.* P
"0!,- ! 4'6C&- - S-$'4, N .
T. =N,$;6- - U:= ? E- 6' ;+- &-)&%,-4$% -4 @2!&- 9.9A"B -,$ )0&06%$&% )0& +-
4'6C&- - S-$'4, N 04, P1, P2, 04 P3. E'66- +- ,N,$;6- - R04C043 "-
&%,-0! -,$ ()"*#+ #,#-!*'%."! 60(, )0, $%.* P "0!,- ! 4'6C&- - S-$'4, N .
V'!, )'!5'4, !$(+(,-& +-, +'2(8!-, $-6)'&-++-, )'!& 5%&(@-& 8!-+8!-, )&')&(%$%,
(4$%&-,,04$, O!4 &%,-0! - <-$&(.
! "#$%&'() *(+,#-(..()
/-, +'2(8!-, $-6)'&-++-, ,'4$ "0)0C+-, - ,)%"(@-& !4 ,N,$;6- O%$0$, W $&04D
,($('4, '! -, ,$&!"$!&-, - R&()X-,. Y4- 1'&6!+- -, +'2(8!-, $-6)'&-++-, %"&($
+-, )&')&(%$%, -, 0&C&-, - "0+"!+. /O0&C&- - "0+"!+ 6'4$&- $'!$-, +-, -7%"!$('4,
)',,(C+-, "'66-4Z04$ P )0&$(& - +O%$0$ (4($(0+. [04, +0 ,!($-3 4'!, &% !(,'4, 4'$&-
%$! - P -!7 )0&6( -, +'2(8!-, $-6)'&-++-, ? /0 +'2(8!- $-6)'&-++- +(4%0(&- /\/ -$
+0 +'2(8!- $-6)'&-++- 0&C'&-,"-4$- E\/.
! !" #$%&'() *)+,$-)..) .&/01&-) #2#
/0 +'2(8!- $-6)'&-++- +(4%0(&- /\/ G<4!-+( IJ]]L )-&6-$ - ,)%"(@-& -, )&'D
)&(%$%, +- +'42 -, "*-6(4, O-7%"!$('4 O!4 ,N,$;6- - $&04,($('4,. E-, )&')&(%$%,
N406(8!-, 1'4$ &%1%&-4"- P !4 $-6), "0!,0+. [-, ')%&0$-!&, ,'4$ 1'!&4(, "(D -,,'!,
)'!& %"&(&- -, %5%4-6-4$, 8!( ,!(5-4$ !4 ,-!+ "*-6(4 - "0+"!+.
^ \'!S'!&, φ1 AGφ1B ? φ1 -,$ 5&0(- 04, $'!, +-, %$0$, - +0 ,%8!-4"- %C!$04$
-4 s0
^ \'!S'!&, φ1 S!,8!OP φ2 Aφ1Uφ2B ? φ1 -,$ 5&0(- 04, $'!, +-, %$0$, S!,8!OP "-
8!- φ2 +- -5(-44-
! ! "#$%&'() *(+,#-(..() /
!" #$%&' "&( )$%' φ1 *+&' ,( )"-". / Fφ1 = vraiUφ1
φ1 '( 0.$*"%.+ 1 ,23-+0( '"%4+&-( / Xφ1 5φ1 ('- 4.+%( *+&' ,23-+- '"%4+&- %#6
#3*%+-(#(&- s07
8& 0.+-%9"(: ,(' 0.$0.%3-3' 43.%;3(' )$&- %&-(.4(&%. 0,"'%(".' $03.+-(".' -(#0$.(,'
5X0 F 0 G $" U7 <$#0$'3' (&-.( ("= /
φ1 '( 0.$*"%.+ %&;&%#(&- '$"4(&- / GFφ1 5+"''% ,$%& 9"( &$"' +,,%$&' *+&' ,+
'39"(&<(: &$"' 0$"..$&' -$">$".' -.$"4(. "& 3-+- 0$". ,(9"(, φ1 ('- 4.+%(7
?+&' ,( )"-".: φ1 '( 0.$*"%.+ -$">$".' / FGφ1 51 0+.-%. *2"& <(.-+%& .+&@: φ1
'(.+ -$">$".' 4.+%(7
A+ #3-B$*( ,+ 0,"' .30+&*"( *( 43.%;<+-%$& *( 0.$0.%3-3' ACA ('- D+'3( '". ,+
<$&'-."<-%$& *" 0.$*"%- 'E&<B.$&%'3 *" 'E'-F#( 1 43.%;(. (- *2"& +"-$#+-( 5+00(,3
!"#$ "% &% '()*+ 7 0(.#(--+&- *( .(<$&&+G-.( *(' '39"(&<(' %&;&%(' 4+,%*+&- "&(
0.$0.%3-3 ACA HI+.*% JKLMNO
P& !"#$ "% &% '()*+ ('- "& +"-$#+-( *23-+-' ;&%: *$&- <(.-+%&' '$&- +06
0(,3' ,(' ," "- &. ))%/" "+#0O P&( '39"(&<( *2"& +"-$#+-( *( QR<B% ('- .(<$&&"(
5$" +<<(0-3(7 '% (,,( <$#0$.-( "& <E<,( <$&-(&+&- +" #$%&' "& 3-+- *2+<<(0-+-%$&O
HS+'-%& TUUJ: V$"4.(". JKKK: V$"4.(". TUUWN "-%,%'( "& +"-.( -E0( *2+"-$#+-(' / ,('
+"-$#+-(' 1 -.+&'%-%$&'O ?+&' <(' +"-$#+-(': ,(' <$&*%-%$&' *2+<<(0-+-%$& 0$.-(&-
'". ,(' -.+&'%-%$&' %&;&%#(&- .(&<$&-.3(' 0,"-X- 9"( '". ,(' 3-+-'O
Y$"' #$&-.$&' #+%&-(&+&- <$##(&- <$&'-."%.( "& +"-$#+-( *( QR<B% 0$". ,('
#$*F,(' #(&-%$&&3' <%6*(''"'O V$&'%*3.$&' "& (=(#0,( 0$". ,+ '-."<-".( *( Z.%0[(
<$##( '"%- /
12(+,.( 1#+" !0% -"2!)"!2% &% 32+/4% &,)2+" & 0- 5.%6%$/5% 78 9#!- )#0-+&,2#0-
!0 )#$/#2"%$%0" "%5 :!% ; <= 0- !0 $#$%0" & 0- 5% >!"!2? %-"@)% :!.+5 %6+-"% $ +-
A $ +- B C<8 D+0-+ !0% >#2$!5% &% EFE %-" 2%/2,-%0",% )#$$% 5 -!+G 0"% ;
φ = F (a ∧G(¬b))
H0 )*%$+0 &.," "- :!+ G,2+I% )%""% >#2$!5% "%2$+0% / 2 !0% -,:!%0)% &% s1 )#$$%
5% -!+G 0" ; s0s1...s0s1...s18 J5!- /2,)+-,$%0"? 0#!- #B"%0#0- ;
(ab)(a¬b)...(ab)(a¬b)...(a¬b)
9#!- &,)2+G#0- %0 IK!2% L8M 5. !"#$ "% &% '()*+ &% 5 >#2$!5% EFE &% )%""% >#2$!5%8
V$&'%*3.$&' "& (=(#0,( 0$". ,( .3'(+" *( \(-.%O Y$"' .(4(&$&' +" 0.$D,F#( *('
\B%,$'$0B('O
! "#$%&'() *+ ,-./)0'1 12( -$ 3.(&45$'&60 7) /678-)1
!"# !" # $%&'()&* +* ,-./0
9:)/%-) ; !"#$%&'!"# () *'!+(,-) %)# ./$(!#!*/)# %) (0 #)12$!" 3454 6) #!7#
'&#)07 %) .)2'$ *!7' () $8$,-) */$(!#!*/) )#2 *0'7 )" 9:7') 343 ;0<4
=!7# #!--)# $"2&')##&# > 7" 1!-*!'2)-)"2 2)( ?7) @ AB 7" -!-)"2 %0"# ()
C727'D 7" */$(!#!*/) *')"% (0 C!7'1/)22) %) :071/) -0$# $( "E07'0 F0-0$# 1)(() %)
%'!$2)A )" 72$($#0"2 (0 C!'-7() %) 6G6 1$8%)##!7# @
f1 = F(hasLeft1 ∧ G(¬hasRight1))
H!$' (E072!-02) %) IJ1/$ )" 9:7') 34K ;0<4
=!7# #!--)# &:0()-)"2 $"2&')##&# > 7" 1!-*!'2)-)"2 2)( ?7) @ AB 7" -!-)"2
%0"# () C727'D 07 %&+72D 7" */$(!#!*/) *')"% (0 C!7'1/)22) %) %'!$2)D *7$# $( (0 ($+,')D
)2 9"0()-)"2D 1)22) C!7'1/)22) )#2 *'$#) *0' #!" L!$#$" %'!$2)A )" 72$($#0"2 (0 C!'-7()
%) 6G6 #7$L0"2) @
f2 = (hasRight1 ∪ (Fork1 ∪ hasLeft2))
H!$' (E072!-02) %) IJ1/$ )" 9:7') 34K ;+<4
!"# !1 # $%&'()&*2 +* ,-./0 )3 f1 *& 43 f2
$5672 )8'06 &6)+%0& +9%:* ;'6(%<* =>= *: )%&'()&* +* ,-./0 ?@'<5*6 AABCD
0< :'%2 6*2&* <) .':2&6%.&0': +% 56'+%0& 2E:./6':02F +% 2E2&7(* G 8F60H*6 *& +9%:
)%&'()&* +* ,-./0!
<.(&45$'&60 %62( =>=
I90< *2& :F.*22)06* +* .':2&6%06* <9)%&'()&* 56'+%0& +):2 2) &'&)<0&F 5'%6 56'%J
8*6 K%* M ² φD <) 6F;%&)&0': +* .*&&* 56'560F&F 5*%& L&6* 56'%8F* 5)6 <9*M02&*:.*
+9%: %:0K%* .':&6*J*M*(5<*! =) (F&/'+* +* 8F60H.)&0': +0&* A> (0 L!(&)A .':202&* G
! ! "#$%&'() *(+,#-(..() //
!"#! "#! $! "%"$! &'(""!)*(*+,- !- ./.! *!.)0 12! -,20 ",-0* 2+0,-0 AM ⊗ A¬φ!* 3 +-*! ,.) ! 0( ",-0* 2"*+,- (2 ) !.+! "%"$! &'(""!)*(*+,- * ,245 67! *# 899:;<
= )( *+ &! $( -5>(*+,- &! $( ?, .2$! @A@ 3 45 +B! C -,20 ",-0* 2+0,-0 $'(2*,.(*!
&! DE"#+ )! .!**(-* &! !",--(F* ! 2- ",-* !G!H!.)$! 3 $( ) ,) +5*5< I-02+*! -,20
) ,"5&,-0 3 $( ",-0* 2"*+,- &! $'(2*,.(*! &! DE"#+ A¬φ )! .!**(-* &! !",--(F* !
2- ",-* !G!H!.)$! &! $( ) ,) +5*5 φ< J2+0 -,20 !K!"*2,-0 $! ) ,&2+* 0%-"# ,-+05
AM ⊗ A¬φ 12+ !0* (200+ 2- (2*,.(*! &! DE"#+< L!2H ),00+M+$+*50 0',K !-* ($, 0
3 -,20C 02+4(-* $'(M0!-"! ,2 $( ) 50!-"! &! "%"$! &'(""!)*(*+,- Nc.3.d. &! ",-* !G
!H!.)$!O &(-0 AM ⊗A¬φ<
! ! "#$%&'( )(*+#,(--( .,/#,(01(2)( 34"
@',)5 (*!2 *!.), !$ 12($+B! $!0 "#!.+-0 ),00+M$!0 3 )( *+ &'2- 5*(* &,--5
6P$( Q! 89RS;< J,2 * (4(+$$! 02 $'( M, !0"!-*C -,20 2*+$+0,-0 $!0 ) 5BH!0 .,&(2H
0 N.,&($+*5 2-+4! 0!$$! &50+>-(-* *,20 $!0 "#!.+-0 ),00+M$!0 3 )( *+ &! $'+-0*(-*
",2 (-*O !* 1 N.,&($+*5 !H+0*!-*+!$$! &50+>-(-* 2- "#!.+- ),00+M$! !* !H+0*(-* 3
)( *+ &! $'+-0*(-* ",2 (-*O< P!0 ) 5BH!0 0,-* *,2T,2 0 02+4+0 &!0 .,&($+*50 NX2 F 2
G ,2 UO &! $( $,>+12! *!.), !$$! $+-5(+ !< J( ",-0512!-*C $( $,>+12! PA@ !0* .2-+!
&!0 ,)5 (*!2 0 * (&+*+,--!$0 &! $( $,>+12! M,,$5!--!C !* &!0 ,)5 (*!2 0 *!.), !$0
12(-*+B50 U EX,AX,EF,AF,EG,AG,EU,AU <
@!0 ?, .2$!0 PA@ 0,-* ",-0* 2+*!0 3 )( *+ &!0 ) ,),0+*+,-0 (*,.+12!0 AP &'2-!
0* 2"*2 ! &! V +)Q!< @!2 0%-*(H! !* $!2 05.(-*+12! 0,-* $!0 02+4(-*!0 Nφ1 !* φ2
0,-* &!0 ?, .2$!0 PA@C () !0* 2-! ) ,),0+*+,- (*,.+12!O U
W M, s ² ap⇐⇒ ap ∈ L(s)
W M, s ² ¬φ1 ⇐⇒M, s 0² φ1
W M, s ² φ1 ∧ φ2 ⇐⇒ (M, s ² φ1) ∧ (M, s ² φ2)
W M, s ² φ1 ∨ φ2 ⇐⇒ (M, s ² φ1) ∨ (M, s ² φ2)
W M, s ² EXφ1 ⇐⇒ ∃π, s0 = s⇒M, s1 ² φ1
W M, s ² AXφ1 ⇐⇒ ∀π, s0 = s⇒M, s1 ² φ1
W M, s ² EFφ1 ⇐⇒ ∃π,∃i ≥ 0, s0 = s⇒M, si ² φ1
W M, s ² AFφ1 ⇐⇒ ∀π,∃i ≥ 0, s0 = s⇒M, si ² φ1
W M, s ² EGφ1 ⇐⇒ ∃π,∀i ≥ 0, s0 = s⇒M, si ² φ1
W M, s ² AGφ1 ⇐⇒ ∀π,∀i ≥ 0, s0 = s⇒M, si ² φ1
W M, s ² Eφ1Uφ2 ⇐⇒ ∃π,∃k ≥ 0, s0 = s⇒ (M, sk ² φ2)∧(∀0 ≤ i < k, (M, si ²
φ1))
W M, s ² Aφ1Uφ2 ⇐⇒ ∀π,∃k ≥ 0, s0 = s ⇒ (M, sk ² φ2) ∧ (∀0 ≤ i <
k, (M, si ² φ1))<
X,20 ($$,-0 ",-0* 2+ ! &!0 ?, .2$!0 &! PA@ ),2 $'!H!.)$! 8<
13(+,.( 4 !"# $% &#'()#('* +* ,'"-.* +/)'"# +%0& $1*2*3-$* 45 6"0&"7 8(*$8(*&
9!'3($*& +* :;< &!0# )%$)($/*& )!33* $*& &("=%0#*& >
? EX({s0}) = {s1} !" EX({s1}) = {s0, s1}#
! "#$%&'() !* +,-.)/'0 01( ,$ 2-(&34$'&5/ 6) .567,)0
AX({s0}) = ∅! "# AX({s1}) = {s0}
E({s0})U({s1}) = {s0, s1} "# E({s1})U({s0}) = {s0, s1}$
A({s0})U({s1}) = {s0, s1} "# A({s1})U({s0}) = {s0}$
8-(&34$'&5/ %51( "9:
! "#$%&'!(%)* +,-.)/%012 32 4$)4$%#(#+ 56 1(%/%+2 1* (,42 32 3)**#2+ !.+($!%(
4)1$ /! $24$#+2*(!(%)* 372*+2-./2+ 37#(!(+8 9/ ! #(# 3#-)*($# :;-2$+)* <=>?@ 012 '2
(,42 32 3)**#2+ !.+($!%( 32"!%( A($2 -1*% 32+ )4#$!(21$+ 2*+2-./%+(2+ ($!3%(%)**2/+
B1*%)*C %*(2$+2'(%)* 2( 3%D#$2*'2E !%*+% 012 32 ($)%+ )4#$!(21$+ +1%"!*(+ F
<8 7%*"!$%!*'2 %&' 3!*+ /72*+2-./2 S1 ⊆ S 31 +)1+G2*+2-./2 32+ #(!(+ +!(%+G
H!%+!*( 1*2 ')*3%(%)* .))/#2**2 ϕ F
S2 = invϕ(S1) = s2 ∈ S1, (M, s2 ² ϕ)
I8 2 '!/'1/ ()** 32 /72*+2-./2 S2 32+ #(!(+ +1''2++21$+ J 4!$(%$ 371* 2*+2-./2
37#(!(+ S1 ⊆ S 4!$ /! $2/!(%)* 32 ($!*+%(%)* R F
S2 = succ(S1) = {s2,∀(s1, s2) ∈ S1 × S, (s1, s2) ∈ R}
K8 2 '!/'1/ 2* !$$%L$2 +,"- 32 /72*+2-./2 S1 32+ #(!(+ 4$#3#'2++21$+ J 4!$(%$ 371*
2*+2-./2 37#(!(+ S2 ⊆ S 4!$ /! $2/!(%)* 32 ($!*+%(%)* R F
S1 = pred(S2) = {s1,∀(s1, s2) ∈ S × S2, (s1, s2) ∈ R}
52+ ($)%+ )4#$!(21$+ +)*( 3#&*%+ J 4!$(%$ 32 /! +($1'(1$2 32 M$%4N2 +1$ /!012//2
)* +)1O!%(2 "#$%&2$ 32+ 4$)4$%#(#+8
! !" #$%&'(')*$+ ,-* .$/)01-* 2-%&$(-..-*
P)1$ ')-4!$2$ /2+ 321Q /)R%012+ 2* (2$-2 32 ')-4/2Q%(#C :S%+(/! <=>I@ ! -)*($#
012 /! "#$%&'!(%)* 32 4$)4$%#(#+ 6 2+( 1* 4$)./L-2 PSPT5;G')-4/2( !"2' 1*2
')-4/2Q%(# (2-4)$2//2 2* 2(‖φ‖) × O(‖ M ‖)C )U ‖ φ ‖ 3#+%R*2 /2 *)-.$2 -!Q%-!/
37)4#$!(21$+ 6 %-.$%01#+ 3!*+ φC 2( ‖ M ‖ /2 *)-.$2 37#(!(+ !''2++%./2+ 31
+,+(L-28 :5/!$N2 <=>VC T$*)/3 <=>>@ )*( -)*($# 012 /! "#$%&'!(%)* 32 4$)4$%#(#+
56 2+( 1* 4$)./L-2 PSPT5;G')-4/2( !"2' 1*2 ')-4/2Q%(# (2-4)$2//2 2* O(‖
M ‖ × ‖ φ ‖) B‖ M ‖ 2+( /2 *)-.$2 37#(!(+ 32 W 2( ‖ φ ‖ /2 *)-.$2 37)4#$!(21$+
%-.$%01#+ 3!*+ φE8
!" #$%&'()*&+, -./0+1&234
! ')-4/2Q%(# '$)%++!*(2 32+ +,+(L-2+ %*H)$-!(%012+ 32-!*32 /! -%+2 2* )21"$2
32 -#(O)32+ !1()-!(%012+ 4)1$ /21$ 8-(&34$'&5/ ;5(.),,)8 2+ 3%!R$!--2+ 32
3#'%+%)*+ .%*!%$2+ BXYYE :X$,!*( <=>VC X$,!*( <==I@ +)*( !3)4(#+ 4)1$ ($!%(2$ 32+
!"! #$%&'()*&+, -./0+1&234 5"
!"#!$%&'$ ()* '+!&,-.*$ /* #).$ *& #).$ !"#)*0*$1 2*#*&/%&'3 !""* )* &!"4+*
/* 5%+,%4)*$ /*$ $6$'7"*$ ('./,($ *$' .& 8% '*.+ +,',-.*3 /* &!"4+*.$*$ $'+. '.+*$
!" #$$ !&' ('( #+!#!$(*$1 9.*)-.*$ 5%+,%&'*$ /* :;; #*.5*&' <'+* 5.*$ !""*
/*$ %+4+*$ !" #$$ != )*$ ('%'$ +*#+($*&'*&' /*$ >!,0 /* 5%)*.+$ /* 5%+,%4)*$ &!&
4!!)(*&&*$1
!"!# $%&'(&))*+ ,* ,-.%+%/0+ 1%0&%(*+ 23$$4
?*$ /,%@+%""*$ /* /( ,$,!&$ 4,&%,+*$ A:;;B C:+6%&' DEFG3 :+6%&' DEEHI $!&' .&
$. 7$ #!.+ )% 5(+,J %',!& /* $6$'7"*$ J&,$1 ?*$ :;; $!&' /*$ $'+. '.+*$ !/%&' /*$
8!& ',!&$ 4!!)(*&&*$1 K)$ #*.5*&' <'+* 5.$ !""* /*$ %+4+*$ != )*$ ('%'$ +*#+($*&L
'*&' /*$ >!,0 /* 5%)*.+$ /* 5%+,%4)*$ 4!!)(*&&*$3 .& !+/+* '!'%) $.+ )*$ 5%+,%4)*$
@%+%&',' )M.&, ,'( /. !/%@* /M.&* 8!& ',!&1 ?*$ '* >&,-.*$ /* #%+'%@* /* $'+. L
'.+*$3 !"4,&(*$ N /*$ "('>!/*$ /* +(/. ',!&$3 !&/.,$*&' N /*$ ,"#)("*&'%',!&$
*0'+<"*"*&' *O % *$ *& #+%',-.* CP,&%'! DEEQ3 R.)@%%+/ DEEEI1 S,&$,3 /*$ 5(+,JL
%',!&$ *0>%.$',5*$ !&' #. <'+* +(%),$(*$ $.+ /*$ $6$'7"*$ !"#+*&%&' /*$ ",)),%+/$
/M('%'$ CP,&%'! DEEQ3 R.)@%%+/ DEEEI AT!,+ 81 )M%+', )* %&'()*!+, -*./! 01/,2+34 5
102067"7/6 "3. #/'*3.% C:.+ > DEEHIB1 ?* #!.5!,+ /M*0#+*$$,!& /*$ :;; *$' $.UL
,$%&' #!.+ "%&,#.)*+ .&* @+%&/* )%$$* /* $6$'7"*$ J&,$ C:.+ > DEEHI1
V!.$ #+($*&'!&$ "%,&'*&%&' !""*&' &!.$ #!.5!&$ !&$'+.,+* /*$ $'+. '.+*$
:;; N #%+',+ /M.&* $'+. '.+* /* W+,#X*1 2!&$,/(+!&$ /($!+"%,$ .&* $'+. '.+* /*
W+,#X* M = (S, S0, R, L) '*) -.* Y
Z S *$' )* !/%@* /*$ ('%'$3 #!.+ $,"#),J*+ &!.$ $.##!$!&$ -.* &!.$ %5!&$
*0% '*"*&' 2m ('%'$1 [' Φ : {0, 1} → S *$' .&* 8!& ',!& /*$ 5* '*.+$
4!!)(*&&*$ %.0 ('%'$1
Z R .',),$* )* "<"* !/%@* -.* S
Z L *$' +*#+($*&'( $(#%+("*&' #!.+ >%-.* #+!#!$,',!& %'!",-.* Lp = {s, |p ∈
L(s)}1
V!.$ %))!&$ ,)).$'+*+ *''* $'+. '.+* #%+ .& *0*"#)*1
674/814 9 0*36+.89*36 !" 679:,7:9/ ./ ;9+<2/ M = (S, S0, R, L) .8,9+7 ."36 !=/>?
/(<!/ @A $"36 ,/ ,"6?! B 3*:6 "C*36 ./:> C"9+")!/6 .=87"76 s0, s1A $/ <!:6B 3*:6 +3?
79*.:+6*36 ./:> C"9+")!/6 "..+7+*33/!!/6B s′0, s
′1B <*:9 ,*./9 ./6 87"76 ./ 6:,,/66/:96A
D+36+B 3*:6 "!!*36 <986/37/9 !" 79"36+7+*3 ./ !=87"7 s0 !=87"7 s1 <"9 !" ,*3E*3,7+*3 5
(a ∧ b ∧ a′ ∧ ¬b′)
F" G*9(:!/ )**!8/33/ <*:9 !" 9/!"7+*3 /37+H9/ /67 .*338/ ,+?./66*:6 5
(a ∧ b ∧ a′ ∧ ¬b′) ∨ (a ∧ ¬b ∧ a′ ∧ ¬b′) ∨ (a ∧ ¬b ∧ a′ ∧ b′)
?*$ #!,&'$ 8!+' /* )% '* >&,-.* /*$ :;;$ $!&' ),$'(*$ ,L/*$$!.$ Y
! "#$%&'() *+ ,-./)0'1 12( -$ 3.(&45$'&60 7) /678-)1
9$:-)$2 7;20&5&'. ! "#$% &'())*+,#&% -$'$& .*+./%*&,(&, +#$. $& #01*, 2*
2#&&/*%3 4*5( +*.6*, 5*% 2#&&/*% 2* )#2*. 2*% 6#275*% ,.7% )#6+(),*%3
"$-52- 7) 5$5#) ! 48(-$* )(5)$5 2#9, :,.* 69% *& )()8* +#$. /;9,*. 2* .*<
)(5)$5*. $& ./%$5,(, 2/1= )#&&$3
4*% ,*)8&9-$*% &#$% +*.6*,,*&, 2* 5$,,*. )#&,.* 5'*>+5#%9#& )#609&(,#9.* %$.,#$,
+#$. 5*% 6#275*% %?6/,.9-$*%3
!"! #$%&$'()* +) ,--
4*.,(9&% %?%,76*% 2?&(69-$*% +*$;*&, :,.* +.9% *& )#6+,* (;*) )* ,?+* 2*
,*)8&9-$*% @A#5B% CDDEF3 4#66* 5* <.* 2* ;(.9(05*% 2*% %?%,76*% /,$29/%
*%, $& G(),*$. ).9,9-$*H 2* <.*$%*% %,.$),$.*% !" #$$ #&, /,/ +.#+#%/*%
@I9&*. CDDDH 49(.2# JKKKF3 L*% M9(N.(66*% 2* M/)9%9#& I$5,9;(5$/% OIMMP *%,
$&* ,*55* %,.$),$.* %?60#59-$* !" #$$3 Q& )#&%/-$*&)*H 5( )#6+()9,/ 2*% 2#&&/*%
67&* = $& ($,.* +.#0576* -$9 *%, 5'*R*, 2*% 2#&&/*% 9&,*.6/29(9.*%3 L( %,.(,/N9*
2* %(,$.(,9#& @49(.2# JKKEF +.#+#%/* (;*) 5*% IMM *%, $&* 0#&&* %#5$,9#& +#$.
)* +.#0576*3 4*,,* %,.(,/N9* 2* %(,$.(,9#& &'*%, +(% %*$5*6*&, (++59)(05* +#$. 5*%
IMM 6(9% ($%%9 +#$. 5*% ($,.*% %,.$),$.*% !" #$$3
M(&% 5* )(2.* 2* +.#1*,% 9&2$%,.9*5%H &#$% (;#&% )#&S$ $&* &#$;*55* %,.$),$.* =
5( TMMH 5*% M9(N.(66*% 2* M/)9%9#&% 2* M#&&/*% OMMMP @4#$;.*$. JKKJF3 L'#0<
1*),9G /,(9, 2* G#$.&9. $& #$,95 U*>905* -$9 +*$, :,.*H ($,(&, -$* +#%%905*H (2(+,/
= 5( ;/.9V)(,9#& 2* ,#$, ,?+* 2* 6#275*% *, -$9 #R.* 2*% )(+()9,/% 2* ,.(9,*6*&,
%9695(9.*% ($> TMM3 W 5( 29R/.*&)* 2*% TMMH 5*% #+/.(,9#&% %$. )*% %,.$),$.*% &*
%#&, +(% +./2/V&9*%H 6(9% $&* )5(%%* 2'#+/.(,*$.%H (++*5/* 8#6#6#.+89%6*%H *%, 9&<
,.#2$9,* +#$. +*.6*,,.* = $& $,959%(,*$. 2* )#&)*;#9. %*% +.#+.*% #+/.(,9#&%3 M(&%
)* 6#275*H 5*% ;(.9(05*% &* %#&, +(% 0##5/*&&*%H *55*% +.*&&*&, 5*$.% ;(5*$.% 2(&%
2*% 2#6(9&*% &#& &/)*%%(9.*6*&, 0#.&/%3 X&* ($,.* )(.(),/.9%,9-$* 9&,/.*%%(&,* *%,
-$'($)$& #.2.* %$. 5*% ;(.9(05*% *%, +./%$++#%/ 2(&% 5( 2/V&9,9#&3 M* +5$%H $&* ;(.9<
(05* +*$, (++(.(Y,.* +5$%9*$.% G#9% 2(&% $& 6:6* )8*69&3 4*,,* +.#+.9/,/ *%, ,.7%
$,95* -$(&2 &#$% 6(&9+$5#&% 2*% %,.$),$.*% 2?&(69-$*% )#66* 5*% V5*%3 L*% ;(.9<
(&,*% OL*% %,.$),$.*% 89/.(.)89-$*%P 2* MMM %#&, M9(N.(66*% 2* M/)9%9#& 2'Q&%*6<
05*% OZMMP @4#$;.*$. JKK[FH *, M9(N.(66*% 2* M/)9%9#& 2'Q&%*605*% \9/.(.)89-$*%
O\ZMMP @]89*..?<I9*N JKK^H ]89*..?<I9*N JKK_F3
M'($,.*% %,.$),$.*% #&, +5$,`, )8*.)8/ = /,*&2.* 5* 2#6(9&* 2'(++59)(,9#& 2*
)*% ,*)8&9-$*% @T(8(. CDDaH b$+,( CDDEH b$+,( CDD^H I($0#.N&* CDDDH c*R*5 CDDDH
I($0#.N&* JKKKF3
!" #$%&'$%( )$*+&( ,% -./+012*+)3
"#$% (;#&% +./%*&,/ 2(&% 5*% %*),9#&% +./)/2*&,*% 5*% 5#N9-$*% ,*6+#.*55*% *, 5(
;/.9V)(,9#& %?60#59-$*3 M(&% )*,,* %*),9#&H &#$% +./%*&,#&% -$*5-$*% ;/.9V)(,*$.%
)#&&$% 2(&% )* 2#6(9&* 0(%/% %$. )*% 2*$> ,*)8&9-$*%3
<=>? @\#5d6(&& CDDCH \#5d6(&& CD_eF *%, $& #$,95 2* ;/.9V)(,9#& 2* +.#+.9/,/%
L]L /).9, *& 5(&N(N* 2* 6#2/59%(,9#& fcgIQLW +#$. 2*% %?%,76*% 29%,.90$/%H
!"! #$%&'$%( )$*+&( ,% -./+012*+)3 45
!"#$%&&! &'( )*# !+),&# ,(,-!# &'( .#('( /0 1%$23'** 4$'5%('6%,(#7 8#$$9: +), '
&#(3,7 $' "!(,;<'6,%* #=#<6,"# # &(%6%<%$#7 # <%33)*,<'6,%*0 >!'*3%,*7: ?@A>
(#76# $,3,6! B $C'*'$D7# # 7D76E3#7 (#&(!7#*6'5$#7 7)( 106 B 107 !6'67 ') 3'F,3)30
A$ #76 ,7&%*,5$# B $C' (#77# !!" #$$%"&'())!*+),$%"&'$- .!&%"&'* !,/0
678 $# 3% #$G<H#<I#)( ?JK 4?D35%$,< J% #$ K#(,;#(9 ' !6! !"#$%&&! &'(
$C!+),&# # $CL*,"#(7,6! # M'(*#-,# J#$$%* ,(,-!# &'( N 3)* M$'(I# 4 K%,(
$C' (#77# !!" #$$---01*+%*+,2*342$ ,)43/+ 3+5$%,6* !,/90 A$ (#&(#* $#7 &(,*<,
# $C'(6,<$# 789,:)/&+ ;)43/ < 3+5&'= # 1020%!.!3% .'4 >39)'47 O8)(<H PQQRS0 M#6
'(6,<$# &(!<%*,7# $C#3&$%, #7 8TT7 &%)( )*# (#&(!7#*6'6,%* <%3&'<6# C#*7#35$#
C!6'670 T#&),7 PQQQ: ?JK #76 (#3&$'<! &'( >)?JK OM,3'66, RUURS +), '77%<,#
$' "!(,;<'6,%* 7D35%$,+)# )6,$,7'*6 )*# (#&(!7#*6'6,%* B 5'7# # 8TT ')F 3!6HG
% #7 # (!7%$)6,%* # &(%5$E3#7 ?VW 4K%,( $C' (#77# !!" #$$'2%,6*&(%!*&!+*&!$9: #6 B
M' #*<#G?JK &%)( $#7 (',7%**#3#*67 &'( '576('<6,%* #6 <%3&%7,6,%* 4K%,( $C' (#77#
!!" #$$---*53',+,&/*+),$%,6* !,/90
679:; 4&%)( 8!)+ .%!&+ ;)43/ + 3+5&'= ?'./9@3( A)( B3/&.:&/&!9 .'4 C&,&'=
OJ,*#( PQQQS9 #76 !"#$%&&! &'( $C!+),&# # $C)*,"#(7,6! # M'$,X%(*,# 4Y,"#(7, #9
,(,-!# &'( .,'*X('*<% M,'( %: #76 5'7! 7)( $#7 JTT7 #6 $#7 6#<H*,+)#7 # 7'6)('6,%*
+), 7%*6 # 5%**#7 7%$)6,%*7 &%)( $)66#( <%*6(# $C#=#6 #7 %**!#7 ,*6#(3! ,',(#7
#7 %**!#7 B $' 8TT0 Z#7 &#(X%(3'*<#7 # ?JVYW &%)( $C'*'$D7# # (!7#')F #
@#6(, %)"(#*6 #7 &#(7&#<6,"#7 # (#<H#(<H# &(%3#66#)7#7 &%)( $# 3% #$G<H#<I,*-
7D35%$,+)#0 A$ #76 ,7&%*,5$# B $C' (#77# !!" #$$---*+%*2+(*342$ +&.(4)$8;?BC$0
<<< OM%)"(#)( RUURS [ T'*7 $# &(%\#6 J]Y?N O8(!'*6 RUU^S: )* &(%\#6 Y>WZ:
-(_<# B $' -('* # `#F,5,$,6! # $' 76()<6)(#: *%)7 '"%*7 3%*6(! $C'&6,6) # #7 TTT
B 6(',6#( #7 &(%-('33#7 O>-)D#* RUUaS <%33# #7 76()<6)(#7 &%)( $' "!(,;<'6,%*
# 7D76E3#7 C!<(,67 #* Zb@ O.,$$,#(7 RUU^S: )* $'*-'-# # &(%6%6D&'-# # H')6
*,"#'): 6(' ),6 #* ZXK O>-)D#* RUUcS: )* $'*-'-# &%)( $' "!(,;<'6,%*0 Z# D.'=.=3
43 %"E+&F+.!&)' "&6)! ZX@ &#(3#6 # 7&!<,;#( $# <%3&%(6#3#*6 ) 7D76E3# B $C', # #
&(%<#77)7 7!+)#*6,#$7 <%33)*,+)'*6 ",' #7 3! ,'70 Z# ZXK #76 )* $'*-'-# X%(3#$
&$)7 7,3&$# #6 &$)7 #d<'<# &%)( $' "!(,;<'6,%* #7 7D76E3#7 (!&'(6,70 A$ #76 ,7&%*,5$#
B $C' (#77# !!" #$$,)63*/&"G*A($%)A!-.(3$HHH$0
6<< OM%)"(#)( RUUeS: =6<< OWH,#((DGJ,#- RUUf: WH,#((DGJ,#- RUU^S [
Z#7 "'(,'*6#7 H,!('(<H,+)#7 # TTT0 A$ #76 !-'$#3#*6 ,7&%*,5$# B $C' (#77#
!!" #$$%)2(+3A)(=3*'3!$"()I3+!%$:2449$J !!" #$$6/%&*+)/)(.4)*342$ A.:&)$<KHH$0
>%)7 "#*%*7 # &(!7#*6#( +)#$+)#7 %)6,$7 # "!(,;<'6,%*0 T'*7 $# <H'&,6(# f:
*%)7 '$$%*7 <%3&'(#( *%6(# %)6,$ '"#< $#7 %)6,$7 # "!(,;<'6,%* $,76!7 <,G #77)70 >%7
5#*<H3'(I7: !6'*6 $#7 (!7#')F # @#6(, &(!7#*6!7 #* 7%)7 7#<6,%* R0P0R: !3%*6(#*6
$C#d<'<,6! # *%7 *%)"#')F X%(3'$,73#7 &%)( $' "!(,;<'6,%* # 3% E$#70
!"#$ %& '(#)!*+, -
!"# $% $&!'()*%+ ",-# !.,"# */0-() 1% '*,2134% 0% 4,0/1(#!)(," !-5 '*("$('!-5
$,"$%')# 0% 1! #)*-$)-*% 0% 6*('7% %) 0- */#%!- 0% 8%)*( ',-* 0%# #9#)34%# */!$)(:#; <
'!*)(* 0% $%# 4,031%#+ ",-# ',-.,"# -)(1(#%* 1%# 1,=(>-%# )%4',*%11%# %) 1%# )%$&"(>-%#
0% 1! ./*(?$!)(," #942,1(>-% ',-* ./*(?%* >-%1>-%# '*,'*(/)/# (")/*%##!")%#;
@" -)(1(#!") 1%# 4A4%# '*("$('%#+ ",# ",-.%!-5 :,*4!1(#4%#+ '*/#%")/# %"
$&!'()*% B+ &/*()%") 1C!.!")!=%# 0%# )%$&"(>-%# 0% 1! ./*(?$!)(," #942,1(>-%; D,-#
4,")*%*,"# 1! $!'!$()/ 0% $%# :,*4!1(#4%# ',-* 4,0/1(#%* 1%# 4,031%# 0!"# 1%
$&!'()*% E %) ',-* ./*(?%* 1%# '*,'*(/)/# 0% #F*%)/+ 1C(")%*G21,$!=% %) 1%# '*,'*(/)/#
0% 1,=(>-%# )%4',*%11%# 0!"# 1% $&!'()*% H;
!"# 1% $&!'()*% I+ ",-# !11,"# $,4'!*%* ",)*% ,-)(1 !.%$ 1%# ,-)(1# 0% ./*(?$!)(,"
1(#)/# 0!"# $% $&!'()*%; D,# 2%"$&4!*7#+ /)!") 1%# */#%!-5 0% 8%)*( '*/#%")/# %" #,-#
#%$)(," J;K;J+ 0/4,")*%") 1C%L$!$()/ 0% ",# ",-.%!-5 :,*4!1(#4%# ',-* 1! ./*(?$!)(,"
0% 4,031%#;
!"#$%&' (
!"#$"%&' (& )* +,!"-$*."/# 0 1*'&
(& 234'
!"#$! %# &'()*+ , -
!" #!$%!#$%!" &!" "'"()*!" &! #+,+$#-(.#! &!" (!#*!" /012"3 !( &!" 4.(5*4(!"
&64#7#!" "58( 7!4.$5.9 +(.&-+!" !8 :;!#"%5<-(= >??@A B44&!# >??CA D5*58 E@@EA
F-GG!#58 >??HA B5.4II48- E@@EA J7&.GG4 E@@KLM ;48" $! $%49-(#!A 85." &58858" &!"
9#-8$-94.N +G+*!8(" ".# G!" "'"()*!" &! #+,+$#-(.#! &!" (!#*!" /012"3A 4-8"- O.! $!.N
&!" 4.(5*4(!" &64#7#!" &! (!#*!" $5*95"+" 94# &!" "'*75G!" P58$(-588!G" 7-84-#!"
!( G4 $58"(48(! ⊥M
Q5." &+$#-R58" +S4G!*!8( G!" 9#-8$-9!" &! G4 R+#-T$4(-58 U G64-&! &! 012" !( &!
(#48"&.$(!.#" &64#7#!" !8 -GG."(#48( 85(#! 9#595" G6+(.&! &. 9#5(5$5G! &6J#7-(#4S!
J#75#!"$!8(M D! 9#5(5$5G! 4 +(+ -8(#5&.-( 94# :;-GG >?C?L !( !"( (#4-(+ $5**! .8
9#57G)*! &! R+#-T$4(-58 4"'8$%#58! &48" :DG4#V! >?C?LM
".//0123
!" #$%&'%()* ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! "+
!"!" #$%&'()*&+, - .)/0 10 /2/*340/ 10 %$5$(%&*6%0 ! ! ! ! ! ! ! ! ! "7
!"!8 #$%&'()*&+, - 9:)&10 10 )6*+4)*0/ 1:)%.%0/ ! ! ! ! ! ! ! ! ! ! ! 88
!, #$-.-'-/) 012$3%.$45) 2$3-$)*')&. 672#8 ! ! ! ! ! ! ! ! ! ! ,9
!8!" ;0/(%&<*&+, 16 <%+*+(+90 =>? ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! 8@
!8!8 #$%&'()*&+, 16 =>? - .)/0 10 =AB/ ! ! ! ! ! ! ! ! ! ! ! ! ! ! 8C
!8! #$%&'()*&+, 16 =>? - 9:)&10 10 )6*+4)*0/ 1:)%.%0/ ! ! ! ! ! ! 8D
! :;$%<'4.)=$* 34*;* *=$ 0)* 7>?* ). 0)* 4=.-@4.)* 014$3$)* A
!"! #$%&'%()* "+
!" #$%&'%()*
!"# $%%!&# '&()!*"')+ %+# ,)-%'.'&$')+# *+# /01#2 *+# $"(!.$(+# *3$)4)+ +( *+#
()$&#*"5(+")# *3$)4)+# ,!") *+# (+).+# 5!.,!#-# ,$) *+# #6.4!%+# 7!&5('!&&+%#
4'&$')+# +( %$ 5!&#($&(+ ⊥8 !"# $%%!&# -9$%+.+&( .!&()+) 5!..+&( &!"# ,!":!&#
5$%5"%+) %$ 5%;(")+ )-<+=':+ +( ()$&#'(':+ +& /01# +( +& ()$&#*"5(+")# *3$)4)+#8 !"#
,!":!&# )-#!"*)+ %+# ,)!4%>.+# *+ :-)'?5$('!& (+%# @"+ %$ ,)!,)'-(- *+ #A)+(-2 %3'&(+)B
4%!5$9+2 %+# %!9'@"+# C/D +( D/D 9)E5+ F 5+((+ 5%;(")+ )-<+=':+ +( ()$&#'(':+8
!"!" #$%&'()*&+, - .)/0 10 /2/*340/ 10 %$5$(%&*6%0
G$&# 5+((+ #+5('!&2 &!"# '&()!*"'#!&# %+# ,)-%'.'&$')+# *+# /01# ,!") *+# (+).+#
5!.,!#-# ,$) *+# #6.4!%+# 7!&5('!&&+%# 4'&$')+# +( %$ 5!&#($&(+ ⊥8 H&#"'(+2 &!"#
.!&()!&# 5!..+&( &!"# ,!":!&# 5$%5"%+) %$ 5%;(")+ )-<+=':+ ()$&#'(':+ +& "('%'#$&(
*+# /01#8
!"!"!" ,-*./0)* 1) 2343'$%.5$) 672,*8
C!..+ *-IF .+&('!&&- 5'B*+##"#2 &!"# )-*"'#!&# &!()+ -("*+ $"= (+).+# 5!.B
,!#-# ,$) *+# #6.4!%+# 7!&5('!&&+%# 4'&$')+# +( %$ 5!&#($&(+ ⊥8 !"# &!(!&# ,$) Fbin
5+( +&#+.4%+ *+ #6.4!%+#8 1!'( "& +&#+.4%+ *-&!.4)$4%+ *+ :$)'$4%+# X 8 T (Fbin,X )
*-&!(+ %3+&#+.4%+ *+ (+).+# 5!&#()"'(# #") 5+# #6.4!%+# +( *!(-# *+# :$)'$4%+#2 +(
T (Fbin) #'9&'?+ %3+&#+.4%+ *+ !"#!$ %&'$ J(+).+# #$&# *+# :$)'$4%+#K8 L&+ #"4#('B
("('!& +#( "&+ 7!&5('!& σ *+ X F T (Fbin,X )2 @"' ,+"( M()+ -(+&*"+ "&'@"+.+&( F
"& +&*!.!),N'#.+ *+ T (Fbin,X )8
9:)0(;) < (') Fbin = {t, i,⊥} ! X = {x, y}* +! !"#! t1 = t(i(⊥,⊥), i(⊥,⊥))
!$ ,- !"#! %&'$. %!/!-01- &! !"#! t2 = t(x, i(⊥, y)) -2!$ /1$ %! %1$3&4*
5'-$)06"'-$ ,-! $,7$ ) , )'- σ = {x← i(⊥,⊥), y ← ⊥} 89')" &1 :;,"! <*=>. 1&'"$ ?
t2σ = t2{x← i(⊥,⊥), y ← ⊥} = t1
ii
t
i
y
t =2
t =1
= { x , y }
i
t
x
!"# O8P Q /+).+ +( #"4#('("('!&
! "#$%&'() *+ ,(&-.&%)/ 0) 1$ 23(&4.$'&5- 6 7$/) 0) 89:/
!" #$%&'&$! p #$() (! '")*" t "%' (! *$' %() N+ ,- %./("!0" 1&2" ε 2.!$'" 3-
#3(% 4-('" #$%&'&$!+ ,5"!%"*63" Pos(t) 2" #$%&'&$!% 25(! '")*" t "%' &!2(0'&1"*"!'
2.7!& #-) Pos(f(t1, t2)) = {ε} ∪ {0.p | "' p ∈ Pos(t1)} ∪ {1.p | "' p ∈ Pos(t2)}8
"' Pos(t) = {ε} /(-!2 t ∈ X $( t =⊥+ 9$&' (! $)2)" #-)'&"3 # %() 2"% #$%&'&$!%+
9$&' p1 "' p2 2"(: *$'% 2" N⋆8 !$(% 2&%$!% /(" p1 # p2 %5&3 ":&%'" ω ∈ N
⋆'"3 /("
p1 = p2.ω+
9& p ∈ Pos(t)8 -3$)% t|p %&;!&7" 3"% %$(%<'")*"% 2" t = 3- #$%&'&$! p "' t[s]p 2.!$'"
3" '")*" $6'"!( #-) 3" )"*#3-0"*"!' 2( %$(%<'")*" t|p = 3- #$%&'&$! p #-) 3" '")*" s+
>$(% 2.!$'$!% .;-3"*"!' #-) t(p) 3" %?*6$3" -##-)-&%%-!' 2-!% t = 3- #$%&'&$! p+
@'-!' 2$!!. (! '")*" t ∈ T (Fbin,X ) "' A (! "!%"*63" 2" %?*6$3"%8 A-&' PosA(t) =
{p ∈ Pos(t) | t(p) ∈ A}+ B-) 0$!%./("!'8 PosF (t) "%' 35"!%"*63" 2" #$%&'&$!% 2" t8 =
04-0(!" 2$!' (! %?*6$3" A$!0'&$!!"3 -##-)-C'+ ,5"!%"*63" 2" #$%&'&$!% !"#$%&! 25(!
'")*" t8 2.!$'. #-) FPos(t)8 "%' 2.7!& 0&<2"%%$(% D FPos(t) = {p ∈ Pos(t) | p.0 /∈
Pos(t)}. ,5"!%"*63" 2" 1-)&-63"% -##-)-&%%-!' (! '")*" t ∈ T (Fbin,X ) "%' 2.!$'.
#-) Var(t)+ B3(% A$)*"33"*"!'8 Var(t) = {t|p | p ∈ PosX (t)}+
;<)=%1) > '$(#$ )"##*+ Fbin = {t, i,⊥} &$ X = {x, y} ,"--& )(#+ ./&0&-1.& 23
4"#+%)*!"#+ .& $&!-& t2 = t(x, i(⊥, y)) (5&, .( !(,%#& )& t2 &+$ t3 6."!+7 ./&#+&-8.&
)& )& 1"+%$%"#+ !"#$%&! FPos(t2) &+$ {10} &$ 19%+:9& Var(t2) = {x, y}7 ./&#+&-8.&
)& 5(!%(8.&+ PosX (t2) &+$ {0, 11} 3
! EF9 R "%' (! "!%"*63" 2" !;<.&+ )& !*=*,!%$9!& (l, r) ∈ T (Fbin,X ) ×
T (Fbin,X )8 .;-3"*"!' 2.!$'. #-) l → r $G Var(r) ⊆ Var(l) "' l '∈ X + ," EF9
R &!2(&' (!" )"3-'&$! 2" ).<.0)&'()" →R %() 2"% '")*"% 2$!' 3- 03H'()" ).I":&1" "'
')-!%&'&1" "%' .0)&'" →⋆R+ B3(% #).0&%.*"!'8 !$(% 2&%$!% /(" t →R t′ %5&3 ":&%'" (!"
#$%&'&$! p 2" Pos(t)8 (!" )J;3" l → r ∈ R "' (!" %(6%'&'('&$! σ : X (→ T (Fbin) '"33"
/(" t|p = lσ "' t′ = t[rσ]p+
;<)=%1) ? >"%$ Fbin = {t, i, r,⊥}7 X = {x, y} &$ 9# $&!-& ,."+ t1 =
t(i(⊥,⊥), i(⊥,⊥))3 4"#+%)*!"#+ 9#& !;<.& r1 = i(⊥,⊥)→ r(⊥,⊥)7 (."!+ #"9+ (5"#+
? 1"++%8%.%$*+ )& !*)9,$%"# ,"--& .&+ +9%5(#$&+ @
t1 →r1 t(r(⊥,⊥), i(⊥,⊥))
t1 →r1 t(i(⊥,⊥), r(⊥,⊥))
4"#+%)*!"#+ 9#& !;<.& r2 = t(x, i(⊥, y) → t(y, r(⊥, x)7 (."!+ %. &0%+$& 9#& +&9.&
1"++%8%.%$* )& !*)9,$%"# ,"--& .( +9%5(#$& @ t1 →r2 t(⊥, r(⊥, i(⊥,⊥)))3
!"! #$%&'%()* +"
!"!"!+ ,-$%.'/0%1& (/$ 23/&/24*) 53/'')**%6%2%0-
! "#$%&'()# *# +! ,-.'/$!"'0& 1!. +2!&!+34# *2!$$#44'5'+'"- #4" 5!4-# 4). +2#&4#67
5+# *# R7*#4$#&*!&"4 *2)& #&4#65+# *# "#.6#48
2#&4#65+# *# R7*#4$#&*!&"4 9!11#+- -:!+#6#&" +! $+;").# .-<#=',# #" ".!&7
4'"',# 0) 1+)4 4'61+# +2#&4#65+# *#4 -"!"4 !$$#44'5+#4> *2)& #&4#65+# *# "#.6#4
E ⊆ T (Fbin) #4" R⋆(E) = {s ∈ T (Fbin) | t→
⋆R s ∧ t ∈ E}8
?). )& #&4#65+# *# "#.6#4 *# Fbin@ &0)4 !,0&4 +#4 1.01.'-"-4 +'&-!'.#4 4)',!&"#4 A
B a(s, s1) + a(s, s2) = a(s, s1 + s2)
B a(s1, s) + a(s2, s) = a(s1 + s2, s)
!,#$ a )& 43650+# *# Fbin8
?0'#&" t )& "#.6# *!&4 T (Fbin) #" s )& #&4#65+# *# "#.6#4 $+048 C0)4 *'.0&4
()# t ∈ s 44'@ #& $0&4'*-.!&" +#4 1.01.'-"-4 +'&-!'.#4 1.-$-*#&"#4@ s 1#)" D".# -$.'"
1!. $#""# 6!&'E.# A s = t1 + . . .+ tn !,#$ ti ∈ T (F ) #" '+ #='4"# j "#+ ()# tj = t8 F&
1.!"'()#@ )&# "#++# ".!&4G0.6!"'0& 4'61+# #4" !11#+-# +! $!&0&'4!"'0&8 C0)4 *-&0"0&4
∅ )& #&4#65+# ,'*#8
C0)4 10),0&4 05"#&'. +2#&4#65+# *# "#.6#4 !$$#44'5+#4 H 1!."'. *2)& "#.6# '&'"'!+
init #& $!+$)+!&" R⋆({init})8 I# 1+)4@ &0)4 10),0&4 -:!+#6#&" ,-.'/#. 4' )& "#.6#
att #4" !""#':&!5+# H 1!."'. *) "#.6# init 1)'4 ()# init→⋆R att8
78)9(2) : !"!#$#% & '(!)!*+'! ,- #$.% $/0!#$#% '(!#%!*/'! 1! 0!2*!% 344!%%5/'!%
& +32052 1. 0!2*! t1 6
R⋆({t1}) =
t(i(⊥,⊥), i(⊥,⊥)),
t(r(⊥,⊥), i(⊥,⊥)),
t(i(⊥,⊥), r(⊥,⊥)),
t(r(⊥,⊥), r(⊥,⊥))
.
7$.% +$."$#% 84252! 893'!*!#0 6
R⋆({t1}) =
t(i(⊥,⊥), i(⊥,⊥)) + t(r(⊥,⊥), i(⊥,⊥)) + t(i(⊥,⊥), r(⊥,⊥)) + t(r(⊥,⊥), r(⊥,⊥))
= t(r(⊥,⊥), i(⊥,⊥) + r(⊥,⊥)) + t(i(⊥,⊥), i(⊥,⊥) + r(⊥,⊥))
= t(t(i(⊥,⊥) + r(⊥,⊥), i(⊥,⊥) + r(⊥,⊥))
:# ;#- #$.% 3"$#% +2$."8 <.! '!% 0!2*!% t(r(⊥,⊥), i(⊥,⊥)), t(i(⊥,⊥), r(⊥,⊥))
!0 t(r(⊥,⊥), r(⊥,⊥)) %$#0 300!59#3/'!% +32 28=84250.2! & +32052 1! t1>
J0). .-4)6#.@ &0)4 .#".0),0&4 #& .-7-$.'").# +2-()',!+#&$# *) *$1!'=4?!4@5#9
10). +#4 434"E6#4 /&'48 I!&4 +! 4#$"'0& K8L8L@ &0)4 *0&&0&4 )&# '++)4".!"'0& *# $#""#
"#$%&'()# 10). +! ,-.'/$!"'0& *) J.0"0$0+# *2M.5'".!:# M.50.#4$#&"8
!"#$%&'( )* +'%,-%$(. /( 0# 12'%3-#&%4, 5 6#.( /( 789.
!"!# $%&'()*+',- . /0*'12 12 *3+,4*+25 10*&6&25
!"# $%&&% #%$&'(") "(*# '"&+(,*'#("# -%# .+/-'0'"!'+%# ,%# !*&(0!&%# ,1!+2+% %&
,%# &+!"#,*$&%*+# ,1!+2+%# .(*+ ,%# &%+0%# 3"'# $(0.(#/# .!+ ,%# #402(-%# 5("$6
&'(""%-# 2'"!'+%# %& -! $("#&!"&% ⊥7 8"#*'&%) "(*# 0("&+("# $(00%"& "(*# .(*9("#
$!-$*-%+ -! $-:&*+% +/;%<'9% &+!"#'&'9% %" *&'-'#!"& ,%# &+!"#,*$&%*+# ,1!+2+%#7
)*:* *: ;<&4=#&( />#'6'(. (& 7'#,./<-&(<' />#'6'(.
• ;<&4=#&( />#'6'(.
=" !*&(0!&% ,1!+2+%# 3"' #*+ Fbin %#& *" >*!,+*.-% A = (Q,Fbin, Qf ,△) (?
Q %#& *" %"#%02-% ,1/&!&# @*"!'+%#A) Qf ⊆ Q %#& *" %"#%02-% ,1/&!&# 3"!*<)
%& △ %#& *" %"#%02-% ,% +BC-%# ,% &+!"#'&'(" ,% &4.% #*'9!"&% D
f(q1(x1), q2(x2))→ q(f(x1, x2)),
(? f ∈ Fbin, q, q1, q2 ∈ Q, x1, x2 ∈ X 7
E1!*&(0!&% ,1!+2+%# #*+ Fbin 5("$&'(""% #*+ ,%# &%+0%# $-(# ,% Fbin7 =" !*6
&(0!&% ,1!+2+%# $(00%"$% !*< 5%*'--%# %& 0("&% .!# F .!# ,% 2!# %" G!*&) !#6
#($'!"& ,*+!"& -% &+!H%& '",*$&'9%0%"& *" /&!& !9%$ $G!>*% #(*#6&%+0%7 I- 5!*&
"(&%+ >*1'- "14 ! .!# ,1/&!& '"'&'!- ,!"# *" !*&(0!&% ,1!+2+%# 3"') 0!'#) .(*+
*" $("#&!"& #402(-% ⊥) *"% +BC-% ,% &+!"#'&'(" %#& ,% -! 5(+0% ⊥ → q(⊥)7
J!+ $("#/>*%"&) -%# +BC-%# ,% &+!"#'&'(" .(*+ $% #402(-% $("#&!"& .%*9%"&
K&+% $("#',/+/%# $(00% -%# !"#$% &'&(&)#$%7 L' -%# #(*#6&%+0%# ,'+%$&# u1, u2
,% t = f(u1, u2) #("& /&'>*%&/# !9%$ -%# /&!&# q1, q2) !-(+# -% &%+0% t #%+! /&'6
>*%&/ .!+ >*%->*%# /&!&# q !9%$ f(q1(x1), q2(x2)) → q(f(x1, x2)) ∈ △7 M(*#
,/3"'##("# 5(+0%--%0%"& -! +%-!&'(" ,% ,/.-!$%0%"& .!+ *" !*&(0!&% ,1!+2+%#
3"'7
L('& A = (Q,Fbin, Qf ,△) *" !*&(0!&% ,1!+2+%# 3"' #*+ Fbin7 E! +%-!&'(" ,%
,/.-!$%0%"& →A %#& ,/3"'% .!+ D L('& t, t′ ∈ T (Fbin ∪Q),
t→A t′ ⇐⇒
∃C ∈ C(Fbin ∪Q),∃u1, u2 ∈ T (Fbin)),
∃f(q1(x1), q2(x2))→ q(f(x1, x2)) ∈ △,
t = C[f(q1(u1), q2(u2))],
t′ = C[q(f(u1, u2))].
.
→⋆A %#& -! $-:&*+% &+!"#'&'9% %& +/;%<'9% ,% →A7
=" &%+0% $-(# t ,!"# T (Fbin) %#& !$$%.&/ .!+ *" !*&(0!&% ,1!+2+%# 3"' A =
(Q,Fbin, Qf ,△) #'
t→⋆A q(t)
.(*+ >*%->*%# /&!&# q ,!"# Qf 7 M(&%N >*% $%&&% ,/3"'&'(" $(++%#.(", F -!
"(&'(" ,1!*&(0!&% ,1!+2+%# 3"' "("6,/&%+0'"'#&% $!+ $% 0(,B-% ,1!*&(0!&%
,1!+2+%# 3"' .%+0%& N/+() *"% (* .-*# ,% +BC-%# ,% &+!"#'&'(" !9%$ -% 0K0% $:&/
C!*$G%7 8" $("#/>*%"$%) $1%#& .(##'2-% ,1!9('+ .-*# ,1*"% +/,*$&'(" F .!+&'+
,* 0K0% &%+0% $-(#7 8&) *" &%+0% $-(# t %#& !$$%.&/ #1'- 4 ! *"% +/,*$&'("
!"! #$%&'%()* +
!"#$% &'(&)* +)* #,-(.&%'/* !'**%0+)*1 2 !"#&%# -) .) &)#$) .+'* )& $)/"/& 2
(/) .'/34(#"&%'/ -) +" 5'#$) q(t) '6 q )*& (/ ,&"& 3/"+7
8) +"/4"4) -9"#0#) L(A) #).'//( !"# A )*& +9)/*)$0+) -) &'(* +)* &)#$)* .+'*
"..)!&,* !"# A7 :/ )/*)$0+) L -) &)#$)* .+'* )*& #).'//"%**"0+) *% L = L(A)
!'(# ;()+;()* "(&'$"&)* -9"#0#)* 3/%* A7 <'(* !'(='/* -%#) ;() .)& )/*)$0+)
-) &)#$)* )*& ,="+(, !"# +9"(&'$"&) -9"#0#)* A7
,-).(/) "0 !"# Fbin = {t, i,⊥}$ %!&'"()*!&' +,-.#!/-#0 (,-*1*0' A =
(Q,Fbin, Qf ,△) ()2&" 3-* 4 Q = {qt, qi, q⊥}5 Qf = {qt}5 0# △ +,0&'0/1+0 (0 *67
8+0' (0 #*-&'"#"!& 0'# (!&&) 9"7(0''!.' 4
{⊥ → q⊥(⊥), i(q⊥(x), q⊥(y))→ qi(i(x, y)), t(qi(x), qi(y))→ qt(t(x, y))}
%0# 0&'0/1+0 30.# :#*0 *03*)'0&#) 3+.' '"/3+0 9!//0 +0 '.";-� 4
{⊥−→ q⊥, (q⊥, q⊥)
i−→ qi, (qi, qi)
t−→ qt}
<!.' (!&&!&' +,0=0/3+0 (0 *)(.9#"!& -;09 +- *0+-#"!& (0 ()3+-90/0&# →A1 4
t1 = t(i(⊥,⊥), i(⊥,⊥)) →A1 t(i(q⊥(⊥), q⊥(⊥)), i(q⊥(⊥), q⊥(⊥))) →A1
t(qi(i(⊥,⊥)), qi(i(⊥,⊥))) →A1 qt(t(i(⊥,⊥), i(⊥,⊥)))$ >&2&5 +0 #0*/0 t1 0'#
-9903#) 3-* A15 c.?.d.t1 ∈ L(A1)$ @."'A.0 t1 0'# +,.&"A.0 #0*/0 -9903#) 3-* A15
-"&'" L(A1) = {t1}
• 1$2&*34'5)4$ 362$7$)*
:/ &#"/*-(.&)(# -9"#0#)* "*.)/-"/& )*& (/ ;(%/&(!+) T =
(Q,Fbin,F′bin, Qf , δ) '6 > )*& (/ )/*)$0+) 3/% -)* ,&"&*? Fbin )& F ′
bin
+)* )/*)$0+)* 3/%* -)* +)&&#)* )/&#,)* )& -)* +)&&#)* *'#&%)*1 *'/& -)* "+!@"A
0)&*? Qf ⊆ Q )*& -)* )/*)$0+)* 3/"(B? )& δ )*& (/ )/*)$0+) -) #C4+)* -)
&#"-(.&%'/ -) &D!) *(%="/&) E
f(q1(x1), q2(x2))→δ q(u), u ∈ TF ′bin[{x1, x2}] F7G1
q(x)→δ q′(u), u ∈ TF ′bin[x] F7H1
⊥ →δ q(u), u ∈ TF ′bin
F7F1
'6 ⊥ ∈ Fbin, f ∈ Fbin, x, x1, x2 ∈ X ? )& q1, q2, q, q′ ∈ Q7
I&"/& -'//, t (/ &)#$) )/&#,? T !'(#*(%& +) !#,.,-)/& E %+ .'$$)/.) )/
#)$!+"J"/& ;()+;()* 5)(%++)* *)+'/ +)* #C4+)* F7F7 K'(# +9%/*&"/.)? *% (/) 5)(%++)
)*& ,&%;()&,) ⊥ )& +" #C4+) ⊥ →δ q(u) )*& -"/* δ? !(%* ⊥ )*& #)$!+"., !"# q(u)7
8" *(0*&%&(&%'/ !'(#*(%& L(*;(92 +" #".%/)7 M% +" #C4+) f(q1(x1), q2(x2))→δ q(u)
)*& -"/* δ? "+'#* T #)$!+".) (/) '..(##)/.) -9(/ *'(*A"#0#) f(q1(t1), q2(t2))
! "#$%&'() *+ ,(&-.&%)/ 0) 1$ 23(&4.$'&5- 6 7$/) 0) 89:/
!" #$ %$"&$ q(u[x1 ← t1, x2 ← t2])' () *+!,-$ (**-""$.*$ /$ #! 0!"1!2#$ xi
/!.3 t $3% "$& #!*4$ !" ti5 6$ *!#*-# *(.%1.-$ 7-3,-89 *$ ,-$ #! "!*1.$ /$ t
3(1% !%%$1.%$5
6$ %"!.3/-*%$-" T /4:.1 #! "$#!%1(. "4;-#1<"$ 3-10!.%$ $.%"$ /$3 !"2"$3 RT =
{(t, t′) ∈ TFbin × TF ′bin|t→⋆
δ q(t′)' (-" ,-$#,-$3 q ∈ Qf}5
=(1% L ⊆ TFbin -. #!.;!;$ /8!"2"$' .(-3 /4:.133(.3 /(.* #8$.3$&2#$ RT (L) =
{t′ ∈ F ′bin|∃t ∈ L, (t, t′) ∈ RT }5
*+;+ + !"#$%& '&!! ()*!$ +,!-./0"
>(-3 (-0(.3 *!#*-#$" #! *#?%-"$ "4@$A10$ %"!.31%10$ $. -%1#13!.% /$3 %"!.3/-*B
%$-"3 /8!"2"$3 :.13 (- 1.:.135 >(-3 "$%"(-0(.3 $. %"!.3/-*%$-"3 /8!"2"$3 :.13 #84,-10B
!#$.*$ /- !"#$%&'#&()*+ $. "4B4*"1%-"$ (-" #$3 3C3%<&$3 :.135
DE(-!77!.1 FGGFH /4.(%$ !" RnT #! *(& (31%1(. /$ RT ' . I(135 J1.31' R⋆
T =⋃
n≥0RnT /4.(%$ #! *#?%-"$ %"!.31%10$ $% "4@$A10$ /$RT 5 DJ2/-##! FGGFH .(-3 "43$.B
%(.3 4;!#$&$.% -. &(C$. /$ *!#*-#$" #! *#?%-"$ %"!.31%10$ /$3 "$#!%1(.3 "4;-#1<"$3
/8!"2"$3 $. -%1#13!.% -. %"!.3/-*%$-" /8!"2"$3 /8+13%(1"$5 K#-3 "4*134&$.%' 4%!.%
/(..4 -. %"!.3/-*%$-" /8!"2"$3 T ' .(-3 ;4.4"(.3 -. .(-0$# %"!.3/-*%$-" /8!"2"$3
1.:.1 H' ! $#4 #$ %"!.3/-*%$-" /8!"2"$3 /8+13%(1"$ %$# ,-$ RH = R⋆T 5
<=)>%1) ;; ,!)- A1 $./0-! /-# "./121# "# $.#3# 4$# 567 ,!)- 0* -1/*8"0&-#01
T = (Q,Fbin,F′bin, Qf , δ) "9:*) 4/1 ; Q = {qt, qi, qr, q⊥}< Fbin = {t, i,⊥}< F ′
bin =
{t, i, r,⊥}< Qf = {qt} #- δ $.#*8# 2$# "# -1/*8)-)!*8 &! # $#8 80)=/*-#8 ;
(⊥,⊥)−−−→ q⊥ (q⊥, q⊥)
(i,i)−−→ qi (q⊥, q⊥)
(i,r)−−→ qr
(qi, qi)(t,t)−−→ qt (qi, qr)
(t,t)−−→ qt (qr, qi)
(t,t)−−→ qt
(qr, qr)(t,t)−−→ qt
>!08 /=!*8 "!*& R⋆T (L(A1)) = R⋆
T ({t1}) = R⋆T ({t(i(⊥,⊥), i(⊥,⊥))}) =
{t(i(⊥,⊥), i(⊥,⊥))< t(r(⊥,⊥), i(⊥,⊥))< t(i(⊥,⊥), r(⊥,⊥))< t(r(⊥,⊥), r(⊥,⊥))}7
?#+0$/1 @!"#$ A'#&()*+ $3% "( (34 *(&&$ -.$ %$*+.1,-$ (-" #! 04"1:*!%1(. /$3
3C3%<&$3 1.:.13 /84%!%3 DL$3%$. MNNO' P(# $" MNNQ' E(-!77!.1 FGGG' J2/-##! FGGRH5
S!.3 *$%%$ %$*+.1,-$' #$3 3C3%<&$3 3(.% &(/4#1343 $% !.!#C343 $. -%1#13!.% /$3
"$ "43$.%!%1(.3 3C&2(#1,-$3 2!34$3 3-" /$3 !-%(&!%$3 T *(.:;-"!%1(.3 /$3 3C3%<&$3
3(.% *(/4$3 !" /$3 &(%3 (- /$3 !"2"$35 U$#! 3-;;<"$ #8-%1#13!%1(. /$ #8!-%(&!%$ /$
&(%V!"2"$ :.1 "4;-#1$" !:. /$ %( "$ "43$.%$" /$3 $.3$&2#$3 /$ *(.:;-"!%1(.3' $%
#8-%1#13!%1(. /- %"!.3/-*%$-" /$ &(%V!"2"$ :.1 "4;-#1$" !:. /$ &(/4#13$" #$3 %"!.31B
%1(.3 $.%"$ /$3 *(.:;-"!%1(.35 W.3-1%$' #$ "(2#<&$ /$ 04"1:*!%1(. 2!34 3-" #8!.!#C3$
/8!**$33121#1%4 $3% "4/-1% !- *!#*-# /$3 *#?%-"$3 /$3 #!.;!;$3 "4;-#1$"3 3-" /$3 %"!.3B
/-*%$-"3 /$ &(%V!"2"$ :.1 "4;-#1$" c.9.d.' 3(1% -.$ "$#!%1(. "4;-#1<"$ R $% -. #!.;!;$
!"! #$%&%'%() *+,$-.&$/0) ,$-%$)1')2& 34,#5 "6
!"#$%& L' ()$*#$+,- R⋆(L)' +. R⋆&-/ $) *$0/# & !1&2%3&4/ ),-%/%3& 5& R6 7, 8 +94
$:;& $& 8$#- "!,! )$ &-/ 5& *+,-/ #% & #,& &8 !-&,/)/%+, 5& $) &$)/%+, R⋆*+;;&
#, / ),-5#*/&# <,%6
=),- $&- ->-/:;&- <,%- &/ %,<,%-' #, *)$*#$ 5&- )**&--%9$&- +# 5?#,& -# 4
)88 +2%;)/%+, @A&#%$$)5& BCCDE &-/ $& *+&# 5& $) /&*F,%G#&6 H, 8&#/ 5!;+,/ &
G#?#, /& ;& natt ,?&-/ 8)- )**&--%9$& 8) $& *)$*#$ 5?#,& -# 4)88 +2%;)/%+, 5&
$?&,-&;9$& 5&- /& ;&- !&$$&;&,/ )**&--%9$&- &/ 8) $) ,+,4)88) /&,),*& 5& natt
I *&//& -# 4)88 +2%;)/%+,6
=),- $) -&*/%+, J6B6J' K+#- 5+,,+,- #,& %$$#-/ )/%+, 5& *&//& /&*F,%G#& 8+# $)
3! %<*)/%+, 5# L +/+*+$& 5?M 9%/ )"& M 9+ &-*&,/6
!" #$%&%'%() *+,$-.&$/0) ,$-%$)1')2& 34,#5
(&//& -&*/%+, ;+,/ & *+;;&,/ $& 8 +/+*+$& 5?M 9%/ )"& M 9+ &-*&,/ N !"" #!$
%&'"! (!)')*)+' OMLP' #, 8 +/+*+$& !-+$3),/ $& 8 +9$:;& 5?&2*$#-%+, ;#/#&$$& 8+#
)**!5& I #,& &--+# *& 8) /)"!&' 8&#/ Q/ & R+ ;)$%-! &, /& ;&-' &, OST- &/ &,
)#/+;)/& 5?) 9 &-6
!"!# $%&'()*+),- ./ *(,+,',0% 123
(+;;& ;&,/%+,,! *%45&--#-' $& OML &-/ #, *% *#%/ )->,*F +,& G#% !-+#/ $& 8 +94
$:;& 5?&2*$#-%+, ;#/#&$$& &, *+,-/ #%-),/ #, ) 9 & 5&- *&$$#$&- 5?) 9%/ &6 (& *% *#%/
!/)%/ %,/ +5#%/ 8) @=%$$ UVWVE6 X$ !/)%/ / )%/! *+;;& #, 8 +9$:;& 5& 3! %<*)/%+,
)->,*F +,& &, @($) Y& UVWVE6
7,& *&$$#$& 5?) 9%/ & ) / +%- *F)Z,&- 5& *+;;#,%*)/%+, 5!,+/!&- 8) C0, C1 &/
Cp6 (F)G#& *F)Z,& -& *+;8+-& 5& 5 -%",)#2' r &/ g' &8 !-&,/),/ #, $",-".' &/
#, /*01)2+"30"4"1' N)*YP N[+% *&//& *&$$#$& )# ,%3&)# 5& 8+ / I $) <"# & J6B N)P &/
)# ,%3&)# 5& / ),-%-/+ I $) <"# & J6B N9P6 7,& 5&;),5& $",-".' -# #,& 5&- *F)Z,&-
C0, C1 &-/ / ),-R! !& )# -& 3&# -# $) *F)Z,& Cp6 M8 :- )3+% &\# #, ack -# Cp'
*&//& *&$$#$& 3) $& 8)--& )# 8 +*&--&# 6 M *& ;+;&,/4$I' *& 8 +*&--&# 8 &,5 $& 5 +%/
5?)**:- 5& $) &--+# *& 8) /)"!&6 ]#),5 %$ <,%/ )3&* $) &--+# *&' #, )#/ & *>*$& 5&
$",-".' ^ /*0 &-/ $),*!6 _, *+;9%,),/ 8$#-%&# - *&$$#$&- 5),- #, ) 9 & 9%,)% &' ,+#-
8+#3+,- R+ ;& #, ) 9%/ & 8+# ,?%;8+ /& G#&$ ,+;9 & 5& 8 +*&--&# -6 L) &2&;8$&
5),- $) <"# & J6B N*P )3&* $& ,+;9 & 5& 8 +*&--&# - N = 4' ,+#- )3+,- N − 1 = 3
*&$$#$&- 5?) 9%/ &6
`& *% *#%/ R+,*/%+,,& &, &2!*#/),/ 5&- /+# - 5?!$%;%,)/%+, a #,& *&$$#$& 5?) 9%/ &-
) 9%/ & &,/ & *&- 5 &,R),/-6 `&- R&#%$$&- 5& $?) 9 & -+,/ 5&- 8 +*&--&# -' G#% 8&#/4
Q/ & 3&#$&,/ )**!5& &, )->,*F +,%& I #,& &--+# *& 8) /)"!&6 `&- K 8 +*&--&# - )#
,%3&)# $& 8$#- 9)- -+,/ ) 9%/ !- 8) K^B *&$$#$&-6 `&- ")",),/- 5& *& ,%3&)#4$I -+,/
) 9%/ !- 8) $& ,%3&)# -#%3),/' &/ )%,-% 5& -#%/&6
! "#$%&'() *+ ,(&-.&%)/ 0) 1$ 23(&4.$'&5- 6 7$/) 0) 89:/
!"# !" # $%&'(%) *+,&-%)&,./ ,0 $/11(1/ ,( 2%3/,( */ 45&) -0 $/11(1/ ,( 2%3/,( */
)&,26%6)5& /) '0 75*81/ */ '/11(1/6
!"!" #$%&'()*&+, -. /01 2 3)45 -5 /674
95(6 ()%1%6526 1/6 6:;-51/6 -%2,%&/6 i< r< t /) b &/64/')%3/;/2) 45(& 1, 64='%>',?
)%52 */6 =),)6 !"#$ %#&'#() *+$ ),-#* /) ),-#* .#",/ 5@ A
# =),) !"# 6%.2%>/ B(/ )5(6 1/6 /2C,2)6 *+(2/ '/11(1/ D5( (2 4&5'/66/(&0 2/ C52)
DC,%)0 &%/2!
# =),) %#&'#() *+ 6%.2%>/ B(/ '/ 4&5'/66/(& D5( (2 */6 /2C,2)6 */ '/))/ '/11(1/0
3/(1/2) ,''=*/& E 1, &/665(&'/ 4,&),.=/!
# =),) ),-#* 6%.2%>/ B(/ '/ 4&5'/66/(& D5( '/))/ '/11(1/0 , 1, &/665(&'/ 4,&),.=/!
# =),) ),-#* .#",/ 6%.2%>/ B(/ 1, &/665(&'/ 4,&),.=/ D5( 1/ F/)520 /6) B(/1B(/
4,&)6 *,26 (2 */6 65(6?,&-&/6 ,(?*/665(6 */ '/ 25/(*!
G),2) *522= (2 =),) %2%)%,1 ),)0" !"# 45(& (2 ;5*81/ */ N = 4 4&5'/66/(&6
u = t(i(i(⊥,⊥), i(⊥,⊥)), i(i(⊥,⊥), i(⊥,⊥))) 5@ )5(6 1/6 4&5'/66/(&6 652) !"# /) 1, '/11(1/
*/ &,'%2/ )%/2) 1/ F/)52 DH5%& 1, >.(&/ ! D,0! 95(6 &/;,&B(526 B(/ '/ )/&;/ /6)
/I,')/;/2) 1, &/4&=6/2),)%52 *( )/&;/ *( ;5*81/ /2 >.(&/ !" D'0!
J/6 &/B(K)/6 652) 4&54,.=/6 */ -,6 /2 L,() F(6B(+E 1, &,'%2/ B(% )%/2) 1/ F/)52 A
i(x, r(y, z)) → r(x, r(y, z)) !"#$
i(r(x, y), z) → r(r(x, y), z) !"%$
i(⊥,⊥) → r(⊥,⊥) !"&$
!"! #$%&%'%() *+,$-.&$/0) ,$-%$)1')2& 34,#5 "6
! "#$%$&'!# &("%()(!# *(& +,-%(& .( +(/"0#(& 1'22(%3(& Rreq45 %( &6&#,)( 2("#
'++$7(+ 8 "! 3#'# ! "# $%&'%( )*+ v = t(r(r(⊥,⊥), r(⊥,⊥)), r(r(⊥,⊥), r(⊥,⊥))) 19:$+ %' ;-"+(
<=< 1>4=
!"# <=< ? '4 3#'# -%:>'% ! "# ),#% u (# >4 3#'# -%:>'% ! "# $%&'%( )*+ v
@( #(+)( ! "# $%&'%( )*+ v (&# '##($-!'>%( 8 2'+#$+ ." #(+)( ! "# ),#% u 2"$& /"(
u→⋆Rreq v= A:"& %$&#:!& *$B.(&&"& %' +3."*#$:! .3#'$%%3( C
u = t(i(i(⊥,⊥), i(⊥,⊥)), i(i(⊥,⊥), i(⊥,⊥)))→r3.6
t(i(r(⊥,⊥), i(⊥,⊥)), i(i(⊥,⊥), i(⊥,⊥)))→r3.6
t(i(r(⊥,⊥), r(⊥,⊥)), i(i(⊥,⊥), i(⊥,⊥)))→r3.6
t(i(r(⊥,⊥), r(⊥,⊥)), i(r(⊥,⊥), i(⊥,⊥)))→r3.6
t(i(r(⊥,⊥), r(⊥,⊥)), i(r(⊥,⊥), r(⊥,⊥)))→r3.5
t(r(r(⊥,⊥), r(⊥,⊥)), i(r(⊥,⊥), r(⊥,⊥)))→r3.4
t(r(r(⊥,⊥), r(⊥,⊥)), r(r(⊥,⊥), r(⊥,⊥))) = v
A:"& +()'+/":!& /"( *( *D()$! !E(&# 2'& "!$/"(5 c.8.d. $% (F$&#( '" ):$!& "!
'"#+( *D()$! .( u 8 v=
A:"& *:!#$!":!& 8 2+3&(!#(+ %(& '"#+(& +,-%(& .( GHI= @' *(%%"%( .( +'*$!( '**:+.(
%' +(&&:"+*( ' '" 2%"& "! (!J'!#5 (# %( 2+$7$%,-( &( 2+:2'-( .( D'"# (! >'& K"&/"E8 "!
.(& 2+:*(&&("+&= I$ %(& .("F (!J'!#& .E"!( *(%%"%( .()'!.(!# %' +(&&:"+*(5 %' *(%%"%(
J'$# .:!* "! *D:$F !:!B.3#(+)$!$&#( C
t(x, r(y, z)) → b(x, t(y, z)) !"#$
t(r(x, y), z) → b(t(x, y), z) !"%$
L'+ *:!&3/"(!#5 .( *(##( J'M:!5 &("% "! 2+:*(&&("+ '6'!# "!( .()'!.( .( %'
+(&&:"+*( %E:>#$(!.+'= N"'!. %( 2+:*(&&("+ ' ;!$ '7(* %' +(&&:"+*(5 $% (!7:$( "!(
+(/"0#( .( %$>3+'#$:! /"$ (&# 2+:2'-3( .( >'& (! D'"#=
b(x, t(⊥,⊥)) → t(x, i(⊥,⊥)) !"&$
b(t(⊥,⊥), z) → t(i(⊥,⊥), z) !"'($
! "#$%&'() *+ ,(&-.&%)/ 0) 1$ 23(&4.$'&5- 6 7$/) 0) 89:/
!"# $"%&#' (' !"#$ % &' '()*! )" *'+!,- +#!%$ .!/0!#+'/',+$ $!,+ 0!$$%1('$
0!"# ",' .'(("(' 2
3 4% +!"$ ('$ ',56,+$ )7",' +'((' .'(("(' $!,+ ', 8+6+ )+&!- .'++' .'(("(' )!,,' ('
*'+!, 9 (6 .'(("(' &!%$%,' 0("$ :6"+' $"%&6,+' '+ &6 8;6('/',+ ', 8+6+ )+&!<
b(x, t(i(y, z), i(u, v))) → t(x, i(i(y, z), i(u, v))) !"##$
b(t(i(x, y), i(u, v)), z) → t(i(i(x, y), i(u, v)), z) !"#%$
3 4% ", )'$ ',56,+$ )7",' +'((' .'(("(' )'/6,)' (6 #'$$!"#.'- 6(!#$ 2
3 $!%+ (78+6+ $"%&6,+ )' .'++' .'(("(' $'#6 !,$!-")*. '+ (' *'+!, &6 9 (6 .'(("('
&!%$%,' 0("$ :6"+' $"%&6,+' ', "+%(%$6,+ )'$ #=;('$ $"%&6,+'$
b(x, t(r(y, z), i(u, v))) → t(x, r(r(y, z), i(u, v))) !"#!$
b(x, t(i(y, z), r(u, v))) → t(x, r(i(y, z), r(u, v))) !"#&$
b(t(r(x, y), i(u, v)), z) → t(r(r(x, y), i(u, v)), z) !"#'$
b(t(i(x, y), r(u, v)), z) → t(r(i(x, y), r(u, v)), z) !"#($
3 $!%+ (78+6+ $"%&6,+ )' .'++' .'(("(' $'#6 "#/!* 0! .6# (6 .'(("(' 6 .:!%$% )'
)!,,'# (' *'+!, 9 ", )'$ ',56,+$ !,$!-")*. ', "+%(%$6,+ )'$ #=;('$ >?<@A !"
>?<BA<
C!"$ ,!+!,$ 06# RTAP(' DE4 .!/0('+ )8F,% ', .'++' $'.+%!,< C!"$ 0!"&!,$
!1+',%# (7',$'/1(' )' +'#/'$ 6..'$$%1('$ ', .6(."(6,+ (RTAP )⋆({u})< G' 0("$- ,!"$
0!"&!,$ 8;6('/',+ &8#%F'# H"' (' +'#/' "#"'& !,$!-")*. v '$+ 6++'%;,61(' 9 06#+%#
)" +'#/' "#"'& )+&! u 0"%$ H"' u→⋆RTAP v<
!"! #$%&'()*&+, -. /01 2 34)&-5 -5 ).*+6)*57 -4)%8%57
C!"$ "+%(%$!,$ (' /I/' &!.61"(6%#' Fbin 6&'. (6 $'.+%!, 0#8.8)',+' c.9.d ('$
$J/1!('$ 1%,6%#'$ i- r- t '+ b '+ (6 .!,$+6,+' ⊥<
C!"$ 6&!,$ )!,. ", 6"+!/6+' )76#1#' F,% U 6..'0+6,+ u .!//' $"%+ 2
⊥−→ q⊥
(q⊥, q⊥)i−→ qi (qi, qi)
i−→ qi (qi, qi)
t−→ qt
C!"$ 6&!,$ 8;6('/',+ ", 6"+!/6+' )76#1#' F,% V 6..'0+6,+ v .!//' $"%+ 2
⊥−→ q⊥
(q⊥, q⊥)r−→ qr (qr, qr)
r−→ qr (qr, qr)
t−→ qt
!K qt '$+ ", 8+6+ 6..'0+8<
9);$(<=)/ > 2& 3'$" *#"! ,$! +'*- &4!5!67&! 7 8(8+!*" 9 &4'$"#6'"! +4' 0 ! U
*4'((!7"! 7'- -!$&!6!*" &! "! 6! u 6')- 8.'&!6!*" $* 8"'" "#"'& )+&! ':!( *4)67# "!
,$!&&! "')&&!; <)6)&') !6!*"= &4'$"#6'"! +4' 0 ! V *4'((!7"! 7'- -!$&!6!*" &! "! 6! v
6')- 8.'&!6!*" $* 8"'" "#"'& !,$!-")*. ':!( *4)67# "! ,$!&&! "')&&!; >!- '$"#6'"!-
+4' 0 !- ')*-) ,$! +!- " '*-+$("!$ - +4' 0 !- ,$) *4'((!7"!*" ,$! u !" v -#*" 7&$-
(#67&!5!-;
!"! #$%&%'%() *+,$-.&$/0) ,$-%$)1')2& 34,#5 "6
!"#$%%& '(()* +,$- ./0-1+21 $+ 2/&+-#$321$/ #4&/"/1- .,$/ /0-,$#/1 %1 ./,"%561
#$ 7!89 !:+-:; %1- 02&2- -,+2 &..1%0- qi, qr, qt, qreq, qrel, qgrant, qm, qrt9 <+2$:2:=161+2;
31- 02&2- -:>+:?1+2 3,661 %1- -$:=&+2- @
• qi @ 7,$- %1- +,1$#- &$A#1--,$- #$ +,1$# 3,$/&+2 &:+-: B$1 31 +,1$# -,+2 !"#9
• qr @ 7,$- %1- +,1$#- &$A#1--,$- #$ +,1$# 3,$/&+2 -,+2 -,:2 !"# -,:2 $#%&#'( )*;
&=13 &$ 6,:+- $+ $#%&#'( )*9 <% C & &$3$+1 ./,.&>&2:,+ &$A#1--,$- #1 31 +,1$#9
• qt @ D1 E12,+ 1-2 -,:2 &$A#1--,$- #1 31 +,1$#; -,:2 %1 E12,+ +1 ",$>1&:2 .&-9
• qreq @ D1 +,1$# 3,$/&+2 1-2 1+ 2/&:+ #1 #16&+#1/ %1 E12,+ .,$/ %$: 6F61 ,$
.,$/ $+ +,1$# &$A#1--,$- #1 31 +,1$#9 G1221 /1B$F21 02&:2 ./,.&>019
• qrel @ D1 E12,+ 1-2 #0.%&30 #1 "&- 1+ H&$2 I .&/2:/ #$ +,1$# 3,$/&+29
• qgrant @ D1 E12,+ 1-2 #0.%&30 #1 H&$2 1+ "&- E$-B$4&$ +,1$# 3,$/&+29
• qm @ D1 E12,+ 1-2 #&+- ,$ &$A#1--,$- 31 +,1$#; %1 E12,+ ",$>1&:2 JG41-2 $+
02&2 &331.20K9
• qrt @ D1 E12,+ 1-2 -,:2 #&+- -,:2 &$A#1--,$- #1 31 +,1$#; c.I.d; :% +4C & .&- #1
3H&+>161+2 3:A#1--$- #$ +,1$# 3,$/&+2 JG41-2 $+ 02&2 &331.20K9
L+ $2:%:-&+2 31- 02&2-; %1- 2/&+-:2:,+- #$ 2/&+-#$321$/ -,+2 %1- -$:=&+21- @
(⊥,⊥)−−−→ q⊥
(q⊥, q⊥)(i,i)−−→ qi (q⊥, q⊥)
(i,r)−−→ qreq (q⊥, q⊥)
(r,r)−−−→ qr
(q⊥, q⊥)(r,t)−−→ qgrant (q⊥, q⊥)
(t,t)−−→ qt (q⊥, q⊥)
(t,i)−−→ qrel
(qi, qi)(i,i)−−→ qi (qr, qi)
(i,i)−−→ qi (qi, qr)
(i,i)−−→ qi
(qr, qr)(i,i)−−→ qr (qreq, qi)
(i,i)−−→ qreq (qi, qreq)
(i,i)−−→ qreq
(qreq, qr)(i,i)−−→ qreq (qr, qreq)
(i,i)−−→ qreq (qr, qr)
(i,r)−−→ qreq
(qr, qi)(i,r)−−→ qreq (qi, qr)
(i,r)−−→ qreq (qr, qr)
(r,r)−−−→ qr
(qr, qi)(r,r)−−−→ qr (qi, qr)
(r,r)−−−→ qr (qreq, qr)
(r,r)−−−→ qreq
! "#$%&'() * +(&,-&%). /) 0$ 12(&3-$'&4, 5 6$.) /) 789.
(qr, qreq)(r,r)−−−→ qreq (qr, qi)
(r,t)−−→ qgrant (qi, qr)
(r,t)−−→ qgrant
(qr, qr)(r,t)−−→ qgrant (qi, qi)
(t,t)−−→ qt (qi, qr)
(t,t)−−→ qt
(qr, qi)(t,t)−−→ qt (qr, qr)
(t,t)−−→ qt (qi, qreq)
(t,t)−−→ qrt
(qreq, qi)(t,t)−−→ qrt (qr, qreq)
(t,t)−−→ qrt (qreq, qr)
(t,t)−−→ qrt
(qi, qgrant)(t,b)−−→ qm (qgrant, qi)
(t,b)−−→ qm (qgrant, qr)(t, b) −→ qm
(qr, qgrant)(t,b)−−→ qm (qt, qi)
(b,b)−−→ qt (qt, qr)
(b,b)−−→ qt
(qr, qt)(b,b)−−→ qt (qi, qt)
(b,b)−−→ qt (qm, qr)
(b,b)−−→ qm
(qm, qi)(b,b)−−→ qm (qr, qm)
(b,b)−−→ qm (qi, qm)
(b,b)−−→ qm
(qt, qreq)(b,b)−−→ qrt (qreq, qt)
(b,b)−−→ qrt (qr, qrt)
(b,b)−−→ qrt
(qrt, qr)(b,b)−−→ qrt (qi, qrt)
(b,b)−−→ qrt (qrt, qi)
(b,b)−−→ qrt
(qi, qrel)(b,t)−−→ qm (qrel, qi)
(b,t)−−→ qm (qr, qrel)
(b,t)−−→ qm
(qrel, qr)(b,t)−−→ qm (qi, qi)
(t,i)−−→ qrel (qi, qr)
(t,r)−−→ qrel
(qr, qi)(t,r)−−→ qrel (qr, qr)
(t,r)−−→ qrel
! "#$%&'()*+$, -(,#, ,+$ .*, /01, *) .*, (+)23()*,
.4($-$*,
!"# $%&&% #%$&'(") "(*# +,-#%"&("# .*%/.*%# 0-,'1$!&%*,# $(""*# 2!"# $% 2(3
4!'"% 5!#-# #*, 2%# 678# %& 2%# !*&(4!&%# 29!,5,%#:
7&:6;< ;<%"%& =>>>) <%"%& =>>?) @%*'//!2% =>>A) <%"%& =>>B) C('$D*& =>>EF
%#& *" %"#%45/% 29(*&'/# +(*, /9!"!/G#% 29!$$%##'5'/'&- #*, 678#: H%& (*&'/ 4!"'+*/%
2%# !*&(4!&%# 29!,5,%# +(*, ,%+,-#%"&%, 2%# %"#%45/%# '"1"'# +(##'5/% 2% &%,4%#:
=$;/) ; %"I%, ?JJK) H/!0%/ =>>?) L%#%M*%, =>>BF %#& *" 4(&%*, 2% ,-3-$,'&*,%
*&'/'#- 2!"# *" $("&%N&% 2% 0-,'1$!&'(": H(44% 6'45*I) L!*2% +%*0%"& O&,% *&'/'#-
+(*, /! 0-,'1$!&'(" 2% #G#&P4%# '"1"'#:
7>= ;C!//!"2 =>>EF Q" /!"M!M% 2-2'- R /! &,!"#S(,4!&'(" 2% #&,*$&*,%# !,3
5(,%#$%"&%# +%,4%& 29%45!,.*%, 2%# $("#&,*$&'("# 2% 1/&,!M% %& 2% ,-3-$,'&*,% 2!"#
2%# /!"M!M%# M-"-,!/'#&%# $(44% T!0! (* H:
!"# /% $D!+'&,% K ) "(*# !//("# $(4+!,%, "(&,% (*&'/ !0%$ /%# (*&'/# 2% 0-,'1$!&'("
/'#&-# 2!"# $%&&% #%$&'(": U(# 5%"$D4!,I#) -&!"& /% +,(&($(/% 6VW %& /%# !*&,%#
4(2P/%# !,5(,%#$%"&# +,-#%"&-# 2!"# /! #%$&'(" X:?) 2-4("&,%"& /9%Y$!$'&- 2% "(#
"(*0%!*N S(,4!/'#4%# +(*, /! 0-,'1$!&'(" 2% 4(2P/%#:
!"#$ %& '(#)!*+, -
!"# $%!&# '&()!*"'( +,# -).+'/'&$'),# *,# 012#3 *,# $"(!/$(,# *4$)5), ,( *,#
()$&#*"6(,")# *4$)5),# -!") *,# (,)/,# 6!/-!#.# -$) *,# #7/5!+,# 8!&6('!&&,+#
5'&$'),# ,( +$ 6!&#($&(, ⊥9 !"# $%!&# .:$+,/,&( /!&(). 6!//,&( &!"# -!"%!&#
6$+6"+,) +$ 6+;("), ()$&#'('%, ,& 012# ,( ,& ()$&#*"6(,")# *4$)5),#9 < -$)(') *, 6,((,
6+;("), ()$&#'('%,3 !& %$ -+"# +!'& -!") ).#!"*), +,# -)!5+=/,# *, %.)'>6$('!& (,+#
?", +$ -)!-)'.(. *, #@),(.3 +4'&(,)A5+!6$:,3 +,# +!:'?",# B0C ,( C0C3 ,(69
D4$"(), -$)(3 &!"# ()!"%!&# ?", +, 6!&();+, *" -)!6,##"# *, ).A.6)'("), ,#( ,&A
6!), '++'#'5+,3 6,+$ ,/-E6F, +4,G6$6'(. *,# (,6F&'?",# (,++,# ?", +, ! "# $ %! &''
HIDDA+'J, 6$6F,K9 L& !"(),3 +,# (,6F&'?",# *4$66.+.)$('!& F$5'(",++,/,&( "('+'#. *$&#
+4$+:!)'(F/, *, #$(")$('!& #") +,# #()"6("),# *, *!&&.,# $ %! &'' #!&( '&"('+'#$5+,#9
<'&#'3 +,# /!('%$('!&# *4"& &!"%,+ !"('+ ?", &!"# $++!&# -)!-!#,) *$&# +$ -$)(',
#"'%$&(, #,)!&( *4"&, -$)( *, 8$6'+'(,) +, 6!&();+, *" -)!6,##"# *, ).A.6)'("),3 *,
),&*), 6!/-+=(,/,&( '&*.-,&*$&( *, +4'/-+./,&($('!& *" /!(,") *, ).A.6)'("),3 ,(
*4$"(), -$)( *4!M)') "& 8!)/$+'#/, 5',& $*$-(. $"N (,6F&'?",# *4$66.+.)$('!&9
!"#$%&! '()*$!
!"#$% && ' ()*#+,% -% ./0/1"$#2"%
3451#$455%6 738.(9
!"#$%&' (
!"#$%&" '& ()*)+,-#.,&
/01+#-011&2"
!"#$! %# &'()*+ , -
!"# $% $&!'()*%+ ",-# '*,',#,"# .%# #/#)01%# 2% *343$*()-*% 5,"$)(,""%.# 6789:#;<
=,-# 1,")*,"# >-% ",)*% 1,20.% ! .! '-(##!"$% 2?%@'*%##(," 2%# #/#)01%# 2% *34
3$*()-*% 689:#; 1!.A*3 .! #(1'.($()3 2%# *0A.%# %) .% 1,2% 2% 5,"$)(,""%1%") .3A0*%4
1%") 2(B3*%") '!* *!'',*) !-@ #/#)01%# 2% *343$*()-*% $.!##(>-%#<
=,-# !..,"# 1%))*% %" 3C(2%"$% -"% #,-# $.!##% 2% #/#)01%# 5,"$)(,""%.#+ .%#
3.31%")!(*%# 6D789:#;+ '*3#%*C!") .! '-(##!"$% 2?%@'*%##(," 2%# #/#)01%# 5,"$)(,"4
"%.# %) 2%# )%$&"(>-%# 2?!$$3.3*!)(," 2%# $!.$-.# !E,-)(##!") F -" ,-)(. 2% C3*(G$!)(,"
#/1E,.(>-% %H$!$%<
D" 5!()+ .%# D789:# #,") 2%# 789:# !/!") .% $I)3 A!-$&% 2% .! *0A.% #(1'.(G3<
% '.-#+ ",-# !..,"# $,")("-%* F #(1'.(G%* .% $I)3 2*,() 2%# *0A.%# 2?D789:# ',-*
>-% .%# #/#)01%# 2% *343$*()-*% #,(%") C*!(1%") 3.31%")!(*%#< =,-# .%# !''%.,"# .%#
3.31%")!(*%# F 2*,()% 69D789:#;< ?-"% '!*)+ (.# '*3#%*C%") 3A!.%1%") .! '-(##!"$%
2?%@'*%##(," 2%# #/#)01%# 5,"$)(,""%.# %) 2?!-)*% '!*) (.# #,") $!'!E.%# 2% 1(%-@
!''.(>-%* 2%# )%$&"(>-%# 2?!$$3.3*!)(," 2%# $!.$-.#<
J! *%.!)(," %")*% .%# #/#)01%# 2% *343$*()-*% .(#)3# $(42%##-# %#) '*3#%")3% %" GA-*%
K<L< J! M0$&% A*!##% #(A"(G% -"% )*!"#5,*1!)(," 2(*%$)% >-!"2 .! M0$&% ',(")(..3% %#)
-"% $,"C%*#(," ("2(*%$)% %" $,"#(23*!") .%# #/#)01%# 5,"$)(,""%.# $,11% -" )/'%
#'3$(!. 2%# #/#)01%# $.!##(>-%#<
!"# !" # $%&'()*+ %+(,% -%. ./.(01%. -% ,2324,)(5,%
6'+. &' 785,% !"9 &%. (,'+.:*,1'()*+. %+(,% ;$<. %( =;$<. .*+( ,%>,2.%+(2%.
>', &%. ?04@%. A'B %( ACB! D' (,'+.:*,1'()*+ -% ;$<. E%,. F=;$<. %.( ,%>,2.%+(2%
>', &' ?04@% A4B! <)1)&'),%1%+(9 &' ?04@% A-B .)8+)7% &' (,'+.:*,1'()*+ -% =;$<. E%,.
F=;$<.9 %( &' ?04@% A%B .)8+)7% &' (,'+.:*,1'()*+ -% F=;$<. E%,. $F=;$<.! F+7+9
&%. ?04@%. >*)+()&&2%. .)8+)7%+( &%. 4*+E%,.)*+. )+-),%4(%. -%. ./.(01%. :*+4()*++%&.
E%,. &% ./.(01% 4&'..)G5%!
D%. (,'+.:*,1'()*+. .*+( >&5. .)1>&%. -'+. +*(,% 2(5-% -% 4'. A;HIB 1'). %+
82+2,'&9 &%. (,'+.:*,1'()*+. .*+( (%4@+)G5%. >*5, (,')(%, (*5. &%. 4'. >*..)C&%. -%.
,08&%. -*++2%.!
!""#$%&
!" #$%&'()% *) +,-,./0&1/) 234.&0344)5% 627+#%8 ! ! ! ! ! ! ! 9:
!"!" #$%&'%()* ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! +,
!"!- .$/&*01$2/3%1& 4) .56* 7)$* 8.56* ! ! ! ! ! ! ! ! ! ! ! ! ! ! "
!"!+ .$/&*01$2/3%1& 4) 8.56* 7)$* .56* ! ! ! ! ! ! ! ! ! ! ! ! ! +
!; 27+#% <5,()4&=0/)% 6<27+#%8 ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !
!-!" #$%&'%()* ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !
!-!- .$/&*01$2/3%1& 4) .56* 7)$* 98.56* ! ! ! ! ! ! ! ! ! ! ! ! ! :
!-!+ .$/&*01$2/3%1& 4) 8.56* 7)$* 98.56* ! ! ! ! ! ! ! ! ! ! ! ;<
!9 <27+#% > ?/30&) 6+<27+#%8 ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! @A
!+!" #$%&'%()* ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ;:
!+!- .$/&*01$2/3%1& 4) 98.56* 7)$* 598.56* ! ! ! ! ! ! ! ! ! ! <=
! B34C)/%034 *)% %$%&'()% D34.&0344)5% C)/% 7+#% ! ! ! ! ! ! E"
! "#$%&'() *+ ,-.'/0). 1) 23435(&'6() 7895'&899):.
!" #$%&'()% *) +,-,./0&1/) 234.&0344)5% 627+#%8
!"# $%&&% #%$&'(") "(*# +,(+(#("# -% ! "#$% &% '()(*'+",'% -./*"+.//%0 12,/*)
"+./30 4%'$ 5%6'+"+/7 8! "%$ 9 2458: $!+!.-% /% #'0*-%, -%# 123# $-!##'4*%#5 6%#
(.7%$&'8# /% $% "(*9%- (*&'- #("& /:*"% +!,& /% 8!$'-'&%, -% $("&,;-% /* +,($%##*# /%
,<=<$,'&*,%) /% ,%"/,% $(0+->&%0%"& '"/<+%"/!"& /% -:'0+-<0%"&!&'(" /* 0(&%*, /%
,<=<$,'&*,%) %& /:!*&,% +!,& /:(?,', *" 8(,0!-'#0% .'%" !/!+&< !*@ &%$A"'4*%# /:!$=
$<-<,!&'(" A!.'&*%--%0%"& *&'-'#< /!"# -:!-B(,'&A0% /% #!&*,!&'(" #*, -%# #&,*$&*,%#
/% /(""<%# ; 03 <==5
C%&&% #%$&'(" $("&'%"& <B!-%0%"& -%# &,!"#8(,0!&'("# %"&,% -%# #D#&>0%# /% ,<=
<$,'&*,% 8("$&'(""%-# %& -%# #D#&>0%# /% ,<=<$,'&*,% $-!##'4*%#5
!"!" #$%&'%()*
1(*& /:!.(,/) "(*# '"&,(/*'#("# /% "(*9%!*@ #D0.(-%# 8("$&'(""%-# !++%-<# "("=
&%,0'"!*@5 E(*# /<"(&("# +!, FNT -:%"#%0.-% /% #D0.(-%# "("=&%,0'"!*@ &%- 4*%
FNT ∩ Fbin = ∅ %& +(*, $A!4*% H ∈ FNT ) -:!,'&< /% H %#& *"%5 F(*, 8!',% -!
/'#&'"$&'(" %"&,% /%# #D0.(-%# 8("$&'(""%-# +-*# 8!$'-%) /<#(,0!'#) #%*-# -%# #D0.(-%#
8("$&'(""%-# /% FNT #%,("& <$,'&# %" -%&&,%# 0!7*#$*-%#5 G" 123 8("$&'(""%- Rλ
%#& *" %"#%0.-% /% ,>B-%# /% -! 8(,0% H(t) → α (H H ∈ FNT ) t ∈ T (Fbin,X ) %&
α ∈ T (Fbin∪FNT ,X )5 G" I123 Rλ '"/*'& *"% ,%-!&'(" /% ,<=<$,'&*,% 8("$&'(""%--%
→Rλ#*, /%# &%,0%# /!"# T (Fbin ∪ FNT ) J u→Rλ
v ##:'- %@'#&% *"% +(#'&'(" p /% u
%& *"% ,>B-% H(t)→ α /!"# Rλ &%--%# 4*:'- %@'#&% *"% #*.#&'&*&'(" σ : X → T (Fbin)
!9%$ u|p = H(t)σ %& v = u[ασ]p5
E(*# /<"(&("# <B!-%0%"& +!, →⋆Rλ
-! $-;&*,% &,!"#'&'9% /% -! ,%-!&'(" /% ,<=
<$,'&*,% 8("$&'(""%--% →Rλ5 6! /<K"'&'(" #*'9!"&% /<$,'& -:%"#%0.-% /% &%,0%# /%
T (Fbin) !$$%##'.-%# +!, -! ,<=<$,'&*,% 8("$&'(""%--% /:*" &%,0% /% T (Fbin ∪ FNT )5
;3<9&'&89 = >R⋆λ? 8.+%/" Rλ ,/ 2458 %" E ⊆ T (Fbin ∪ FNT ) "%0 >,% E ∩
T (Fbin) = ∅? @A%/ %$B0% &% "%'$% 3**% +B0% &% T (Fbin) ; C3'"+' &% E % " &(/."(
C3' R⋆λ(E) %" % " &(D/+ *+)&% ., E
R⋆λ(E) = {β ∈ T (Fbin)|α→
⋆Rλ
β ∧ α ∈ E}.
2)0$(@6). A F/ ! "#$% &% '()(*'+",'% -./*"+.//%0 3 "'.+ &+G('%/*% C3' '3CC.'"
3,H ! "#$% &% '()(*'+",'% *03 +>,% E
I? 8+$C0+*+"( &% '#70% 13J%* &% /./)"%'$+/3,H &% FNT :?
K? L.&% &% -./*"+.//%$%/" E @3 '#70% &% 2458 H(t) → α % " 3CC0+*3B0% ,' ,/
"%'$% &% 03 -.'$% H(t) %,0%$%/" + 0% "%'$% " /% *./"+%/" 3,*,/ /./)"%'$+/30?
M? N30*,0 &% "%'$% 3**% +B0% 13J%* 0% *./"'O0% &, C'.*% , &% '()(*'+",'% C3'
&% /./)"%'$+/3,H:?
6%# +,(+,'<&<# -'"<!',%# /%# &%,0%# #("& /<7L +!,-< L -! #%$&'(" M5N5 O!'"&%"!"&)
#*, *" %"#%0.-% /% &%,0%# /% Fbin∪FNT ) "(*# !9("# -! +,(+,'<&< -'"<!',% #*'9!"&% J
P H(s1) +H(s2) = H(s1 + s2)
!"! #$%&'()% *) +,-,./0&1/) 234.&0344)5% 627+#%8 9:
!"# H $% &'()*+" ," FNT - .% /0 123$"4 $%" 1"++" 10 %&5*0( 12*% &2(/+" "&1 //"+6"
+ # %*%2& 12*%-
7*$& ++*%& 2++$&10"0 #"11" &10$#1$0" / 0 $% "8"(/+"-
;<)(=5) "> !"#$%!""&'&"$ (& )*+,
+!%$ Rλ -& )*+ ./%01"$ 2
Rλ =
F (x)→ H(G(x))
H(x)→ a(x, x)
G(x)→ 0(⊥,⊥)
G(x)→ 1(⊥,⊥)
.
34&".&'5-& (& $&6'&. 1##&..%5-&. R⋆λ({⊥}) &.$ #1-#/-7 2
8 &" '!(& 9!"#$%!""&- 2
F (⊥)→ H(G(⊥))
→ H(0(⊥,⊥)) + H(1(⊥,⊥))
→ a(0(⊥,⊥), 0(⊥,⊥)) + a(1(⊥,⊥), 1(⊥,⊥))
8 &" '!(& "!":9!"#$%!""&- ;#!''& -& '!(& (& 9!"#$%!""&'&"$ (&. )*+. #-1.:
.%</&.= 2
F (⊥)→ H(G(⊥)
→ H(0(⊥,⊥)) + H(1(⊥,⊥)) + a(G(⊥), G(⊥))
→ a(0(⊥,⊥), 0(⊥,⊥)) + a(1(⊥,⊥), 1(⊥,⊥))
+ a(0(⊥,⊥), 1(⊥,⊥)) + a(1(⊥,⊥), 0(⊥,⊥))
31 (%>76&"#& &.$ (1". -& '!(& "!":9!"#$%!""&-?H(x) → a(x, x) &.$ 1@@-%#15-&
'A'& .% t = G(⊥) #!"$%&"$ /" "!":$&6'%"1- G,
+)(?/@1)% 9 +% /" )*+ &.$ -%"71%6& B1/#C& &$ (6!%$&? "!/. @!/0!". '!"$6&6
</& -4&".&'5-& (&. 7$1$. 1##&..%5-&. .1". "!":$&6'%"1/D (& #& )*+ &" /$%-%.1"$ /"
'!(& (& 9!"#$%!""&'&"$ (&. )*+. #-1..%</&.,
#=,.0A.?&034 *1 =/3&3.35) *BC/D0&/?E) C/D3/)%.)4& )4 27+#
9 %& + &"#12*% :-;4 %*$& !*%& (*%106 #*(("%1 %*$& /*$!*%& 10 21"0 +" <=>4 $%
/0*1*#*+" 06&*+! %1 +" /0*)+?(" ,@"8#+$&2*% ($1$"++" /*$0 ##6,"0 A $%" 0"&&*$0#"
/ 01 B6"4 "% <CD&-
E"11" &"#12*% (*%10" #*(("%1 +" <=> /"$1 F10" 5*0( +2&6 "% G<CD& &*$& 5*0("
,@$% /0*)+?(" ," # +#$+ ,@ ##"&&2)2+216-
D*21 FNT $% "%&"()+" ," &'()*+"& %*%H1"0(2% $8 1"+& 3$" FNT = {H,Arbiter}-
I@"%&"()+" Fbin "&1 #"+$2 ,6J%2 , %& + &"#12*% :-;4 c.A.d.4 {i, r, t, b,⊥}-
! "#$%&'() * +,-'./)- 0) 12324(&'5() 6784'&788)9-
1.:9)- :282(2)-* !"#$%& '()*& +,'-#*& .& /01 α → β 2&'# 34'& 5,+6&'47&
&+ %+ 8/01 '()*& .& *# -#+7('& 2%76#+4& 9 H(α)→ β #6&5 α, β ∈ T (Fbin,X ):
;#' &<&-=*&> Rreqλ )?+?'?&2 @ =#'47' .&2 '()*&2 .& '?A?5'74%'& BC:DE> BC:FE &4
BC:GE .,++?&2 .#+2 *# 2&547,+ C:H:
H(i(x, r(y, z)))→ r(x, r(y, z)) BD:IE
H(i(r(x, y), z))→ r(r(x, y), z) BD:HE
H(i(⊥,⊥))→ r(⊥,⊥) BD:CE
1.:9)- 0) 4&(459$'&78 Rcirλ * ;,%' $%& *&2 '()*&2 )?+?'?&2 2,7&+4 #==*75#J*&2>
+,%2 #6,+2 J&2,7+ .K%+ &+2&-J*& .& '()*&2 .& 8/01 5,--& 7+.7$%? 57A.&22,%2 9
H(a(x, y))→ a(H(x), y),
H(a(x, y))→ a(x,H(y))|a ∈ Fbin,H ∈ FNT
!&2 '()*&2 +,%2 =&'-&44&+4 .& 57'5%*&' *& +,+A4&'-7+#* H @ 4,%2 *&2 &+.',742
.% 4&'-& .,++?:
1.:9)- 0) ;7&8' 6&<) RFPλ * L& 5#*5%* .& =,7+4 M<& &24 27-%*? =#' *&2 '()*&2
2%76#+4&2 9
Arbiter(x)→ x
Arbiter(x)→ Arbiter(H(x))|Arbiter,H ∈ FNT
!&2 '()*&2 +,%2 =&'-&44&+4 .& )?+?'&' *& +,+A4&'-7+#* H #%4#+4 $%K,+ # J&A
2,7+:
N,%2 =,%6,+2 ,J4&+7' *K&+2&-J*& .& 4&'-&2 #55&227J*&2 &+ 5#*5%*#+4
(Rreqλ )⋆({Arbiter(u)}): O& =*%2> +,%2 =,%6,+2 ?)#*&-&+4 6?'7M&' $%& *& 4&'-& ! "#
$%&'%( )*+ v &24 #44&7)+#J*& @ =#'47' .% 4&'-& ! "# ),#% u =%72 $%& u→⋆Rreqλ
∪RcirλRFPλ
v:
L& 4&'-& ! "# $%&'%( )*+ v &24 #44&7)+#J*& @ =#'47' .% 4&'-& ! "# ),#% u =%72 $%&
u→⋆ v: N,%2 *724,+2 57A.&22%2 *# '?.%547,+ .?4#7**?& 9
Arbiter(u) = Arbiter(t(i(i(⊥,⊥), i(⊥,⊥)), i(i(⊥,⊥), i(⊥,⊥))))→RFPλ
Arbiter(H(t(i(i(⊥,⊥), i(⊥,⊥)), i(i(⊥,⊥), i(⊥,⊥)))))→⋆Rcirλ
Arbiter(t(i(H(i(⊥,⊥)), i(⊥,⊥)), i(i(⊥,⊥), i(⊥,⊥))))→r4.3
Arbiter(t(i((r(⊥,⊥)), i(⊥,⊥)), i(i(⊥,⊥), i(⊥,⊥))))→RFPλ
Arbiter(H(t(i(r(⊥,⊥), i(⊥,⊥)), i(i(⊥,⊥), i(⊥,⊥)))))→⋆
...
Arbiter(t(r(r(⊥,⊥), r(⊥,⊥)), r(r(⊥,⊥), r(⊥,⊥))))→RFPλ
t(r(r(⊥,⊥), r(⊥,⊥)), r(r(⊥,⊥), r(⊥,⊥))) = v
N,%2 +,4,+2 =#' RTAPλ *& 8/01 5,-=*&4 .?M+7 &+ 5&44& 2&547,+:
!"! #$%&'()% *) +,-,./0&1/) 234.&0344)5% 627+#%8 "
!"!# $%&'()*%+&,-*' ./ $01( 2/%( 3$01(
!"# $%#&"'!(# )*%('+(*(' ,-+./0+##%1%'2 $+ 3456# +' $+ 456#7 8* /0!/!#%'%!(
9 )!('0+ :"-"( /*# $+ 02;2&0%'"0+ "'%,%#*(' R /+"' <'0+ #%)",2 /*0 "( 3456 Rλ
*1+& "( #+", (!(;'+0)%(*, Fl→r7
9/3:3%0&034 " !"# R $% &' ($) T (Fbin,X ) × T (Fbin,X )* α ∈ T (Fbin)* β ∈
T (Fbin) +# l→ r ∈ R, !"# Rλ -+ .&' #+- /$+
Rλ = {Fl→r(l)→ r} ∪
Fl→r(a(x, y))→ a(Fl→r(x), y),
Fl→r(a(x, y))→ a(x, Fl→r(y))|
a ∈ Fbin
.
0"%("*
α→{l→r} β ⇔ Fl→r(α)→⋆Rλ
β.
+)(;/<1)% 1- 23$# %!#+) /$+ 43%( Fl→r(α)→⋆Rλ
β +(# -3 5-6#$)+ #)3%("#"7+ 4+ -3
)+-3#"!% 4+ )8985)"#$)+ 2!%5#"!%%+--+ :!$) /$+ α→{l→r} β +(# 8#3;-"+ +<35#+=+%# $%+
2!"(,
9/)1=) " 69/)1=) %1..04.&) 6>30/ .?! *;4% 5@;44)A) B88 ,
> α→{l→r} β ⇒ Fl→r(α)→⋆Rλ
β ?
+-!% -3 48@%"#"!%* "- +<"(#+ $%+ :!("#"!% p 4+ α +# $%+ ($;(#"#$#"!% σ : X →
T (Fbin) #+--+( /$+ α|p = lσ +# β = α[rσ]p, !"# α′$% #+)=+ 5!%(#)$"# 5!==+
-+ ($"73%# ? α′ = α[Fl→r(α|p)]p,
A-3")+=+%#* +% $#"-"(3%# (+$-+=+%# -+( )BC-+( 4+ 5")5$-3#"!%* c.D.d. -+( )BC-+(
4+ -3 2!)=+ Fl→r(a(x, y)) → a(Fl→r(x), y) +# Fl→r(a(x, y)) → a(x, Fl→r(y))*
%!$( 37!%(
Fl→r(α)→⋆Rλ
α′. =>7>?
+-!% -3 5!%(#)$5#"!% 4+ α′* "- +<"(#+ $%+ )BC-+ C8%8)8+ 4+ Rλ* c.D.d Fl→r(l)→
r* $%+ ($;(#"#$#"!% σ : X → T (Fbin) +# $%+ :!("#"!% p #+--+( /$+ α′|p =
Fl→r(l)σ, E% 5!%(8/$+%5+* %!$( :!$7!%( 5!%(#)$")+ -+ #+)=+ α′[rσ] /$" +(#
8C3- D β :3) -3 5!%(#)$5#"!% 4+ α′, F!$( 37!%( 4!%5 ?
α′ →Rλβ. =>7@?
0"%("* +% $#"-"(3%# GH,HI +# GH,JI* %!$( :!$7!%( 484$")+ /$+ Fl→r(α)→⋆Rλ
β,
K3) 5!%(8/$+%#* α→{l→r} β ⇒ Fl→r(α)→⋆Rλ
β,
> α→{l→r} β ⇐ Fl→r(α)→⋆Rλ
β ? EL+5#"7+=+%#* :!$)($"7!%( +% "%4$5#"!% ($)
-+ 5M+="% 4+ )8985)"#$)+ /$" %!$( 5!%4$"# D -3 )+7+%4"53#"!%,
A%(#%B ,* '0*(#C!0)*'%!( =/02#+('2+ /*0 ,* DE&F+ =*? $+ ,* GH"0+ >79?B 2'*(' ,+
02#",'*' /0%(&%/*, $+ &+''+ #+&'%!(B +#' $!((2 /*0 ,+ 'F2!0E)+ 9 c.I.d. :"+ /!"0 '!"#
456 RB (!"# /!"1!(# &!(#'0"%0+ "( 3456 Rλ #%)",*(' R +( "'%,%#*(' $+# (!(;
'+0)%(*". Fl→r /!"0 &F*:"+ l→ r ∈ R +' "( (!(;'+0)%(*, G /!"0 ,+ /!%(' G.+7
! "#$%&'() * +,-'./)- 0) 12324(&'5() 6784'&788)9-
:#27(./) ; !"# R $% &' ($) T (Fbin,X )×T (Fbin,X )* E ⊆ T (Fbin)+ !"# l→ r
$%, )-./, 0, R ,# Rl→r /, 1&' #,/( 2$,
Rl→r = {Fl→r(l)→ r} ∪
Fl→r(a(x, y))→ a(Fl→r(x), y),
Fl→r(a(x, y))→ a(x, Fl→r(y))|
a ∈ Fbin
.
!"# RFP /, 1&' #,/ 2$,
RFP = {G(x)→ G(Fl→r(x)) | l→ r ∈ R} ∪ {G(x)→ x}.
!"# Rλ /, 1&' #,/ 2$,
Rλ =⋃
l→r∈R
Rl→r ∪RFP .
3"%("*
R⋆(E) = R⋆λ(E
′)
45,6 E′ = {G(t)|t ∈ E}+
1)/$(<5)- = 7/ 84$# %!#,) 2$, /4 6/9#$), ):;,<"5, ,# #)4%("#"5, 0, &' ,(# ("=$/:,
>4) RFP = {G(x)→ G(Fl→r(x)) | l→ r ∈ R} ∪ {G(x)→ x}+
>()5?) ! @>()5?) -544&84') @A7&( 4B* 0$8- 9C$88)D) EFF +
? R⋆(E) ⊆ R⋆λ(E
′) @ !"# α ∈ E ,# β ∈ R⋆({α})+ ,/!% /4 0:A%"#"!% 0, R⋆* "/
,<"(#, l1 → r1, . . . ln → rn ∈ R ,# t0, . . . , tn #,//,( 2$, α→{l0→r0} t0 →{l1→r1}
. . . →{ln−1→rn−1} tn−1 →{ln→rn} tn = β. ! "#$% Rl0→r0 &!'()*+ )!, -./0*1
+*./, 2++!/03!, ,(4-*54!, 6 )2 (&.(.,*+*./ 78
9/ :4/4&2)*,2/+ -! (&.-!,,3,; /.3, .<+!/./, )! -=!'*/ 0! &414-&*+3&! -*1
0!,,.3, > G(t0) →⋆Rλ
G(t1) . . . →⋆Rλ
G(tn) →G(x)→x tn = β8 ?2& -./,4@3!/+;
β ∈ R⋆λ(E
′)8
A*/,*; /.3, (.3B./, 0403*&! @3!
R⋆({α}) ⊆ R⋆λ({G(α)}). !"#$
C R⋆λ(E
′) ⊆ R⋆(E) > D+30*./, R⋆λ({G(α)})8
C %3((.,./, @3E3/! &F:)! 0! )2 G.&'! G(x)→ G(Fl→r(x)) !,+ 2(()*@34!8 A*/,*;
G(α)→G(x)→G(Fl→r(x)) G(Fl→r(α)))8 H! ()3,; 0E2(&F, )2 (&.(.,*+*./ 7; /.3,
(.3B./, 0403*&! @3!
R⋆({α}) ⊇ R⋆λ({G(α)}). !"%$
C %3((.,./, @3E23-3/! &F:)! 0! )2 G.&'! G(x) → G(Fl→r(x)) !,+ 2(()*@34!8
?2& -./,4@3!/+; 23-3/! &F:)! 0!
⋃
l→r∈RRl→r (!3+ I+&! 2(()*@34! 8
E3/*@3! +!&'! 2++!*/+ 0! T (Fbin) !,+ 0./- .<+!/3 !/ 2(()*@32/+ )2 &F:)!
G(x)→ x8 A*/,*; α ∈ R⋆λ({G(α)})8 HE2(&F, )2 045/*+*./; α ∈ R
⋆({α})8
?.3& -./-)3,*./; (.3& +.3, α ∈ E; 0! JK8LM !+ JK8NM; /.3, .<+!/./, @3!
R⋆(E) = R⋆λ(E
′)8
!"! #$%&'()% *) +,-,./0&1/) 234.&0344)5% 627+#%8 9
!"!# $%&'()*%+&,-*' ./ 0$12( 3/%( $12(
!"# $! #%&'()" *+,&,-%"'%. ")/# !0)"# *+)/0, 1/% #)(' /" 234. ($ %5(#'% /"
6234 &)++%#*)"-!"'7 8! +,&(*+)1/% %#' *)#,% -!"# &%''% #%&'()"7
8! '+!"#9)+:!'()" ;*+,#%"',% *!+ $! <=&>% ;?@ -% $! AB/+% C7D@ %#' -)"",% *!+
$% $%::% D c.E.d *)/+ /" 6234 Rλ #!'(#9!(#!"' /"% '%$$% &)"-('()". ")/# *)/0)"#
&)"#'+/(+% /" 234 R &)++%#*)"-!"' 7
:)(() " !"# $% &'( Rλ )$* T (Fbin ∪ FNT ,X ) × T (Fbin ∪ FNT ,X )+ !"# ,-
'( )$* T (Fbin,X ) × T (Fbin,X ) #-,) .$- /!$* 012.$- *34,- l → r ∈ R5 ", -6")#-
$% )!$) -%)-78,- &'( Rl→r ⊂ Rλ #-, .$-
Rl→r = {Fl→r(l)→ r} ∪
Fl→r(a(x, y))→ a(Fl→r(x), y),
Fl→r(a(x, y))→ a(x, Fl→r(y))|
a ∈ Fbin
.
-# ", -6")#- 942,-7-%# $%- *34,- rFP ∈ RFP #-, .$-
rFP = G(x)→ G(Fl→r(x))
-# G(x)→ x ∈ RFP #-,) .$-
⋃
l→r∈R
Rl→r ∪RFP = Rλ.
:"%)"5
R⋆λ(E
′) = R⋆(E)
2;-0 E ⊆ T (Fbin) -# E′ = {G(t)|t ∈ E}+
;/)1<) 9 6;/)1<) %1..04.&)8 $//!)!%) $% #-, '( 0!%)#*$"# 0!77- <2%) ,-
,-77- =+ -,!% ,- #19!*37- =5 %!$) !8#-%!%) R⋆λ(E
′) = R⋆(E)+
F'!"' -)"", /" 234. ($ %5(#'% ')/G)/+# /" 6234 &)++%#*)"-!"'7 H!(# $! +,I
&(*+)1/% "J%#' *!# #)/0%"' *)##(?$%7 3%B!+-)"# $J%5%:*$% -)"", &(I-%##)/#7
=>)(?5) "9 >!%)"<9*!%) $% &'( .$" #*2;2",,- 0!77- $% ?,#*24-
OnlyOneRequest <-) 9#2#) <$ ':@ 2A2%# $%- )-$,- <-72%<- <- *-))!$*0-
.$" -)# <9BC /*!/249- B$).$DC ,2 *20"%-+ :"%)"5
Rλ =
OnlyOne(t(r(x, y), i(z, t)))→ t(OnlyOne(r(x, y)), i(z, t))
OnlyOne(t(i(x, y), r(z, t)))→ t(i(x, y), OnlyOne(r(z, t)))
OnlyOne(r(r(x, y), i(z, t)))→ r(OnlyOne(r(x, y)), i(z, t))
OnlyOne(r(i(x, y), r(z, t)))→ r(i(x, y), OnlyOne(r(z, t)))
OnlyOne(r(⊥,⊥))→ r(⊥,⊥)
.
!"#$%&'( ) *+,&-.(, /( 01213'%&4'( 5673&%677(8,
! "#$%&'($ )*' +$ ,-. /0& 12"3(&2""$ +$ !* 454$ 4*"&67$8 9*&' "20' )20:2"'
+;<"&7 0"$ )72)7&;(; OnlyOneRequest '07 α ∈ T (Fbin) ($!!$ /0$ )207 (20( p, p′ ∈
Pos(α)= '& α(p) = r, p '= p′= *!27' p ( p′ 20 p′ ( p8 >*"' !* '0&($= "20' *!!2"' 7$:$"&7
? 3$ '0@$( ? !* '$3(&2" !"#$%& '("))&%%*+*#*,- ./01 #&% *!2"1*"!,% +0 3A*)&(7$ B8
>*"' !* '$3(&2" C8C= "20' *!!2"' 3A$73A$7 0"$ 32":$7'&2" &"+&7$3($ $" 32"'&+;7*"(
!$' 'D'(64$' 12"3(&2""$!' 3244$ 0" (D)$ ');3&*! +$' 'D'(64$' 3!*''&/0$'8
!" #$%&' ()*+,-./01,' 2(#$%&'3
!"!# $%&'(&)*+
>*"' 3$(($ '$3(&2"= ;(*"( +2""; 0" ,-. R '07 T (Fbin,X ) × T (Fbin,X ) $( 0"
($74$ t ∈ T (Fbin)= "20' 42"(72"' /0$ 3#$'( )2''&E!$ +$ 3*!30!$7 $%*3($4$"( !$ 454$
$"'$4E!$ +$ ($74$' *33$''&E!$' $" 0(&!&'*"( 0" F,-. +$ !* 1274$ )*7(&30!&67$ c.?.d.
F,-. ;!;4$"(*&7$8
91:7%&%67 ; 3/*, Rλe 0! 45637 Rλe &%, '*, '(8,1& -#-9&!,"*1& :-;"#&9&!, "..&#-
<4563= %&0#&9&!, %* )>"?0& 1@;#& '& Rλe &%, 0!& '&% A/19&% )*B'&%%/0% C
D7 H(a(x, y)) → α "2&) a ∈ FbinE H ∈ FNT E x, y ∈ X E x '= yE α ∈ T (Fbin ∪
FNT ,X ) &, Var(α) = {x, y}
F7 H(⊥)→ α "2&) α ∈ T (Fbin ∪ FNT )
G7 H(a(x,⊥))→ x "2&) x ∈ X E H ∈ FNT &, a ∈ Fbin7
G" 1*&(= !$' GF,-.' +$ !* +;<"&(&2" H= '2"( '0I'*44$"( $%)7$''&:$' )207 !*
');3&<3*(&2" +0 4;3*"&'4$ +$ 7;J;37&(07$ )207 0"$ 76K!$ l→ r ∈ R L
M8 +$ (720:$7 0"$ )2'&(&2" )207 !* 7;J;37&(07$=
H8 +$ :;7&<$7 '& l 32"327+$ *:$3 !$ '20'J($74$ 3207*"(= )0&'
N8 +$ 3*!30!$7 !* '0E'(&(0(&2" 7;'0!(*"( σ= $(
C8 +$ 7$)!*3$7 !$ '20'J($74$ )*7 rσ8
*$13%:3#&%67 /4 $'6&6368( /<='>%&'#?( ='>6'(,3(7& (7 @5A0*
>*"' !* '$3(&2" N8H= "20' *:2"' 42"(7; 3244$"( "20' )20:2"' (7*&($7 !$ ,OP
$" ,-.'8
Q$(($ '$3(&2" 42"(7$ 3244$"( !$ ,OP )$0( 5(7$ 1274*!&'; $" GF,-.' '20' 1274$
+#0" )72E!64$ +$ 3*!30! +#*33$''&E&!&(;8
.2&( FNT 0" $"'$4E!$ +$ 'D4E2!$' "2"J($74&"*0% ($! /0$ FNT =
{H,R,RT, TI, I, Arbiter}8 R#$"'$4E!$ Fbin $'( 3$!0& +;<"& +*"' !* '$3(&2" N8H= c.?.d=
{i, r, t, b,⊥}8
!"! #$%&' ()*+,-./01,' 2(#$%&'3 4
!" #$%&!" '! #()(*#+,-#! ./0123 ./042 !, ./052 '677(!" '87" &8 "!*,+67 /09 "67,
,#87":6#;(!" *6;;! &!" #$%&!" '<=>?@A Rreqλe *6;;! +7'+B-( *+)'!""6-"0
H(i(x, y)) → r(x,R(y)) !"#$
H(i(x, y)) → r(R(x), y) !"%$
R(r(x, y)) → r(x, y) !"&'$
R(⊥)→ ⊥ !"&&$
C6-" +&&-",#67" &8 #()(*#+,-#! '! &8 *67D%-#8,+67 "-+E87,! t0 = H(t(i(⊥,⊥), i(⊥
,⊥)))0 F-*-7! #$%&! ;!7,+677(! *+)'!""-" 7! G!-, H,#! 8GG&+B-(!0 F+7"+3 8-*-7 ,!#;!
'! T (Fbin) 7<!", 8**!""+I&! '! H(t(i(⊥,⊥), i(⊥,⊥)))0
C6-" 8E67" '67* I!"6+7 '! #$%&!" G6-# "G(*+D!# &8 *+#*-&8,+67 '- "J;I6&!H 8E87,
'<8GG&+B-!# &!" #$%&!" .10K23 0 0 0 3 .10LL20 M!" #$%&!" Rcirλe "67, '677(!" *+)'!""6-" !,
"67, '(D7+!" G6-# *N8B-! "J;I6&! '! Fbin "8-: ⊥0
H(i(x, y))→ i(H(x), y) !"&($
H(i(x, y))→ i(x,H(y)) !"&)$
H(r(x, y)) → r(H(x), y) !"&!$
H(r(x, y)) → r(x,H(y)) !"&*$
H(b(x, y))→ b(H(x), y) !"&+$
H(b(x, y))→ b(x,H(y)) !"&,$
H(t(x, y)) → t(H(x), y) !"&#$
H(t(x, y)) → t(x,H(y)) !"&%$
O8+7,!787, 76-" +&&-",#67" &<-,+&+"8,+67 '!" #$%&!" .10L9230 0 0 3 .10LP20 =7 8GG&+)
B-87, &8 #$%&! .10L92 .#!"G0 .10L/22 "-# t03 &! ,!#;! t′1 = t(H(i(⊥,⊥)), i(⊥,⊥))
.#!"G0 t′′1 = t(i(⊥,⊥),H(i(⊥,⊥)))2 !", 6I,!7-0 =7 8GG&+B-87, &8 #$%&! .10K2 !, &8
#$%&! .10LL2 "-# t′1 .#!"G0 t′′123 76-" 6I,!767" &! ,!#;! t
′2 = t(r(⊥,⊥), i(⊥,⊥)) .#!"G0
t′′2 = t(i(⊥,⊥), r(⊥,⊥))20
C6,!Q B-! t′2 !, t′′2 7! *67,+!77!7, 8-*-7 "J;I6&! '! FNT 0 <=>?@A 7! G!-, '67*
G&-" "<8GG&+B-!#0 F+7"+3 76-" +7,#6'-+"67" &! "J;I6&! Arbiter '67, &!" "(;87,+B-!"
"67, '(*#+,!" G8# &!" #$%&!" .10/R2 S .10//20 F-,#!;!7, '+,3 *! "J;I6&! 76-" G!#;!, '!
#()(*#+#! 8-,87, B-! 76-" 8E67" I!"6+70 !" #$%&!" +7'+B-(!" *+)'!""6-" #!;G&+""!7,
&8 "G(*+D*8,+67 '- ?FT G8# -7 =>?@A0
! "#$%&'() * +,-'./)- 0) 12324(&'5() 6784'&788)9-
H(t(x, y)) → b(x,RT (y)) !"#$%
H(t(x, y)) → b(RT (x), y) !"#&%
RT (r(x, y)) → t(x, y) !"##%
H(b(x, y)) → t(x, TI(y)) !"#'%
H(b(x, y)) → t(TI(x), y) !"#!%
TI(t(x, y)) → i(I(x), I(y)) !"#(%
TI(t(x, y))→ r(R(x), I(y)) !"#)%
TI(t(x, y))→ r(I(x), R(y)) !"#*%
I(i(x, y)) → i(x, y) !"#+%
I(⊥) → ⊥ !"#,%
Arbiter(t(x, y)) → t(x, y) !"'$%
Arbiter(t(x, y))→ Arbiter(H(t(x, y))) !"'&%
Arbiter(b(x, y)) → b(x, y) !"'#%
Arbiter(b(x, y)) → Arbiter(H(b(x, y))) !"''%
!"#$%#!#$ &'#(")*+'#( ,% $%+-% ). )*/.$ t0 = Arbiter(t(i(⊥,⊥), i(⊥,⊥)))0
1% t0 %$ %# !22,"3.!#$ ,%( +45,%( 67089:; 6709<:; 670=: %$ 67099:; ,% $%+-% t1 =
Arbiter(t(i(⊥,⊥), r(⊥,⊥))) %($ &!,&.,*0 ># !22,"3.!#$ ,%( +45,%( 670<?: %$ 670<<:
)% t1; ,% $%+-% t2 %($ &!,&.,* !@%& t2 = Arbiter(b(i(⊥,⊥), t(⊥,⊥)))0 A'$%B 3.% )%
t1 6+%(20 t2:; #'.( 2'.@'#( *5!,%-%#$ !22,"3.%+ ,! +45,% 6708?: 6+%(20 708<: %$ 2."(
'/$"%#$ ,% $%+-% t(i(⊥,⊥), r(⊥,⊥)) 6+%(20 b(i(⊥,⊥), t(⊥,⊥)):0 C%( )%.D $%+-%( -%#E
$"'##*( &"E)%((.( +%2+*(%#$%#$ )%.D &'#F5.+!$"'#( @+!"-%#$ !&&%(("/,%(0
A'.( 2'.@'#( '/$%#"+ ,G%#(%-/,% )% $%+-%( !&&%(("/,%( H 2!+$"+ ). $%+-% ! "#
$%#& %# &!,&.,!#$ (Rreqλe )
⋆({Arbiter(u)})0 1% 2,.(; #'.( 2'.@'#( *5!,%-%#$ @*+"F%+
3.% ,% $%+-% ! "# '&()&* $+, v %($ !$$%"5#!/,% H 2!+$"+ ). $%+-% ! "# $%#& u 2."( 3.%
u→⋆Rreq
λev0
>#F#; #'.( #'$'#( 2!+ RTAPλe ,G>IJKL &'-2,%$ )*F#" %# &%$$% (%&$"'#0 A'.(
+%-!+3.'#( 3.% RTAPλe %($ .# )%( /%#&M-!+N( .$","(*( )!#( ,% &M!2"$+% =0
1)/$(:5)- ! -.&* /!**$0#& %& *$1)#&' #" '2,#& H(x)→ α(x) /"' #&* 3#31&+ "$'&*4
5!)* '"$ !+* %&)6 7"* %& 8"'$"0#& x %& 7& & '2,#& 9
: H(⊥)→ α(⊥)4
: H(a(x, y)) → α(a(x, y))4 "8&7 x ∈ X ; H ∈ FNT ; a ∈ Fbin; α(x), α(⊥) &
α(a(x, y)) ∈ T (Fbin ∪ FNT ,X ).
<!)' *$1/#$=&'; +!)* 7!+*$%3'!+* %3*!'1"$* ()& #" '2,#& H(x) → α(x) &* 3#31&+>
"$'&4 <"' 7!+*3()&+ ; #&* '2,#&* %.?@ABC* %& /!$+ =6& D4EF G D4EE /&)8&+ H '&
37'$ &* 7!11& #&* *)$8"+ 9
Arbiter(x)→ x
Arbiter(x)→ Arbiter(H(x))
C! $+!#(O'+-!$"'# %($ *@")%#$% )!#( #'$+% *$.)% )% &!( 6JPQ: -!"( %# 5*#*+!,;
,! $+!#(O'+-!$"'# )% JKL( @%+( >IJKL( %($ $%&M#"3.% 2'.+ $+!"$%+ $'.( ,%( &!(
2'(("/,%( )%( +45,%( )'##*%(0
!"! #$%&' ()*+,-./01,' 2(#$%&'3 4
!"!" #$%&'()$*%+,)& -. #/0' 1.$' 23#/0'
!"# $%&#'()!(# * '(#'+,-'# .' %/0-'# .123456# .' 7!(7!%.8(7' Rl→rcheck9 .' 7!(:
#)%"7);!( .18%,%' .' <8%;8,-'# Rl→rσ 9 .' )'#) .' #",#);)");!( Rl→r
GS ') .' #",#);)");!(
Rl→rσ−apply .' -8 #!"# #'7);!( *=>=>=? @ -8 #!"# #'7);!( *=>=>=*9 ') +!()%'% 7!++'() -'#
");-;#'% $!"% #;+"-'% -8 %&:&7%;)"%' .'# 456# .8(# -8 #!"# #'7);!( *=>=>=A=
B'# $%;(7;$'# .' 7C8D"' '(#'+,-' .' %/0-'# #!() -;#)&# 7;:.'##"# E
?= 5/0-'# .123456# .' 7!(7!%.8(7' Rl→rcheck '#) "( 23456 (!"# $'%+'))8() .'
<&%;F'% #; "( )'%+' .' T (Fbin) !" !#$% 8<'7 l= G"8(. -8 7!(7!%.8(7' '()%' t
') l (1'#) $8# $!##;,-'9 (Rl→rcheck)
∗({F εl→r(t)}) = ∅=
>= 5/0-'# .123456# .' 7!(#)%"7);!( .18%,%' .' <8%;8,-'# Rl→rσ '#) "( 23456
(!"# $'%+'))8() .' 7!(#)%";%' "(' -;#)' !%.!((&' .' )'%+'# ;(.'H&# $8% .'#
<8%;8,-'# .' l= I'))' -;#)' '#) %'$%&#'()&' $8% "( )'%+' ') #&+8();D"'+'()
8" 7!"%8()9 (!"# $!"<!(# 7!(#;.&%'% 7'))' -;#)' 7!++' -8 #",#);)");!( %&#"-:
)8() ." $8# !" !#$&" %= 6!;) t′ -' )'%+' %&#"-)8() .' (Rl→rcheck)
∗({F εl→r(t)})=
(Rl→rσ )∗(F l→r
copy(t′)) +/(' @ -1"(;D"' )'%+' t′′ %'$%&#'()8() -8 #",#);)");!( %&:
#"-)8() .' -8 !" !#$&" % '()%' t ') l=
J= 5/0-'# .123456# .' )'#) .' #",#);)");!( Rl→rGS '#) "( 23456 #$&7;F8() -8
<&%;F78);!( .1"(' #",#);)");!( ,;'(:K!%+&'= 2L'7);<'+'()9 #; l (1'#) $8# -;(&8;%'
8-!%# "(' <8%;8,-' x 8$$8%8M) .'"H K!;# 8" +!;(#= 6; )!")'# -'# !77"%%'(7'# .' x
$8%)80'() -' +N+' )'%+' 7!++' <8-'"% 8-!%# -8 #",#);)");!( '#) ,;'(:K!%+&'=
(Rl→rcheck)
∗(F l→rcheck(t
′′)) = {t′′} '#) -8 #",#);)");!( ,;'(:K!%+&'9 ∅ #;(!(=
*= 5/0-'# .123456# .' #",#);)");!( Rl→rσ−apply '#) "( 23456 #$&7;F8() -18$$-;:
78);!( .' -8 #",#);)");!( ,;'(:K!%+&' %&#"-)8() .' t ') l #"% -' )'%+' r= O;(#;9
(Rl→rσ−apply)
∗(F l→rrewrite(t
′′)) = {rσ}=
2( ");-;#8() -'# * '(#'+,-'# .' %/0-'# .123456#9 (!"# #!++'# 78$8,-'# .'
#;+"-'% -8 %&:&7%;)"%' .1"( 456=
!"!"!5 %67),' 89(#$%& 8, :;-:;18/-:, < Rl→rcheck
Rl→rcheck '#) "( 23456 (!"# $'%+'))8() .' <&%;F'% #; "( )'%+' .' T (Fbin) !"'
!#$% 8<'7 l= B18$$-;78);!( .' 7') 23456 '#) %'(<!P& $8% -8 $%&#'(7' ." #P+:
,!-' (!(:)'%+;(8- F εl→r= 2)8() .!((& "( )'%+' t ∈ T (Fbin) 7!(7!%.8() 8<'7 l9
(Rl→rcheck)
∗({F εl→r(t)}) '#) {t
′} !Q
R $!"% )!") p ∈ PosF (l)9 t(p) = t′(p) S
R $!"% )!") p ∈ PosX (l)9 t′(p) = ⊕x,p ∈ Fbin S
R $!"% )!") p ∈ FPos(l)\PosX (l)9 ') $!"% )!") p′ ∈ Pos(t|p)9 t(p.p
′) = t′(p.1.p′)=
B' #P+,!-' ⊕x,p '#) "( +8%D"'"% #;0(;F8() D"' -' )'%+' #!"# 7' +8%D"'"% '#) -8
<8-'"% D"' -8 <8%;8,-' x @ -8 $!#;);!( p .8(# l !,);'() D"8(. -8 7!(7!%.8(7' 8$$8%8M)=
G"8(. -8 7!(7!%.8(7' '()%' t ') l (1'#) $8# $!##;,-'9 (Rl→rcheck)
∗({F εl→r(t)}) = ∅=
!)'T D"'9 @ 7' $!;()9 (!"# (' <&%;F!(# $8# -' 78# !Q .'"H <8%;8,-'# ;.'();D"'#
.!;<'() $8%)80'% -8 +N+' <8-'"%= U!(7 (Rl→rcheck)
∗({F εl→r(t)}) = ∅ #'"-'+'() #; t ') l
#!() #)%"7)"%'--'+'() .;L&%'()#=
! "#$%&'() * +,-'./)- 0) 12324(&'5() 6784'&788)9-
:2;8&'&78 < =Rl→rcheck> !"# l→ r $%& '()*& +& T (Fbin,X )×T (Fbin,X ), -!$. +/%!0
#!%. 12' Rl→rcheck *& 3456 7!%.#'$"# 8 12'#"' +& l 7!99& .$"# :
;, F pl→r(x) → ⊕l|p(x) ∈ R
l→rcheck ." p ∈ PosX (l) &# !< ⊕l|p &.# $% .=9>!*& .1/7"2*
?$& %!$. 7!%."+/'!%. +@A#'& +2%. Fbin
B, F pl→r(a(x, y)) → a(F p.0
l→r(x), Fp.1l→r(y)) ∈ Rl→r
check ." p ∈ Pos(l) \ FPos(l) &#
a = l(p)
C, F pl→r(⊥)→⊥∈ Rl→r
check ." p ∈ FPos \ PosX (l)
?@)/%9) A D!%."+/'!%. *2 '()*& a(b(x,⊥), c(d(⊥, y), z))→ r 7!99& $%& '()*& +&
'/0/7'"#$'&E x, y, z ∈ X &# Var(r) ⊆ {x, y, z}, F!$' +&. '2".!%. +& *".">"*"#/E %!$.
+/%!#!%. 7& #&'9& 12' l, G"%."E
Rl→rcheck =
F εl→r(a(x, y))→ a(F 1
l→r(x), F2l→r(y))
F 1l→r(b(x, y))→ b(F 11
l→r(x), F12l→r(y))
F 11l→r(x)→ ⊕x(x)
F 12l→r(⊥)→⊥
F 2l→r(c(x, y))→ c(F 21
l→r(x), F22l→r(y))
F 21l→r(d(x, y))→ d(F 211
l→r(x), F212l→r(y))
F 211l→r(⊥)→⊥
F 212l→r(x)→ ⊕y(x)
F 22l→r(x)→ ⊕z(x)
.
! "#$ %! &'!()!*"&! %! +,-&!) %! +./.0+1$#+! Rl→rcheck 2+.)!($. 01/%!))#) !)$
%! 3.+14!+ 5#'#( $!+*! t %6((. !)$ %! &7 86+*! )6#971$.! c.:.d l; <( 722&15#7($
&'<=>?@ %! &'!A!*2&! BC )#+ &! $!+*! F εl→r(t) %,)5#! (6#) 6"$!(6() #( 261($ 4A!
%! &'!()!*"&! %! $!+*!) )#00!))!#+) %7() T (Fbin)D )1 0!$ !()!*"&! !)$ 31%!D 7&6+)
t (! 06(06+%! 27) 73!0 l; @1(6(D &'!()!*"&! %! $!+*!) )#00!))!#+) !)$ 3+71*!($ #(
)1(-&!$6( %6$. %# $!+*! t *7+5#. : &7 26)1$16( %!) 37+17"&!) %7() l 73!0 #( )E*"6&!
)2.017& ⊕x 6F x !)$ #( 37+17"&! %! l;
!"!"!" #$%&'( )*+,-#. )' /01(234/2501 )*6373' )' 863567&'( 9 Rl→rσ
Rl→rσ !)$ #( <=>?@ (6#) 2!+*!$$7($ %! 06()$+#1+! #(! &1)$! 6+%6((.! %! $!+*!)
1(%!A.) 27+ %!) 37+17"&!) %! l; G!$$! &1)$! !)$ +!2+.)!($.! 27+ #( $!+*! !$ ).*7(/
$15#!*!($ 7# 06#+7($D (6#) 26#36() 06()1%.+!+ 0!$$! &1)$! 06**! &7 )#")$1$#$16(
+.)#&$7($ %# 27) !" !#$%" &; @61$ t′ &! $!+*! +.)#&$7($ %! (Rl→rcheck)
∗({F εl→r(t)});
(Rl→rσ )∗(F l→r
copy(t′)) *,(! : &'#(15#! $!+*! t′′ +!2+.)!($7($ &7 )#")$1$#$16( +.)#&$7($
%! &7 !" !#$%" & !($+! t !$ l;
H6#) 1($+6%#1)6() &7 (6$16( %'%#'#& $& (%#)%'*&+ 06()$+#1$! : 27+$1+ %'#( $!+*!;
I6#+ )1*2&14!+D &!) 37+17"&!) %! l )6($ +!(6**.!) %7() %!) 37+17"&!) xi 73!0 i .$7($
#( !($1!+;
!"! #$%&' ()*+,-./01,' 2(#$%&'3 4
5*6-0.07- !"# t $% #&'(& )& T (Fbin,X )* !"# x0, . . . , xn +&, -.'"./+&, .00.1
'.",,.%# ).%, t )& 2.$34& 5 )'!"#&* 67.'/'& )& -.'"./+&, -")& 8.'/'& )& -.'"./+& &%
/'&9: )& t; )<%!#< 0.' ⊤X (t); &,# )<=%" 0.' +& #&'(& ,$"-.%# > ⊤(⊕x0(⊥),⊤(⊕x1(⊥
), . . . ,⊤(⊕xn(⊥),⊥⊤) . . .))*
(8,+9), :; !"# t $% #&'(& )& T (Fbin,X ) #&+ ?$& t = a(b(x0,⊥), c(d(⊥, x1), x2))*
@"%,";
⊤X (t) = ⊤(⊕x0(⊥),⊤(⊕x1(⊥),⊤(⊕x2(⊥),⊥⊤))).
A!%,")<'&' +& #&'(& t; ,!"# t′ &,# +& #&'(& a(b(a(⊥,⊥),⊥), c(d(⊥,⊥), a(⊥,⊥)))*
A+."'&(&%#; t =+#'& t′*
!"# !$%&'(# "' )*+,- '!". /�&%%1'% 2& 3!'.%#"(#& 451#$#& 2& 61#(1$4&. ('7
.%1'3(8 2& t9 '!". 16!'. 45('%&'%(!' 2& 28:'(# "' *+,- %&4 ;"&9 45"'(;"& %�&
133&..($4& 2& t′ &.% ⊤(a(⊥,⊥),⊤(⊥,⊤(a(⊥,⊥),⊥⊤)))<
=1 2(>3"4%8 (3( &.% ;"& '!". 6!"4!'. .&"4&0&'% 2&. )*+,-.< =& /!('% 348 &.% 4&.
.?0$!4&. ⊕x< )@&3%(6&0&'%9 t′′3!'%(&'% 2& %&4. .?0$!4&.< !"# 3A1;"& .!".7%�&
s 2& t′′ %&4 ;"& s = ⊕x,p(s′)9 '!". /!"6!'. 45('%&#/#8%&# 2& 41 01'(B#& ."(61'%& C
/!"# 41 ."$.%(%"%(!' σ %&44& ;"& t = t′σ9 σ(x) = s′< D!". /!"6!'. 2!'3 3!'.(28#&#
4&. .?0$!4&. ⊕x 3!00& 2&. 01#;"&"#. /!"# ('(%(14(.&# 41 3!/(& 2& σ(x)<
!"# #8."0&# 41 .(%"1%(!' 3!"#1'%&9 /!"# 2&"E %�&. t ∈ T (Fbin) &% t′ ∈
T (Fbin,X )9 /!"# 4&.;"&4. (4 &E(.%& "'& ."$.%(%"%(!' σ : X → T (F) %&44& ;"& t′σ = t9
F< '!". /!"6!'. 3!'.%#"(#& "' %�& t′′ 2!%8 2&. 01#;"&"#. 1?1'% 4& 4(&' 16&3 σ G
H< I&. 01#;"&"#. .!'% 8J14&0&'% 21'. 451#$#& 2&. 61#(1$4&.⊤X (t′) &% 4&"# /!.(%(!'
21'. ⊤X (t′) &.% 3!''"& /"(.;"& t′ &.% 2!''8<
=& $"% 2& 45)*+,- 2!''8 21'. 41 28:'(%(!' K &.%
F< 2& 68#(:&# ;"5"' %�& t′ &.% 2& 41 L!#0& 2!''8& MtN
H< 2& 01#;"&# 21'. t′ 41 /!.(%(!' 2& 61#(1$4&. 2& t &%
O< 2& 3!/(&# .?0$!4& /1# .?0$!4& 3A1;"& 614&"# ."# 2&. 01#;"&"#. 21'. 451#$#&
2&. 61#(1$4&. t<
5*6-0.07- ; 2Rl→rσ 3 !"# l → r $%& 'B2+& )& T (Fbin,X ) × T (Fbin,X )* !"#
x0, . . . , xn +7&%,&(/+& )& -.'"./+&, .00.'.",,.%# ).%, l &# ?$" 0&$# C#'& +$ )& 2.$34&
5 )'!"#& ).%, l* @"%,"; +7DEFG Rl→rσ−compute &,# )<=%" 3!((& ,$"# >
! F l→rcopy(x)→ Dcopy(RWx0 (RWx1 (. . . RWxn (⊤(F ε
t (x),⊤X (t))) . . .)))
"! RWxi (x)→ RWxi (RWxi (x))
#! RWxi (x)→ NMAxi (x)
$! {NMAxi (a(x, y)) → a(NMAxi (x), NMAxi (y)) | a ∈ Fbin}
%! {NMAxi (⊕xj (x))→ ⊕xj ,(x) | xi &= xj}
! "#$%&'() *+ ,-.'/0). 1) 23435(&'6() 7895'&899):.
! NMAxi (⊥)→⊥
"! NMAxi (⊤(x, y)) → ⊤(NMAxi (x), NMAxi (y))
#! NMAxi (⊥⊤) →⊥⊤
$! RWxi (x) → RWxi,a(x) %&'( a ∈ Fbin
)*! RWxi,a(⊤(x, y)) → ⊤(Rxi,a(x), SWxixi,a(y)) %&'( a ∈ Fbin
))! {SWxx0,a
(⊤(x, y)) → ⊤(Wx,a(x), y)} ∪ {SWxxi,a
(⊤(x, y)) → ⊤(x, SWxxi−1,a
(y)) | i > 0 ∧ i ≤ n∧ x ∈
Var(t)}
)+! Rx,a(g(x, y)) → g(Rx,a(x), y) %&'( g ∈ Fbin
),! Rx,a(g(x, y)) → g(?x(x), Rx,a(y)) %&'( g ∈ Fbin
)-! Rx,a(⊕x(x)) → R⊕x,a (x) %&'( a ∈ Fbin
).! R⊕x,a (a(x, y)) → a(⊕x(x),⊕x(y))
) ! R⊕x,⊥(⊥) →⊥
)"! Wx,a(g(x, y)) → g(Wx,a(x), y) %&'( g ∈ Fbin
)#! Wx,a(g(x, y)) → g(?⊕x (x),Wx,a(y)) %&'( g ∈ Fbin
)$! Wx,a(⊕x(⊥)) → a(⊕x(⊥),⊕x(⊥))
+*! Wx,⊥(⊕x(⊥)) →⊥
+)! Dcopy(⊤(x, y)) → D′copy(⊤(y,D(x))
++! D′copy(⊤(x,⊥)) → x
+,! D(x) → D(D(x))
+-! D(⊥) →⊥
+.! D(x) → D′(x)
+ ! D′(g(x, y)) → g(D′(x), D′(y))
+"! D′(⊥) →⊥
+#! D′(g(x,⊥)) → x
+$! ?⊕x (g(x, y)) → g(?⊕x (x), ?⊕x(y)) %&'( g ∈ Fbin '/ x ∈ Var(t)
,*! ?⊕x (⊥) →⊥
!"# $%&'()!*# +, '-+, $,# './+,# $, '%0%&'(1"', 2'%#,*1%,# $3*# +3 $%4*(1(!* 56
73 './+, 86 ,#1 "*, './+, 2'(*&(23+, !" "*, &!92!#(1(!* $,# !2%'31(!*#6 :,11,
'./+, +3*&, +3 &!2(, +3 )3+,"' !;1,*", 23' &<3=", )3'(3;+, 32'.# 3)!(' 93'="%, =", +,
1,'9, #1!&>% $3*# +3 )3'(3;+, x6 ?*, &!2(, $" 1,'9, %1(=",1% ,#1 %13;+(, @ +3 '3&(*,
$, +A3';', $, )3'(3;+,#6
;<)0%:) => !"# t′ $% #%&'% a(b(a(⊥,⊥),⊥), c(d(⊥, b(⊥,⊥)), c(⊥,⊥))) %# (!"# t $%
#%&'% a(b(x0,⊥), c(d(⊥, x1), x2)).
)* +&%'",&% &,-$% %(# .!/0
Fcopy0t(x)→
Dcopy(RWx0(RWx1(RWx2(⊤(Fεt (x),⊤(⊕x0(⊥),⊤(⊕x1(⊥),⊤(⊕x2(⊥))))))))).
)* &1210&"#3&% +%3# 4#&% &1*$"(1% (3& $% #%&'% Fcopy(t′) %# +3"( $% #%&'%
Dcopy(RWx0(RWx1( RWx2(⊤(Fεt (t
′),⊤(⊕x0(⊥),⊤(⊕x1(⊥),⊤(⊕x2(⊥)))))))))5
6/ 3#"$"(*/# $%( &,-$%( .% Rtcheck .!//1%( .*/( $7%8%'+$% 9:; /!3( !<#%/!/( $%
#%&'% (3"=*/# >
Dcopy(RWx0(RWx1(RWx2( ⊤(t′′,⊤(⊕x0(⊥),⊤(⊕x1(⊥),⊤(⊕x2(⊥),⊥⊤))))))))
!? t′′ %(# $% #%&'% a(b(⊕x0(a(⊥,⊥)),⊥), c(d(⊥,⊕x1(b(⊥,⊥))),⊕x2(c(⊥,⊥))))5
!"! #$%&' ()*+,-./01,' 2(#$%&'3 45
! "#$%& '( )*+,-.& /0102 2304&!0 *!) 5& %! ,3*-& 5& %! 4!%&0" )63,7+& 5!2) %!
4!"-!8%& xi 932,6-322&( :;&,6-4&<&26= 5!2) %& <&-%%&0" ,!)= 02& !**%-,!6-32 5& RWxi
,3""&)*325 > %! ,3*-& 5102 )?<83%&= !-2)- > ,3*-&" &26-#"&<&26 %! 4!%&0" )63,7+&
5!2) xi= 230) !432) 8&)3-2 51!06!26 <3-2) 50 23<8"& 51!**%-,!6-32) /0& %& 23<8"&
5& *3)-6-32) 932,6-322&%%&) 5!2) %! 4!%&0"(
(6,+7), 58 !"#$%&' () !)*+) t′′ #) (,)-)+.() .*/0/#)&!1 '"* () '2+3%() 4%&0!$%&&)(
⊕x2 5 $( 2 6 !*%$' .%'$!$%&' 4%&0!$%&&)(()' 0%**)'.%&! 6"- '2+3%()' c5 ⊥ )! ⊥1 7$&'$
(6 *89() RWx2(x) → RWx2(RWx2(x)) #):*6 ;!*) 6..($<"/) 6" +%$&' #)"- 4%$' .%"*
"&) 0%.$) 0%+.(8!) #) (6 :6()"* '!%0=/) '"* ⊕x2 1
>"$'5 &%"' .%":%&' %3!)&$* () !)*+) t′′′ =
Dcopy(RWx0(RWx1(RWx2(RWx2(RWx2(⊤(t′′,⊤(⊕x0(⊥),⊤(⊕x1(⊥),⊤(⊕x2(⊥
),⊥⊤))))))))))1
! "#$%& @( &A*"-<& /0& %! ,3*-& 5& %! 4!"-!8%& xi &)6 6&"<-2+ 5#) /01-% 21? !
!0,02 <!"/0&0" ,3""&)*325!26 > xi 5!2) %1!"8"& 5& 4!"-!8%&)( &) "#$%&) @= B= C &6 D(
%!2,&26 02& "&,E&",E& 5!2) %1!"8"& 5& 4!"-!8%&) 50 <!"/0&0" ,3""&)*325!26 > ,E!/0&
4!"-!8%& xi( :2 9!-6= )1-% &A-)6& 02 6&% <!"/0&0"= *0-)/0& ,& ,!) &)6 *!) 6"!-6+ *!" %!
"#$%& B= !%3") %& )?<83%& 932,6-322&% NMAxi 2& *&06 *!) F6"& &;!,+ &6 *0-) !0,02
6&"<& )0,,&))&0" &2 T (Fbin) 5& 6&% 6&"<& &A-)6&( G!" ,32)+/0&26= 02 2304&!0 *!)
5& ,3*-& 5&4"!-6 F6"& %!2,+(
&) "#$%&) H > IJ 5+,"-4&26 5102& *!"6= ,3<<&26 02 )?<83%& &)6 %0 5!2) %&
*"&<-&" +%+<&26 5& %1!"8"& 5& 4!"-!8%&) &6 51!06"& *!"6= ,3<<&26 ,& )?<83%& &)6
)-<0%6!2+<&26 +,"-6 5!2) %1!"8"& 5& 4!"-!8%&) .2!%& &%%&K<F<&( G%0) *"+,-)+<&26=
%! "#$%& H( )*+,-.& /0& %! %&,60"& &6 %1+,"-60"& )-<0%6!2+&) )326 3"5322+&) *30" 02
)?<83%& a 5& Fbin( :2 9!-6= -% ? ! 02& 6&%%& "#$%& *!" )?<83%& 5!2) Fbin L⊥ ,3<*"-)M(
&) "#$%&) IN > IB &A*%-/0&26 ,3<<&26 230) 4+"-.32) /0& %& )?<83%& )0-4!26 >
%-"& 5!2) %& 6&"<& )63,7+ 5!2) %! 4!"-!8%& x &)6 4"!-<&26 ,&%%& )0**3)+& -26+$"+& 5!2)
%& )?<83%& 932,6-322&% Rx,a( G%0) *"+,-)+<&26= Rx,a &)6 "+,0")-4&<&26 !**&%+ 5!2) %&
63* 50 6&"<& 5& %1!"8"& 5& 4!"-!8%&( O36&P /0& %! "#$%& II( )*+,-.& /0& 230) ,E&",E32)
5!2) %! *!"6-& 5"3-6& 5102 )?<83%& )&0%&<&26 )1-% 21? ! !0,02 )?<83%& ⊕x 5!2) )!
*!"6-& $!0,E&( :2,3"& 02& 93-)= )0**3)32) /0& ,&66& "#$%& &)6 !**%-/0+& /0!25 -% ? !
&2,3"& %& )?<83%& ⊕x 5!2) %! *!"6-& $!0,E&( !-2)-= )0-432) %&) "#$%&) 'Q &6 @N 230)
43?32) /0& %& )?<83%& ?⊕x !**!"!R6"! &2,3"& 5!2) 630) %&) 6&"<&) )0,,&))&0") &6 *0-)
!0,02 6&"<& 5& T (Fbin) )&"! *3))-8%& 5& 5+"-4&"( ! %&,60"& 50 )?<83%& 932,6-322&%
)0-4!26 &)6 )*+,-.+& *!" %&) "#$%&) I' &6 I@( :;&,6-4&<&26= %&) )+<!26-/0&) 5& Rx,a
)326 /0& %& )?<83%& )0-4!26 > %-"& 53-6 F6"& a( S#) /0& %& <!"/0&0" 5& 4!"-!8%& x
&)6 <-) 5!2) %! "#$%& I'= %& *!) )0-4!26 &)6 5& 4+"-.&" /0& %& )?<83%& )0-4!26 &)6
&;&,6-4&<&26 a = &6 )- ,1&)6 ,& ,!)= %&) <!"/0&0") )0-4!26) )326 +6!8%-) 5-"&,6&<&26
)0" ,& )?<83%&( & *"3,&))0) 51+,"-60"& 5!2) %1!"8"& 5& 4!"-!8%&) &)6 *"3,E& 5& ,&%0-
! "#$%&'() *+ ,-.'/0). 1) 23435(&'6() 7895'&899):.
!"#$% "$& '(()(* (+), -)' . /0( -)' 1' 2+#-)')# '(% %#3)4! 5(6!"$7! 6+# 1+ #081' 9:;*
)< <3)4'+) %'#2' '(% "#!! c.=.d a(⊥,⊥)>
?+ 78)#' @>A $11)(%#' 1+ "36$' ' 1+ 4+1')# (%3"B!' +<( 1+ 4+#$+C1' x2> ?' 6#3"'(()(
'(% 1' 2D2' 63)# 1'( 4+#$+C1'( x1 '% x0>
E% 7<+1'2'<%* 0( -)' 1+ "36$' '( 4+1')#( (%3"B!'( +<( '( 4+#$+C1'( '(%
%'#2$<!'* 1' %'#2' +) %36 ' 1F+#C#' ' 4+#$+C1'( '(% 3C(310%'> G13#(* 1'( #081'( 9H
= AI ' 1+ !7<$%$3< J !"#$4'<% "322'<% !%#)$#' 1' %'#2' 3C(310%' '% "322'<%
#'%3)#<'# = 1F+#C#' ' 4+#$+C1'( 7<+1 $<(%+<"$!>
*+!+!+; 2/<:). 1=>7?2, 1) ').' 1) .6@.'&'6'&89 A Rl→rGS
Rl→rGS '(% )< EKLMN (6!"$7+<% 1+ 4!#$7"+%$3< F)<' ()C(%$%)%$3< C$'<&,3#2!'>
EO'"%$4'2'<%* ($ l <F'(% 6+( 1$<!+$#' +13#( )<' 4+#$+C1' x +66+#+P% ')Q ,3$( +)
23$<(> N$ %3)%'( 1'( 3"")##'<"'( ' x 6+#%+8'<% 1' 2D2' %'#2' "322' 4+1')# +13#(
1+ ()C(%$%)%$3< '(% C$'<&,3#2!'> (Rl→rcheck)
∗(F l→rcheck(t
′′)) = {t′′} '(% 1+ ()C(%$%)%$3<
C$'<&,3#2!'* ∅ ($<3<> Rl→rGS %#+$%' "R+-)' 4+#$+C1' +66+#+$((+<% 61)( F)<' ,3$(
"322' 2'<%$3<<! "$& '(()(>
B3C9&'&89 D ERl→rGS F !"# l → r $%& '()*& +& T (Fbin,X ) × T (Fbin,X ), !"#
x0, . . . , xn *&- ./'"/0*&- +& l +& )/$12& 3 +'!"#&, !"# Indexes *4&%-&50*& +& 1!$6&-
+4"%+&7&- +& ./'"/0*& 1!''&-6!%+/%# 3 */ 585& ./'"/0*& "%"#"/*&, !"# Rl→rGS *4 9:;<
'=-&'.= 3 .='">&' ?$& +&$7 ./'"/0*&- !%# *& 585& #&'5& -$' *4/'0'& +& ./'"/0*&- "%@
-#/%#/%=, A"%-"B Rl→rGS &-# 1!56!-= 6/' +&- '()*&- -$"./%#&- C
! F l→rcheck
(x) → CForin,jn (CForin−1,jn−1 (. . . CFori1,j1 (x) . . .)) "#$% Indexes ={(i1, j1), . . . , (in, jn)}
&! PutMarksi(⊤(x, y)) → ⊤(x, PutMarksi−1(y)) '()* i > 1
+! PutMarks0(⊤(x, y)) → ⊤(⊕=(x), y)
,! CFori,j(x) → Test=i,j (PutMarksi(PutMarksj(x)))
-! Test=i,j (x) → Test=i,j (RRi,j(x))
.! Test=i,j (x) →?x,p(x)
/! {RRi,j(x) → RRi,j,g(x)} "#$% g ∈ Fbin
0! {RRi,j,g(⊤(x, y)) → ⊤(x,RRi−1,j−1,g(y))} "#$% g ∈ Fbin $1 i > 0
2! {RR0,j,g(⊤(x, y)) → ⊤(Rg(x), Rj−1,g(y))} "#$% g ∈ Fbin $1 i > 0
3! {RRi,g(⊤(x, y)) → ⊤(x,Ri−1,g(y))} "#$% g ∈ Fbin $1 i > 0
! {RR0,g(⊤(x, y)) → ⊤(Rg(x), y)} "#$% g ∈ Fbin $1 i > 0
&! {Rg(g(x, y)) → g(Rg(x), y), Rg(g(x, y)) → g(?⊕= (x), Rg(y))} "#$% g ∈ Fbin
+! Rg(⊕=(x)) → Rg,⊕= (x)
,! Rg,⊕= (g(x, y)) → g(⊕=(x),⊕=(y))
-! R⊥,⊕= (⊥) →⊥
*+!+!+* 2/<:). 1=>7?2, 1) .6@.'&'6'&89 A Rl→rσ−apply
Rl→rσ−apply '(% )< EKLMN (6!"$7+<% 1F+661$"+%$3< ' 1+ ()C(%$%)%$3< C$'<&,3#2!'
#!()1%+<% ' t '% l ()# 1' %'#2' r> G$<($* (Rl→rσ−apply)
∗(F l→rrewrite(t
′′)) = {rσ}>
!"! #$%&' ()*+,-./01,' 2(#$%&'3 45
!"#$% &'( &)( &)( &*( +,( -*
⊕x0
a
RWx2
RWx2
RWx1
RWx0
Dcopy
RWx2
RWx2
RWx1
RWx0
Dcopy
RWx1
RWx0
Dcopy
RWx1
RWx0
Dcopy
RWx2RWx2
RWx2
RWx1
RWx0
Dcopy
RWx2RWx2
RWx2
RWx1
RWx0
Dcopy
c
⊕x2⊕x2
⊥ ⊥
RWx2
RWx1
RWx0
Dcopy
RWx2
RWx1
RWx0
Dcopy
⊕x2
c
⊥ ⊥
c
⊤
⊤
⊤
⊤⊕x1
⊕x0
⊥
⊥
⊥
⊥⊤
a
c
⊥⊤
⊤
⊤
⊤
⊤⊕x1
⊥
⊕x0
⊥
c
⊕x2⊕x2
⊥ ⊥
Rx2,⊥(a)
c
⊥⊤
⊤
⊤
⊤
⊤⊕x1
⊥
⊕x0
⊥
c
⊥
⊥
c
⊥ ⊕x2
⊥
a
c
⊥⊤
⊤
⊤
⊤
⊤⊕x1
⊥
⊕x0
⊥
c
⊥ ⊥
c
⊥ ⊥
Rx2,c(⊕x2)
Wx2,c(⊕x2)
Wx2,⊥(⊕x2)
a
⊥
⊥ ⊥ b
⊥ ⊥
⊕x1
⊥ d ⊕x2
c
⊥ ⊥
b c
RWx2RWx2RWx2
RWx2
RWx1
RWx0
Dcopy
⊤
⊤
⊤
⊤
⊕x2
⊕x1
⊕x0
⊥
⊥
⊥
⊥⊤
c
⊥⊤
⊤
⊤
⊤
⊤⊕x1
⊥
⊕x0
⊥
c
⊕x2
⊥ ⊥
c
⊕x2⊕x2
⊥ ⊥
a
c
⊥⊤
⊤
⊤
⊤
⊤⊕x1
⊥
⊕x0
⊥
c
⊥ ⊕x2
⊥
c
⊥ ⊕x2
⊥
a
c
⊥⊤
NMAx2(⊤)
⊤
⊤
⊤⊕x1
⊥
⊕x0
⊥
c
⊥ ⊥
c
⊥ ⊥
a
c
⊥⊤
⊤
⊤
⊤
⊤⊕x1
⊥
⊕x0
⊥
c
⊥ ⊥
c
⊥ ⊥
Rx2,⊥(a)
!"#$% .( +( '( /( )
Wx2,⊥(⊕x2 )
!"#$% .'( .)( .*( &,( +*
!"#$% &( -( 0( *( .,( -+
!"#$% &'( &)( &+( &*( +,( -*
!"#$% &( -( 0( *( .,( -)
!"# !" # $%&'( )( *+ ,+*(-. /0%123( )+4/ x2
! "#$%&'() !* +,-'./)- 0) 12324(&'5() 6784'&788)9-
:2;8&'&78 < =Rl→rσ−apply> !"# l → r $%& '()*& +& T (Fbin,X ) × T (Fbin,X ), -"%."/
Rl→rσ−apply 0!%#"&%# *&. '()*&. .$"12%#&. 3
! F l→rrewrite(x) → D⊤(RW rew
x0(RW rew
x1(. . . RW rew
xn(⊤(rσl→r, x)) . . .)))
"! RW rewxi
(x) → RW rewxi
(RW rewxi
(x))
#! RW rewxi
(x) → NMAxi (x)
$! RW rewxi
(x) → RW rewxi,a
(x) %&'( a ∈ Fbin
)! RW rewxi,a
(⊤(x, y)) → ⊤(Wxi,a(x), SRxi,a(MSxixi
(?⊕xiy))) %&'( a ∈ Fbin
*! {MSzx0
(⊤(x, y)) → ⊤(⊕z(x), y)}∪{MSzxi
(⊤(x, y)) → ⊤(x,MSzxi−1
(y)) | i > 0∧i ≤ n∧z ∈ Var(r)}
+! {SRx0,a(⊤(x, y)) → ⊤(Rx0,a(x), y)} ∪ {SRxi,a(⊤(x, y)) → ⊤(x, SRxi−1,a(y)) | i > 0 ∧ i ≤ n}
,! D⊤(⊤(x, y)) → D⊤(⊤(x,D(y))
-! D⊤(⊤(x,⊥)) → x
.! D′(⊤(x, y)) → ⊤(D′(x), D′(y))
! D′(⊤(x,⊥⊤)) → x
"! D′(⊤(x,⊥)) → x
!"# $%%!&# '()%*+"', %-'&#'./%' 0' ,12%'# 3!,"*4'# 0$&# %$ 056&*4*!& 78 9$
,12%' :8 #)53*6' +"' 4!"4'# %'# ;$,*$/%'# $))$,$*##$&4 0$&# r 0!*;'&4 <4,' ,'.)%$35'#
)$, %'",# ;$%'",# #4!3=5'# 0$&# %-$,/,' 0' ;$,*$/%'8 !4'> +"' %'# ;$,*$/%'# 0' r #!&4
,'.)%$35'# )$, 0'# .$,+"'",# '4 3' 4',.' '#4 54$/%* $" )!#*4*!& 4!).!#4 0' %-$,/,'
0' ;$,*$/%'#8 9'# ,12%'# ? '4 @ 053,*;'&4 +"' %$ 3!)*' 0-"&' ;$,*$/%' 0!&&5' 0!*4 <4,'
A$*4' B"#+"-C 3' +"-$"3"& .$,+"'", 0' 3'44' ;$,*$/%' $))$,$D4' 0$&# %-$,/,' 4!4$%8
E!", "&' ;$,*$/%' 0!&&5'F %$ ,12%' G %$&3' %$ 3!)*' )!", 3H$+"' #I./!%' 0' Fbin8 J"
3!&4,$*,' 0' %$ 056&*4*!& KF r )'"4 3!&4'&*, +"'%+"'# A!*# %$ .<.' ;$,*$/%'F '4 )"*#F %$
3!)*' 0' 3H$+"' ;$,*$/%' &!& %*&5$*,' 0!*4 <4,' 4,$*45' ),"0'..'&48 L& )$,4*3"%*',F
#* &!"# $;!&# %-*&4'&4*!& 0' 3!)*', ;$,*$/%' $),1# ;$,*$/%'F &!"# 0';!&# $%!,# &!"#
$##",', +"' "&' &!";'%%' 3!)*' &' )'"4 )$# 3!..'&3', $;$&4 %$ 6& 0' %$ ),5350'&4'8
L& A$*4F 3-'#4 A$3*%' 0' %' 4,$*4', )!", "&' ;$,*$/%' 0!&&5' x8 LM'34*;'.'&4F '&
;5,*6$&4 %$ ),5#'&3' 0" #I./!%' ⊕x 0$&# %-$,/,' 0' ;$,*$/%' ,5'% N%-$,/,' 0' ;$,*$/%'
#$&# %' ),'.*', 5%5.'&4OF !& )'"4 0*,' #* %$ %'34",' '#4 '& ,!"4'8 P1# +"-"&' 3!)*'
0-"&' ;$,*$/%' 0!&&5' '#4 4',.*&5'F #-*% I $ "&' &!";'%%' 3!)*' C A$*,' )!", %$ .<.'
;$,*$/%'F $%!,# "& .$,+"'", '#4 54$/%* 0$&# %-$,/,' 0' ;$,*$/%'# '& "4*%*#$&4 %'# ,12%'#
K '4 Q8 9' )$# #"*;$&4 '#4 0' ,'3!..'&3', %$ %'34",' )!", %$ 3!)*'8 R' ),!3'##"#
'#4 *.)%5.'&45 )$, %$ ,12%' 78 P1# +"' 3H$+"' ;$,*$/%' 0' r '#4 #"/#4*4"5' )$, %'",#
;$%'",# 3!,,'#)!&0$&4'#F %-$,/,' 0' ;$,*$/%'# '#4 054,"*4F '& "4*%*#$&4 0'# ,12%'# SF TF
:UF ::F :? 3!..' 0'# ,12%'# ?@F ?GF ?KF ?QF ?7F ?S 0' %$ 056&*4*!& K8 V*&$%'.'&4F
%-*$&3' 0' r '#4 !/4'&"'8
!*?*?* @($8-A7(/$'&78 0) @1+- B)(- C6@1+-
L& 3!+"'&3'F '& "4*%*#$&4 %'# G '&#'./%'# 0' ,12%'# 056&*# 3*W0'##"# $;'3 "&
'&#'./%' 0' ,12%'# 0-LVXYZ 0' 3*,3"%$4*!& Rl→rvisit 4'% +"'
Rl→rvisit = {Fl→r(a(x, y))→ a(Fl→r(x), y), Fl→r(a(x, y))→ a(x, Fl→r(y))}
'4 "& '&#'./%' 0' ,12%'# 0-LVXYZ 0' ,'%$&3' Rl→rTV 4'% +"'
Rl→rTV = {Fl→r(x)→ F l→r
rewrite(Fl→rcheck(F
l→rcopy( F
εl→r(x))))}F
&!"# #!..'# 3$)$/%'# 0' #*."%', %$ ,5W53,*4",' 0-"& XYZ8
!"! #$%&' ()*+,-./01,' 2(#$%&'3 44
!"# $%&'$("!)# ("% *%# $+,*%# -./0123 -% 45$4"*'65!) Rl→rvisit !"# $%&'%()
(% ( *% +,&+"-%& -% ! )(%&', .- Fl→r / (!"# -%# % *&!,(# *" (%&'% *! 0 1". *
-%# &23-%# *456789 *% &%-. +% Rl→rTV %#( " % +!'$!#,(,! *% : ! )(%&', .";
F l→rrewrite, F
l→rcheck, F
l→rcopy %( F ε
l→r <+!&%##$! *. ( .=%+ : % #%'>-%# *% &23-%# *0? ,# +,)
*%##"#@A
5 +! #01"% +%B !"# .=! # -% &0#"-(.( $!"& " $.# *% &0)0+&,("&% +!''% #",( C
!"#"$%&%"' ( !"# R $% &' ($) T (Fbin,X ) × T (Fbin,X )* α ∈ T (Fbin)* β ∈
T (Fbin) +%, l→ r ∈ R- ./ 01"(#0 $% 23&' Rλe #0/ 4$0
α→{l→r} β ⇔ Fl→r(α)→∗Rλe
β.
!)*+) , - !)*+) $*..%'.&) -/"%! .01 23'$ 453'')6) 788 50 6!"%# 7/8 0(# ,0
7!%(#)$")0 $% 23&' Rλe 4$" "96/890%#0 /0 6)!70(($( ,0 )8:87)"#$)0-
50 23&' Rλe 0(# 7!%(#)$"# 7!990 /0 ($";+%# <
Rλe = Rl→rvisit ∪R
l→rTV ∪Rl→r
check ∪Rl→rσ ∪Rl→r
GS ∪Rl→rσ−apply
= α →{l→r} β ⇒ Fl→r(α) →∗Rλe
β < 2>07#";090%#* F l→rrewrite(t) →
∗Rl→rσ−apply
rσ ∈
T (Fbin)- 2% 7!%(84$0%70* α[F l→rrewrite(t)]p →∗
Rl→rσ−apply
α[rσ]p ∈ T (Fbin) 0#
6$"(4$0 α[rσ]p = β* β ∈ R∗λe({Fl→r(α)}).
= α →l→r β ⇐ Fl→r(α) →∗Rλe
β < 5+ 6)0$;0 6!$) 70 7+( 0(# 6)!7?0 +$ 7+(
("9"/+")0 #)+"#8 ,+%( /+ 6)0$;0 ,0 /+ 6)!6!("#"!% @ ,+%( /0 (0%( 4$0 /0( 6+( ,0
)8:87)"#$)0 (!%# !),!%%8( 0# ,860%,+%#( ,$ #0)90 "%"#"+/ c.A.d Fl→r(α)-
B"%("* 6!$) )8($98* "/ 01"(#0 $%0 6!("#"!% ,0 α 0# $%0 ($C(#"#$#"!% σ : X →
T (Fbin) #0//0( 4$0 α|p = lσ* β = α[rσ]p 0# Fl→r(α)→∗Rλe
β-
D!$( 6!$;!%( ,8,$")0 4$0 α→l→r β ⇒ Fl→r(α)→∗Rλe
β 4$" 7!%7/$# /+ 6)0$;0-
5 30 0&.-,#. ( -% &0#"-(.( +,)*%##"#B !"# !>(% ! # -% +.-+"- *% -4% #%'>-% *%#
0(.(# .++%##,>-%#B 0(. ( 03.-%'% ( -% &0#"-(.( *% -. (&. #D!&'.(,! <+@ *% -. ?3"&%
:AEA
9:;"!<=) ( !"# R $% &' ($) T (Fbin,X )× T (Fbin,X ) 0# E ⊆ T (Fbin)- B"%("*
"/ 01"(#0 $% 23&' Rλe 0# $% (E9C!/0 G ∈ FNT #0/( 4$0 <
R∗(E) = R⋆λe(E
′),
!F E′ = {G(α)|α ∈ E}.
!)*+) > G#+%# ,!%%8 /H23&' Rλe =⋃
l→r∈R(Rl→rvisit ∪ R
l→rTV ∪ Rl→r
check ∪ Rl→rσ ∪
Rl→rGS ∪ Rl→r
σ−apply) ∪ RFP +;07 RFP = {G(a(x, y)) → G(Fl→r(a(x, y))) | l → r ∈
R ∧ a ∈ Fbin} ∪ {G(a(x, y)) → a(x, y) | a ∈ Fbin}* /+ 6)0$;0 0(# #)I( 6)!7?0 A 70//0
,$ #?8!)I90 @-
F. # +%((% #%+(,! B !"# .=! # $&!"=0 1"% #!,( " 789B ,- %;,#(% " 56789
+!&&%#$! *. (A 5 #",(%B / $.&(,& *%# 56789#B !"# $!"=! # -%# +! =%&(,& =%&# *%#
789# +!''% *. # -. #%+(,! :AEA
! "#$%&'() *+ ,-.'/0). 1) 23435(&'6() 7895'&899):.
!"!# $%&'()*%+&,-*' ./ 0$12( 3/%( 40$12(
!"#$%& $' ()$*%( +! ,-(.$! /( 0123* 4(5* 0123* (! 6#**#!% 6#5 '(* 78,-(*
9:; (% 9/; /( '# <=+5( >?@? A+%5(.(!% /$%& B(6(!/#!%& '# %5#!*"C5.#%$C! /$5(,%( /(
'# <=+5( >?@ 9:; /( 0123 4(5* 123 !D(*% 6#* %C+EC+5* "#$*#:'(& !C+* /(4C!* #'C5*
65C,F/(5 /( .#!$85( $!/$5(,%(?
BC!*$/F5C!* +! ()(.6'( G
;<)0%:) => !"!#$#% & '(!)!*+'! ,- ."!/ '! 01 2 OnlyOneRequest 3(401 2
/$55!%+$#6.#7 !%7 '! %89".#7 :
Rλe =
OnlyOne(t(x, y))→ t(OnlyOne(R(x)), I(y))
OnlyOne(t(x, y))→ t(I(x), OnlyOne(R(y)))
OnlyOne(r(x, y))→ r(OnlyOne(R(x)), I(y))
OnlyOne(r(x, y))→ r(I(x), OnlyOne(R(y)))
OnlyOne(r(⊥,⊥))→ r(⊥,⊥)
R(r(⊥,⊥))→ r(⊥,⊥)
I(i(⊥,⊥))→ i(⊥,⊥)
.
HC+* 65C6C*C!* /#!* ,(%%( *(,%$C! +!( %5#!*"C5.#%$C! /$5(,%( /+ 0123 4(5*
+! (!*(.:'( /( 58='(* /( 0123* I+$ *C!% *+J*#..(!% ()65(**$4(* 6C+5 '# *6FK
,$<,#%$C! /+ .F,#!$*.( /( 5FKF,5$%+5(? 1C+% /D#:C5/& !C+* .C!%5C!* I+D+! 6#* /(
5FKF,5$%+5( +%$'$*#!% Rλ 9c.L.d +!( 58='( 0123 /( '# "C5.( H (t) → α; 6(+% M%5(
*$.+'F 6#5 +! 0123 Rλe ?
@? /( %5C+4(5 +!( 6C*$%$C! 6C+5 '# 5FKF,5$%+5(&
N? /( 4F5$<(5 *$ H (t) ,C!,C5/( #4(, '( *C+*K%(5.( ,C+5#!% (! ,C!*$/F5#!% H
,C..( +! *O.:C'( *6F,$#' (! Fbin& 6+$*
P? /( ,#',+'(5 '# *+:*%$%+%$C! 5F*+'%#!% σ& (%
>? /( 5(.6'#,(5 '( *C+*K%(5.( 6#5 ασ?
Q# %5#!*"C5.#%$C! 9/; /( '# <=+5( >?@ (*% /C!!F( 6#5 '# 65C6C*$%$C! *+$4#!%(?
?(8%8.&'&89 @ 2$97 Rλ 8# 01 2 α ∈ T (Fbin); <' !)9%7! 8# 401 2 Rλe 7!' =8!
{α} ∪⋃
H∈FNT ,β∈Reachables
(R⋆λ({H(β)})) = {α} ∪
⋃
H∈FNT ,β∈Reachables
(Rλe⋆({H(β)})).
?()6A) ! >7.#7 6$##? '(401 2 Rλe @ '. +5!8"! !%7 75A% +5$/B! & /!''! 6! '. +5$+$C
%979$# D;
3! +$9#7 /'? !%7 6! /$#%75895! 8# 401 2 Rλe =89 9*+'?*!#7! '! +5$/!%%8% 6!
5?C?/59785! !# /$#%96?5.#7 H /$**! 8# %E*F$'! %+?/9.' !# Fbin !7 !# 9G#$5.#7 '!%
5AG'!% 6! /95/8'.79$#;
3! 401 2 Rλe !%7 /$#%75897 /$**! '! %89".#7 :
Rλe = RH(t)→α
TV ∪RH(t)→α
check ∪RH(t)→ασ ∪R
H(t)→α
GS ∪RH(t)→α
σ−apply
!"! #$%&'( ) *+,-./ 0&#$%&'(1 23
!" #$%&'( ) *+,-./ 0&#$%&'(1
!"# $%%!&# '()#*&+*( ,$&# -*++* #*-+.!& "&* #!"# -%$##* ,* #/#+01*# 2!&-+.!&&*%#
)%)1*&+$.(*#3 %*# )%)1*&+$.(*# 4 ,(!.+* 567896:#;< =>"&* '$(+3 .%# '()#*(?*&+ )@$%*A
1*&+ %$ '".##$&-* ,>*B'(*##.!& ,*# #/#+01*# 2!&-+.!&&*%# *+ ,>$"+(* '$(+ .%# #!&+
-$'$C%*# ,* 1.*"B $''%.D"*( ,*# +*-E&.D"*# ,>$--)%)($+.!& ,*# -$%-"%#< F% *#+ '!##.C%*
,* +($!(1*( ,>"& 7896: ?*(# ,*# 67896:#<
!"!# $%&'(&)*+
G*++* #*-+.!& '()#*&+* "&* #!"# -%$##* ,* #/#+01*# 2!&-+.!&&*%# )%)1*&+$.(*#3
%*# )%)1*&+$.(*# 4 ,(!.+* 567896:#;3 '()#*(?$&+ %$ '".##$&-* ,>*B'(*##.!& ,*# #/#A
+01*# 2!&-+.!&&*%#< =>"&* '$(+3 %>.1'%)1*&+$+.!& ,>"& 1!+*"( ,* ()A)-(.+"(* '!"(
%*# 67896:# *#+ '%"# #.1'%* D"* %*# $"+(*# 896:#3 &!+$11*&+ %$ '(!-),"(* ,*
-!&-!(,$&-* ,*# +*(1*#< =>$"+(* '$(+3 %>"+.%.#$+.!& H*B.C%* ,*# &!&A+*(1.&$"B *&
67896:# '*"+ $"@1*&+*( %>*I-$-.+) ,* %$ +*-E&.D"* ,* -$-E* !" #$$ *+ %>$%@!A
(.+E1* ,* #$+"($+.!&<
7+$&+ ,!&&) "& 7896: Rλe 3 &!"# $%%!&# %* +($.+*( '$( %>$&$%/#* ,* -$#<
J< H(⊥) → β $?*- β ∈ T (Fbin ∪ FNT ) K G* +/'* *#+ -!&#.,)()* -!11* "&
%&'(%) *+,- . K Rλe→A<
L< H(a(x,⊥))→ x $?*- x ∈ X 3 H ∈ FNT $&, a ∈ Fbin K G* +/'* *#+ -!&#.,)()*
-!11* "& %&'(%) *+,- # K Rλe→B<
M< H(a(x, y))→ α $?*- a ∈ Fbin3 H ∈ FNT 3 x, y ∈ X 3 α ∈ T (Fbin ∪ FNT ,X ) *+
Var(α) = {x, y} K G* +/'* &>*#+ '$# *&-!(* /!/0-1*"23- $" -N+) ,(!.+ -!11*
%*# $"+(*#< O$( -!&#)D"*&+3 &!"# +($.+!&# ,*# +/'*# ,* H(a(x, y)) → α '!"(
#.1'%.P*( %* -N+) ,(!.+ ,* -* +/'* ,* (0@%*<
!"# $%%!&# ,.#+.&@"*( +(!.# +/'*# ," -N+) ,(!.+ ,* %$ (0@%* H(a(x, y))→ α $?*-
a ∈ Fbin3 H ∈ FNT 3 x, y ∈ X 3 α ∈ T (Fbin ∪ FNT ,X ) *+ Var(α) = {x, y}< Q*# ,*"B
'(*1.*(# 67896: +/'* F *+ +/'* FF D". -!&#*(?*&+ %* #/1C!%* a $'(0# $?!.( $''%.D")
%* &!&A+*(1.&$% H #!&+ $''*%)# 4516-37/ !" 8530-< R$%@() %* +/'* +(0# #.1'%* D".
(*1'%.+ C.*& %*# -!&,.+.!&# '!"( %*# +*-E&.D"*# ,>$--)%)($+.!& C$#)*# #"( %>$%@!(.+E1*
,* #$+"($+.!&3 %* 67896: +/'* F $''$($S+ +(0# #!"?*&+3 &!+$11*&+ ,$&# %$ -.(-"%$A
+.!&< Q* 67896: +/'* FF *#+ )@$%*1*&+ .1'!(+$&+* -$( .% &!"# '*(1*+ ,* -!1'!#*(
"&* #".+* ,>$"+$&+ ,>!')($+.!&# D"* &!"# ?!"%!&#< Q* +(!.#.01* 67896: +/'* FFF
D". &!"# '*(1*++*&+ ,* -E$&@*( %* %*++(* -!"($&+* a3 ,* '*(1"+*( ,*"B #!"#A+*(1*#
5-!&-!(,$&+ $?*- x *+ y;3 *+ ,>$T!"+*( "&* %*++(* 4 %>*B+)(.*"( ," +*(1* -!"($&+ *#+
$''*%) 151 9516-37/ !" 8530- <
9!"+ ,>$C!(,3 &!"# -!11*&U!&# '$( %* 67896: +/'* F 5$''*%) 4516-37/ !"
8530- :; %* +/'* %* '%"# #.1'%* ,*# (0@%*# ,* +/'* H(a(x, y)) → α< R$%@() %* +/'*
+(0# #.1'%* D". (*1'%.+ C.*& %*# -!&,.+.!&# '!"( %*# +*-E&.D"*# ,>$--)%)($+.!& C$#)*#
#"( %>$%@!(.+E1* ,* #$+"($+.!&3 *%%*# $''$($.##*&+ +(0# #!"?*&+3 &!+$11*&+ ,$&# %$
-.(-"%$+.!&<
! "#$%&'() *+ ,-.'/0). 1) 23435(&'6() 7895'&899):.
;3<9&'&89 ! !"# $ %&'( ) *
Rλe (+% ,-% !"# $ %&'( ) +- %./%(+ 0(+ 1230(+ 4&45% 67%8 34/69( H (a(x, y)) +.5%
+./+ :.1;(
< H (a(x, y))→ a (x, y)
< H (a(x, y))→ a(G (x), y)
< H (a(x, y))→ a(x,D (y))
4=(6 H,G,D ∈ FNT , a ∈ Fbin*
>./+ ,85.%.5+ '41 Rλe→I *
!"#$%&!&!'() '*"+ ,-.'$++*'+ /! 012304 (56! 77) 8"$ !+( !'9*:! ;66!/- ?.5@
+(1=8 04 :.1;() !( 8"$ '*"+ 6!:&!( ,<!#-9"(!: "'! 9*&6*+$($*' ,! ,!"# ;"(:!+ *6-:;=
($*'+> ?<!+( "'! :%@/! (:%+ $&6*:(;'(! 9;: !//! '*"+ 6!:&!( ,! 9*&6*+!: "'! +"$(!
,<;"(;'( ,<*6-:;($*'+ 8"! '*"+ A*"/*'+>
;3<9&'&89 = !"# $ %&'( ))*
Rλe (+% ,-% !"# $ %&'( )) +- %./%(+ 0(+ 1230(+ 4&45% 67%8 34/69( H (a(x, y)) +.5%
+./+ :.1;(
< H (a(x, y))→ G (D (a(x, y)))
4=(6 H,G,D ∈ FNT , a ∈ Fbin*
>./+ ,85.%.5+ '41 Rλe→II *
1' "($/$+;'( +!"/!&!'( ,!+ :%@/!+ ,! (56! 7 !( 77) '*"+ 6*"A*'+ +$&"/!: /; :%@/!
H (a(x, y))→ a(G (x), D (y)) 9*&&! /!+ +"$A;'(!+ B
C H (a(x, y))→ G′ (D′ (a(x, y)))
C G′ (a(x, y))→ a(G (x), y)
C D′ (a(x, y))→ a(x,D (y))>
D*"+ ;//*'+ ,-.'$: 9$=,!++*"+ /!+ :%@/!+ 6:$&$($A!+ 012304 (56! 777 8"$ '*"+
6!:&!((!'( ,! 9E;'@!: /; /!((:! 9*":;'(!) ,! 6!:&"(!: ,!"# +*"+=(!:&!+) !( ,<;F*"(!:
"'! /!((:! G /<!#(-:$!": ," (!:&! 9*":;'(>
;3<9&'&89 >? !"# $ %&'( )))*
Rλe (+% ,-% !"# $ %&'( ))) +A-0 (B-+%( /5( 1230( %(00( C/(
< H (a(x, y))→ b(x, y)
< H (a(x, y))→ a(y, x)
< H (a(x, y))→ a(⊥, a(x, y))
4=(6 H ∈ FNT et a, b ∈ Fbin*
>./+ ,85.%.5+ '41 Rλe→III *
,%35&<5$'&89 16 @AB )9 2C7@2,
?!((! +!9($*' &*'(:! 9*&&!'( /! 3HI 6!"( J(:! K*:&;/$+- !' 012304+ +*"+
K*:&! ,<"' 6:*L/%&! ,! 9;/9"/ ,<;99!++$L$/$(->
M!+ :%@/!+ ,! :-=-9:$(":!NO>PQR !( NO>PPR ,*''-!+ !' +!9($*' O>S 3HI !' 12304
+*'( ,-FG -/-&!'(;$:!+ G ,:*$(!>
!"! #$%&'( ) *+,-./ 0&#$%&'(1 23
!"# $%&# '(# !&$)(# *!# "%" +'+,("$!-)(#. "%&# *)+%"# /0!1%)/ &"( '-#$( /(#
"%"2$(),-"!&3 FNT "+*(##!-)(#4 5&-# *6!7&( H ∈ FNT 8! (9(*$&() &"( %:+)!$-%"
+'+,("$!-)(4 5!) (3(,:'(. '(# );<'(# /( )+2+*)-$&)( =>4?@ ($ =>4A@ /%""+(# /!"# '!
#(*$-%" >4B #%"$ $)!"#C%),+(# *%,,( '(# );<'(# /( DEFGDH Rreqλe →∪
-"/-7&+(# *-2
/(##%
H(i(x, y)) → H4.8(H′4.8(i(x, y))) !"#!$
H4.8(i(x, y)) → r(x, y) !"#%$
H′4.8(i(x, y)) → i(x,R(y)) !"#&$
H(i(x, y)) → H4.9(H′4.9(i(x, y))) !"#'$
H4.9(i(x, y)) → r(x, y) !"#($
H′4.9(i(x, y)) → i(R(x), y) !"#)$
H-,-'!-)(,("$. "%&# !8%"# '(# );<'(# /( *-)*&'!$-%" Rcirλe →∪ = Rcir
λe 4 I(# );<'(#
-"/-7&+(# *-2/(##%&# )(,:'-##("$ '! #:+*-J*!$-%" /& GK5 :!) &" DEFGDH4
H(t(x, y)) → H4.20(H′
4.20(t(x, y))) !"!*$
H4.20(t(x, y)) → b(x, y) !"!+$
H′4.20(t(x, y)) → t(x,RT (y)) !"!,$
H(t(x, y)) → H4.21(H′
4.21(t(x, y))) !"!#$
H4.21(t(x, y)) → b(x, y) !"!!$
H′4.21(t(x, y)) → t(RT (x), y) !"!%$
RT (r(x, y)) → t(x, y) !"!&$
H(b(x, y)) → H4.23(H′
4.23(b(x, y))) !"!'$
H4.23(b(x, y)) → t(x, y) !"!($
H′4.23(b(x, y)) → b(x, TI(y)) !"!)$
!"%*$
H(b(x, y)) → H4.24(H′
4.24(b(x, y))) !"%+$
H4.24(b(x, y)) → t(x, y) !"%,$
H′4.24(b(x, y)) → b(TI(x), y) !"%#$
H(t(x, y)) → H4.25(H′
4.25(t(x, y))) !"%!$
H(t(x, y)) → H′′′4.25(H′′
4.25(t(x, y))) !"%%$
H′′′4.25(t(x, y)) → i(x, y) !"%&$
H′4.25(t(x, y)) → t(I(x), y) !"%'$
H′′4.25(t(x, y)) → t(x, I(y)) !"%($
H(t(x, y)) → H4.26(H′
4.26(t(x, y))) !"%)$
H(t(x, y)) → H′′′4.26(H′′
4.26(t(x, y))) !"&*$
H′′′4.26(t(x, y)) → r(x, y) !"&+$
H′4.26(t(x, y)) → t(R(x), y) !"&,$
H′′4.26(t(x, y)) → t(x, I(y)) !"&#$
!"&!$
! "#$%&'() *+ ,-.'/0). 1) 23435(&'6() 7895'&899):.
H(t(x, y)) → H4.27(H′
4.27(t(x, y))) !"#$%
H(t(x, y)) → H′′′4.27(H′′
4.27(t(x, y))) !"##%
H′′′4.27(t(x, y)) → r(x, y) !"#&%
H′4.27(t(x, y)) → t(I(x), y) !"#'%
H′′4.27(t(x, y)) → t(x,R(y)) !"#(%
I(i(x, y)) → i(x, y) !"&)%
I(⊥) → ⊥ !"&*%
Arbiter(t(x, y)) → t(x, y) !"&+%
Arbiter(t(x, y)) → Arbiter(H(t(x, y))) !"&,%
Arbiter(b(x, y)) → b(x, y) !"&!%
Arbiter(b(x, y)) → Arbiter(H(b(x, y))) !"&$%
!"# $!%!$# &'( RTAPλe →∪ )* +,-.+/ 0!1&)*% 234$5 *$ 0*%%* #*0%5!$6 !"#
&!"7!$# !8%*$5( )9*$#*18)* 2* %*(1*# '00*##58)*# : &'(%5( 2" %*(1* ! "# $%#& *$
0')0")'$% (Rreqλe →∪
)⋆({Arbiter(u)})6 ;* &)"#< $!"# &!"7!$# 3=')*1*$% 73(54*( >"*
)* %*(1* ! "# '&()&* $+, v *#% '%%*5=$'8)* : &'(%5( 2" %*(1* ! "# $%#& u &"5# >"*
u→⋆Rreq
λe →∪
v6
!"!# $%&'()*%+&,-*' ./ 01$23( 4/%( 201$23(
?0%"*))*1*$%< 2*# +,-.+/#< &(3#*$%3# &'( )*# 234$5%5!$# @< A *% BC #!$% #"DE
5#'11*$% *F&(*##57*# &!"( )' #&30540'%5!$ 2" 130'$5#1* 2* (3E30(5%"(* &!"( "$*
(G=)* ,-.+/ 'H'$% 0I%3 ='"0J* H (a(x, y))→ α ∈ Rλe K
B6 2* %(!"7*( "$* &!#5%5!$ &!"( )' (3E30(5%"(*<
L6 2* 73(54*( #5 H (a(x, y)) 0!$0!(2* '7*0 )* #!"#E%*(1* 0!"('$%< &"5#
M6 2* 0')0")*( )' #"8#%5%"%5!$ (3#")%'$% σ< *%
N6 2* (*1&)'0*( )* #!"#E%*(1* &'( ασ '7*0 α O;*# #"8#%5%"%5!$# 2*# 7'(5'8)*#
&*"7*$% P%(* 1")%5&)53*#Q6
,$ 0!$#3>"*$0*< $!"# '7!$# )* (3#")%'% )5#%3 05E2*##!"#6 !"# (*1'(>"!$# >"*
)*# #H#%G1*# 2* (3E30(5%"(* &(515%57*# )5#%3# 05E2*##"# #!$% )5$3'5(*# 2(!5%*# *% #!$%
0'&'8)*# 2* #51")*( $951&!(%* >"*) #H#%G1* ,-.+/ )5$3'5(* 2(!5%* !" 1P1* $!$
)5$3'5(* 2(!5%*6
R' %('$#S!(1'%5!$ O*Q 2* )' 4="(* N6B *#% 2!$$3* &'( )' &(!&!#5%5!$ #"57'$%*6
;(8%8.&'&89 * -!$ Rλe )+ ./01- & α ∈ T (Fbin)2 3# &4$* & %&* 1./01-* Rλe→I 5
Rλe→II 5 Rλe→III 5 Rλe→A & Rλe→B 6&+ 7'&8 Rλe→∪9 &#* ()&
{α}∪⋃
H∈FNT ,β∈Reachables
(R⋆λe({H(β)})) = {α}∪
⋃
H∈FNT ,β∈Reachables
(Rλe⋆→∪({H(β)})).
! ! "#$%&'()#$ *&( (+(,-.&( /#$0,)#$$&1( %&'( 234( 56
7'&8%& 9 :7'&8%& (800)$0,& :;#)' 0/! *<$( 1=<$$&>& ?@@ !"#! $%##& '()
*+,-*.) Rλe→I / Rλe→II / Rλe→III / Rλe→A (! Rλe→B/ '" 01(23( ()! !14) 01%56(
7 5(''( $( '" 01%0%)8!8%# 9:
;( 0%8#! 5'& ()! $( 5%#)!1281( 2# *+,-*. <28 8=0'&=(#!( '( 01%5())2) $( 1&>
&518!21(:
;( *+,-*. ()! 5%#)!128! 5%==( '( )283"#! ?
Rλe→∪ = RH (a(x,y))→α
TV ∪RH (a(x,y))→α
check ∪ RH (a(x,y))→ασ ∪R
H (a(x,y))→α
σ−apply
!"# $%&'$!(# "( )*)+,-) ,!"% ./-&'%)% -& ,%!,!#'$'!( ,%./.0)($)1
A>&.B1& 6C +!"#! $%##&( 2#( 14@'( +,-*. $( '" A%1=( H(a(x, y)) −→
b(a(x,⊥), a(⊥, y)): +''( ()! )8=2'&( 0"1 '() 14@'() *+,-*.) 5%==( )28! ?
B H (a(x, y))→ H”(H ′(a(x, y)))
B H ′ (a(x, y))→ b(x, y)
B H” (b(x, y))→ b(G (x), D (y))
C%21 D(x)→ a(⊥, x)/ #%2) "3%#) 2# (#)(=D'( $( 14@'() )283"#!() ?
B D (⊥)→ a(⊥,⊥)
B D(c(x, y))→ A(D′(c(x, y))) ∀c ∈ FbinB A (c(x, y))→ a(x, y)
B D′ (c(x, y))→ c(⊥, c(x, y))
B D′ (⊥)→ a(⊥,⊥)
C%21 G(x)→ a(x,⊥)/ #%2) A"8)%#) 2# "00(' $( E ?
B G (⊥)→ a(⊥,⊥)
B G(c(x, y))→M(D(c(x, y))) ∀c ∈ FbinB M (a(x, y))→ a(y, x)
2($"'$'3)+)($4 (!"# !5$)(!(# -) %.#"-$&$ ,-"# 6.(.%&-4 .$&($ -& /!+5'(&'#!( 0)#
$%&(#7!%+&$'!( 8,%.#)($.)# ,&% -)# 9:/;)# 8/< )$ 8)< 0) -& =6"%) >1? ! "#$%&'($ )*+
,&%%-.% / 0-.# (-.( 12345! '-.% "-.6-'% )-'%(#.*#& ,&% 412345% $7.*689&'(% "-.#
9:8))&%%*;*9*($ %.# T (Fbin)< 1' #$%.=$! (-.% 345 "&.( >(#& %*=.9$ "8# 9&% 412345%<
!"#$%&' ( !"# R $% &' ($) T (Fbin,X ) × T (Fbin,X ) *# α ∈ T (Fbin)+
,"%("- ". */"(#* $% '01&' Rλe→∪ #*. 2$*- 3!%("45)!%( Reachables = {α} ∪⋃
H∈FNT ,β∈Reachables(Rλe
⋆→∪({H(β)}))- R
⋆(α) = Reachables+
)$'*+' , *.!% .*( 6)!6!("#"!%( 7 *# .8 #95!):;* <+
! "#$%&'()#$ *&( (+(,-.&( /#$0,)#$$&1( %&'( 234(
?8'% )&((& %&)(*-'! '-(#& 8=;*(*-' &%( ,& )@&#)@&# .'& )-'6&#%*-' *',*#&)(& &'
)-'%*,$#8'( 9&% %A%(B=&% C-')(*-''&9% )-==& .' (A"& %"$)*89 ,&% %A%(B=&% )98%%*7.&%<
199& &%( #&"#$%&'($& "8# 9&% DB)@&% "-*'(*99$&% ,& 98 EF.#& G<H<
! "#$%&'() *+ ,-.'/0). 1) 23435(&'6() 7895'&899):.
!"# Rλ $% &'( )$* T (Fbin∪FNT )+ ,!$) -!%)#*$")!%) $% %!$./0$ '( Rconv+
'!$# 1203!*14 52/%)/635/ 1/ )763!5/) T (Fbin) 8 90*#"* 1/ FNT /# Fbin /)# 1:%!#:
90* Fconv/)# 1:;%" -"<1/))!$) =
> ?!$* -@0A$/ )763!5/ 3"%0"*/ a ∈ Fbin B7 -!69*") ⊥C4 0D!$#/* a ∈ Fconv
> ?!$* -@0A$/ )763!5/ $%0"*/ H ∈ Fbin4 0D!$#/* h ∈ Fconv
#/5 A$/ @ /)# 3"%0"*/
/# h $∈ FbinE%)$"#/4Rconv /)# -!%)#*$"# 90* 1/) *FG5/) 1/Rλ /% */6950H0%# #!$) 5/) )763!5/)
$%0"*/) H(...) 90* 1/) )763!5/) 3"%0"*/) h(...,⊥) !I ⊥ .0 */695"* 5/ )!$)<#/*6/ 1*!"#
1/ h %!% $#"5"):+
;<)0%:) != !"#$%&'!"# () *+,- %) (.)/)01() 234 56$#76) OnlyOne ∈ FNT )#8
8'9%6$8 : onlyone ∈ Fconv; 9$"#$ () +,- Rconv !<8)"6 )#8 =!00) () #6$>9"8 ?
Rconv =
onlyone(t(r(x, y), i(z, t)),⊥)→ t(onlyone(r(x, y),⊥), i(z, t))
onlyone(t(i(x, y), r(z, t)),⊥)→ t(i(x, y), onlyone(r(z, t),⊥))
onlyone(r(r(x, y), i(z, t)),⊥)→ r(onlyone(r(x, y),⊥), i(z, t))
onlyone(r(i(x, y), r(z, t)),⊥)→ r(i(x, y), onlyone(r(z, t),⊥))
onlyone(r(⊥,⊥),⊥)→ r(⊥,⊥)
.
J05@/$*/$)/6/%#4 5/ 9*!35F6/ 1:)!*60") ."/%# 1$ 6!1/ 1/ K!%-#"!%%/6/%#+
L0%) 5/ -0) 1/ 52/M/695/ NO4 5/ 6!1/ 1/ K!%-#"!%%/6/%# 1/ -/ '( /)# )"6"50"*/
0$ 6!1/ 1$ &'( + P/9/%10%#4 10%) 5/ -0) 1/ 52/M/695/ QN4 %!$) 0.!%) !3#/%$ 5/)
*:)$5#0#) 1"R:*/%#)+
E% *:)$6:4 %!$) 0.!%) 9*!$.: A$/ 9!$* #!$# '( 4 "5 /M")#/ $% &'( -!**/)9!%<
10%#+ ?!$* 50 *:-"9*!A$/4 )" $% &'( )0#")K0"# 50 -!%1"#"!% 1/ 50 )/-#"!% 4 "5 /M")#/
$% '( -!**/)9!%10%#+ L20$#*/ 90*#4 )" -/ &'( -!%#"/%# A$/ 1/) *FG5/) */)9/-#/%#
5/ 6!1/ 1/ K!%-#"!%%/6/%# 1/ '( 4 %!$) 9!$.!%) #*0%)K!6/* ./*) $% '( 90* $%/
60%"F*/ "%1"*/-#/+
!"#$ %& '(#)!*+, -
!"# $% $&!'()*%+ ",-# !.,"# '*,',#/ 0%# #1#)23%# 4% */5/$*()-*% 6,"$)(,""%0#7
8,-# 3,")*,"# 9-% ",)*% 3,420% ! 0! '-(##!"$% 4:%;'*%##(," 4%# #1#)23%# 4% */5
/$*()-*% %) 9-:(0 %#) <(%" !4!')/ = 0:/)-4% 4% '*,'*(/)/# 4% #>*%)/ 4% %) 4% '*,'*(/)/#
4% 0,?(9-% )%3',*%00% 4% #1#)23%# '*/#%")/% %" 4/)!(0 = 0! '!*)(% @@@7
8,-# !.,"# 3(# %" /.(4%"$% -"% #,-# $0!##% 4% #1#)23%# 6,"$)(,""%0#+ 0%# /0/3%"5
)!(*%#+ '*/#%*.!") 0! '-(##!"$% 4:%;'*%##(," 4%# #1#)23%# 6,"$)(,""%0# %) 4%# )%$&5
"(9-%# 4:!$$/0/*!)(," 4%# $!0$-0# !<,-)(##!") = -" ,-)(0 4% ./*(A$!)(," #13<,0(9-%
%B$!$%7 8,-# %;'0(9-,"# $,33%") (0# #,") /.!0-/# !- $&!'()*% C7
% '0-#+ ",-# !.,"# 3(# %" /.(4%"$% -"% #,-# $0!##% 4% #1#)23%# 6,"$)(,""%0#
/0/3%")!(*%#+ 0%# /0/3%")!(*%# = 4*,()%+ '*/#%*.!") 0! '-(##!"$% 4:%;'*%##(," 4%# #1#5
)23%# 6,"$)(,""%0# %) 4%# )%$&"(9-%# 4:!$$/0/*!)(," 4%# $!0$-0#7
!"# 0! '!*)(% %;'/*(3%")!0% D%" $&!'()*% E F+ ",-# !00,"# $,3'!*%* ",)*% ,-)(0
<!#/ #-* ",)*% 6,*3!0(#3% #,() !.%$ 4%# ,-)(0# 4% */5/$*()-*% )%0# 9-% G(3<-H %)
I!-4%+ #,() !.%$ 4%# ,-)(0# 4% ./*(A$!)(," )%0# 9-% JK@8+ 8-JIL+ JIMNG+ OJ +
%)$ !A" 4% 3,")*%* ",# '%*6,*3!"$%# $,3'/)()(.%#7
!"#$%&' (
!"#$"%&'( )*+ +,+%-.*+
/#/.*(%"&0*+
!"#$! %# &'()*+ , -
!"# $% &'!()*+% (+,&,-%"*. "/0# !1/"# (+/(/#, $%# #2#*34%# -% +,5,&+)*0+%
6/"&*)/""%$# 789:;#<. $%# ,$,4%"*!)+%# 7=89:;#< %* $%# ,$,4%"*!)+%# > -+/)*%
7:=89:;#< (+,#%+1!"* $! (0)##!"&% -?%@(+%##)/" -%# #2#*34%# 6/"&*)/""%$#A
!"# &% &'!()*+%. "/0# (+,#%"*/"# $%# !$B/+)*'4%# -?,1!$0!*)/" &$!##)C0%# (/0+ $%#
#2#*34%# -% +,5,&+)*0+% ,$,4%"*!)+%# 7=89:;# /0 :=89:;#<A D! +%$!*)/" %"*+% $%#
#2#*34%# -% +,5,&+)*0+% $)#*,# &)5-%##0# %* $%# !$B/+)*'4%# -?,1!$0!*)/" %#* (+,#%"*,%
(!+ -%0@ E3&'%# -% !"#$ > %&'()'*+,- %* -% # !"#$ > %&'()'*+,- -!"# $! FB0+%
GAHA D%# !0*+%# 7$%# &$!##)C0%# %* $%# 89:;#< -/)1%"* (!##%+ (!+ 0"% *+!"#6/+4!*)/"
#)4($% 7(+/(/#,% !0 &'!()*+% (+,&,-%"*< !1!"* $?,1!$0!*)/" (!+ "/# !$B/+)*'4%#A
I% &'!()*+% 1! ,*0-)%+ $! &!(!&)*, -?/(*)4)#!*)/" -% C0%$C0%# *2(%# -% +3B$%# ,$,5
4%"*!)+%# > -+/)*%A D! +%$!*)/" %"*+% $%# :=89:;# %* $%# !$B/+)*'4%# -?/(*)4)#!*)/"
%#* (+,#%"*,% (!+ $! E3&'% -% # !"#$ > $'*).'*+,- -!"# $! FB0+% GAHA J1)-%45
4%"*. $?%K&!&)*, -% &%**% /(*)4)#!*)/" -,(%"- -0 "/4L+% -% +3B$%# -% +,5,&+)*0+%#
!(($)&!L$%#. -% $?)4(/+*!"&% -% &%# +3B$%# !)"#) C0% -0 ")1%!0 -% #24,*+)% -0 4/-3$%
-/"",A
D% 4,&!")#4% -?,1!$0!*)/" -% "/# !$B/+)*'4%# 7&$!##)C0%# %* /(*)4)#,#<. C0) +%5
#(%&*% L)%" $% 4/-% 6/"&*)/""%$. %#* )$$0#*+, (!+ $?,*0-% -% &!# 79MN< #/0# 6/+4%
-?0" (+/L$34% -% &!$&0$ -?!&&%##)L)$)*,A
!"# GAH O :%$!*)/" %"*+% -%# #2#*34%# -% +,5,&+)*0+% %* $%# !$B/+)*'4%# -?,1!$0!*)/"
".//0123
!" #$%&'()*+, -./01$21)(&3 4$155(62, ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! 78
!9 :;)(+(51)(&3 ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! 8"
!"!# $%&'(&)*+, &%-. '& /)0&)12*- 3- /&)(0&)*+, ! ! ! ! ! ! ! ! ! ! ! 4#
!"!" 5'2+0*)67- 381%&'(&)*+, +9)*7*/1- ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! 4
!"! #$%&'()*+, -./01$21)(&3 4$155(62, 78
!" #$%&'()*+, -./01$21)(&3 4$155(62,
!"# $%&&% #%$&'(") "(*# +,-#%"&("# .%# !./(,'&01%# 23-4!.*!&'(" $.!##'5*%# +(*,
.%# #6#&71%# 2% ,-8-$,'&*,% -.-1%"&!',%# 9:;<=># (* =:;<=>#?@ A%# !*&,%# 9.%#
$.!##'5*%# %& .%# ;<=>#? 2('4%"& +!##%, +!, *"% &,!"#B(,1!&'(" #'1+.% 9+,(+(#-% !*
$0!+'&,% +,-$-2%"&? !4!"& .3-4!.*!&'(" +!, "(# !./(,'&01%#@
>('& *" :;<=> Rλe 96 $(1+,'# *" =:;<=>? 5*' '"2*'& *"% ,%.!&'(" 2% ,-8
-$,'&*,% →Rλe#*, 2%# &%,1%# E ⊆ T (Fbin)) .3(CD%$&'B %#& 2% $!.$*.%, .! $.E&*,%
&,!"#'&'4% -$,'&% R⋆λe(E)@ ("$ $%& :;<=> Rλe "% $("&'%"& 5*% 2%# ,7/.%# 2% .!
B(,1%
F@ H(a(x, y)) → α !4%$ a ∈ Fbin) H ∈ FNT ) x, y ∈ X ) x $= y) α ∈ T (Fbin ∪
FNT ,X ) %& Var(α) = {x, y}
G@ H(⊥)→ β !4%$ β ∈ T (Fbin ∪ FNT )
H@ H(a(x,⊥))→ x !4%$ x ∈ X ) H ∈ FNT %& a ∈ Fbin@
I!'"&%"!"& "(*# 2(""("# 5*%.5*%# !./(,'&01%# 5*' %J+.'5*%"&
F@ $(11%"& !++.'5*%, .3%"#%1C.% 2% ,7/.%# Rλe #*, .% &%,1% H(s) %&
G@ $(11%"& $!.$*.%, .% +('"& KJ%@
L(*# +,-#%"&("# .3!./(,'&01% F 5*' $!.$*.% *" +!# 2% ,-8-$,'&*,% %" &,!'8
&!"& $0!5*% $!# 2% B(,1% 2% ,7/.% -.-1%"&!',%@ >' .! ,7/.% %#& C'%" 2% .! B(,1%
H(a(x, y)) → α) .3!./(,'&01% F 4! B!',% !++%. M .3!./(,'&01% G +(*, &,!'&%, "3'18
+(,&% 5*%. α ∈ T (Fbin ∪ FNT ,X )@ A% 2%,"'%, &,!'&% .% $!# (N '. %J'#&% .%# ,7/.%# 2%
+('"& KJ% H(a(x, y)) → a(x, y),H(a(x, y)) → H(H ′(a(x, y))) ∈ Rλe @ >' $3%#& C'%"
$% $!#) .3!./(,'&01% H #%,! !++%.-@
#$%&'()*+, " O!.$*. 2% HRλe(s)
9'/4&3-()(&3 : s ⊆ T (Fbin)) H ∈ FNT %& Rλe *" %"#%1C.% 2% ,7/.%# 23:;<=>#
! res← ∅
"! 5( ⊥ ∈ s %& H(⊥)→ β ∈ Rλe 1$&'5
#! ,%# +← β
$! ;3 5(
%! <&2' a(u, v) ∈ s %& H(a(x, y))→ α ∈ Rλe =1(',
&! ,%# +← αRλe(u, v)
'! ;3 <&2'
(! <&2' a(u,⊥) ∈ s %& H(a(x,⊥))→ x ∈ Rλe =1(',
)! ,%# +← u
*! ;3 <&2'
! <&2' a(u, v) ∈ s %& H(a(x, y)) → a(x, y),H(a(x, y)) → H(H ′(a(x, y))) ∈ Rλe
=1(',
"! ,%# +← FPH′
Rλe({a(u, v)})
#! ;3 <&2'
$! ',)&2'3,' res
! "#$%&'() *+ ,-$./$'&01 2)3 343'56)3 7.76)1'$&()3
8.90(&'#6) : !"#$" %& αRλe(u, v)
;(7<012&'&01 = u, v ∈ T (Fbin)' α ∈ T (Fbin ∪ FNT ,X ) &( Rλe $) &)*&+,"& %&
-./"&* %012345*
! 3& 6α = ⊥7 $.0(3
"! ()'0/(1)( ⊥
#! 3&101 3& 6α = x7 $.0(3
$! ()'0/(1)( u
%! 3&101 3& 6α = y7 $.0(3
&! ()'0/(1)( v
'! 3&101 3& 6α = H(α′) !8&# H ∈ FNT 7 $.0(3
(! ()'0/(1)( HRλe(α′Rλe
(u, v))
)! 3&101 3& 6α = a(α1(x, y), α2(x, y)) !8&# a ∈ Fbin7 $.0(3
*! ()'0/(1)( a(α1Rλe(u, v), α2Rλe
(u, v))
! >1 3&
9!)* "0!"/:-;(<+& =' "! ):(!(;:) αRλe(u, v) 8;*;(& "& #>(? %-:;( %& H(a(x, y))→
α ∈ Rλe &( *$,*(;($& -&*@&#(;8&+&)( "&* 8!-;!,"&* x, y @!- u &( v %!)* αA 9& @"$*'
*; $) *B+,:"& ):)C(&-+;)!" %& FNT &*( -&)#:)(-? @&)%!)( "0&D@":-!(;:)' HRλe&*(
!@@&"?A &"! /!-!)(;( "& +?#!);*+& %0?8!"$!(;:) &) +:%& E:)#(;:))&"A & @-:#&**$*
&*( %?#-;( %!)* "0!"/:-;(<+& FA G& %&-);&- (-!;(& "& #!* :HRλe )0&*( @!* ";)?!;-& %-:;(&A
I:$* !"":)* ;""$*(-&- #&( !"/:-;(<+& @!- ):(-& ?($%& %& #!* 3JKA
?@)6%.) :A !"# $%&'(&)*$& +& ,-.$&( +& ,/0/1,"#2,& Rreqλe 345678 345978 345:;7 &#
345::7 <"'(" =2& $&( ,-.$&( +& 1",12$<#"!' Rcirλe > 345:?78 555 345:97 +& '!#,& /#2+& +&
1<( @AB +<'( $< (&1#"!' 45?5 A2 +/*2#8 '!2( C!2$!'( 1<$12$&, 2' D<( +& ,/0/1,"#2,&
E!'1#"!''&$ &' 2#"$"(<'# $%<$.!,"#F)& : >
HR
req
λe ∪Rcirλe
→∪
(u) = H(t(i(i(⊥,⊥), i(⊥,⊥)), i(i(⊥,⊥), i(⊥,⊥))))
G' <DD$"=2<'# $&( ,-.$&( +& 1",12$<#"!' Rcirλe 8 '!2( #,!2C!'( +&( #&,)&( 1"0+&((!2( >
t(H(i(i(⊥,⊥), i(⊥,⊥))), i(i(⊥,⊥), i(⊥,⊥))) + t(i(i(⊥,⊥), i(⊥,⊥)),H(i(i(⊥,⊥), i(⊥,⊥))))
G# D2"( $&( (!2( #&,)&( <H<'# H I $< ,<1"'& 1!'#"'2&'# I J#,& (")D$"K/( &' 2#"$"(<'#
$& Rcirλe >
t((i(H(i(⊥,⊥)), i(⊥,⊥)), i(i(⊥,⊥), i(⊥,⊥))) + t((i(i(⊥,⊥),H(i(⊥,⊥))), i(i(⊥,⊥), i(⊥,⊥))) +t(i(i(⊥,⊥), i(⊥,⊥)), i(H(i(⊥,⊥)), i(⊥,⊥))) + t(i(i(⊥,⊥), i(⊥,⊥)), i(i(⊥,⊥),H(i(⊥,⊥)))).
A 1& )!)&'#8 $&( ,-.$&( Rcirλe +&C"&''&'# <DD$"1<*$&(8 '!2( <$!,( <##&".'!'( 2' &'0
(&)*$& +& #&,)&( 1$!( 1"0+&((!2( >
t(i(r(⊥,⊥), i(⊥,⊥)), i(i(⊥,⊥), i(⊥,⊥))) + t(i(i(⊥,⊥), r(⊥,⊥)), i(i(⊥,⊥), i(⊥,⊥))) +t(i(i(⊥,⊥), i(⊥,⊥)), i(r(⊥,⊥), i(⊥,⊥))) + t(i(i(⊥,⊥), i(⊥,⊥)), i(i(⊥,⊥), r(⊥,⊥)))
AD,-( $< 1<'!'"(<#"!'8 HRreq
λe∪Rcir
λe(u) =
t(i(r(⊥,⊥), i(⊥,⊥)) + i(i(⊥,⊥), r(⊥,⊥)), i(i(⊥,⊥), i(⊥,⊥))) +t(i(i(⊥,⊥), i(⊥,⊥)), i(r(⊥,⊥), i(⊥,⊥)) + i(i(⊥,⊥), r(⊥,⊥)))
!"! #$%&'()*+, -./01$21)(&3 4$155(62, 78
#$%&'()*+, 9 !"#$" %& FPHRλe
(s)
:'/4&3-()(&3 ; s ⊆ T (Fbin)' H ∈ FNT &( Rλe $) &)*&+,"& %& -./"&* %012345*
! res1 ← ∅ 6
"! res2 ← s 6
#! )13)62, res1 $= res2 <1(',
$! res1 ← res2 6
%! res2+← HRλe(res2)
&! =3 )1362,
'! ',)&2'3,' res2
57+!)(89$&+&)(' %!)* #&( &:&+;"&' )<$* ;<$=<)* #<)*8%7-&- "& *>+,<"& H
#<++& $) +!-9$&$- 8)%89$!)( ? 9$&" &)%-<8( )<(-& *>*(.+& %& -7@7#-8($-& &*( &)
(-!8) %& (-!8(&-A B$(-&+&)( %8(' #C!9$& (&-+& u %& "0&)*&+,"& %& (&-+&* %<))7 s
*;7#8D!)( $) 12345' H(u) *;7#8D& 9$& "& (&-+& u %<8( E(-& +8* ? F<$-A G!- #<)@
*79$&)(' )<$* !;;"89$<)* Rλe *$- H(u) %!)* "& ,$( %08+;"7+&)(&- "! +8*&@?@F<$-
!$(!)( 9$& ;<**8,"&A
B$(-&+&)( %8(' "&* !"/<-8(C+&* H &( I !;;"89$&)( "& *>+,<"& H *$- #C!9$& (&-+&
%& "0&)*&+,"& %& (&-+&* #"<* &( !;;"89$&)( "& *>*(.+& %& -7@7#-8($-& !$(!)( 9$&
;<**8,"&A
J" K!$( *!=<8- 7/!"&+&)( 9$& "! *7+!)(89$& %$ *>+,<"& K<)#(8<))&" H &*(' %!)*
$) #&-(!8) *&)*' %7D)8& /-L#& ? "012345 Rλe ' ;<$- $) &)*&+,"& %& (&-+&*A J" &*(
8)(7-&**!)( %& #!"#$"&- (<$(&* "&* #<)D/$-!(8<)* ;<**8,"&* &) !;;"89$!)( H *$- s ⊆
T (Fbin) $) )<+,-& )<)@,<-)7 %& K<8* &( !;;"89$&- Rλe !$(!)( %& K<8* 9$& ;<**8,"&'
c.?.d.A FPRλe(s)A 58 $) ;<8)( D:& &*( <,(&)$' "0&)*&+,"& %& (&-+&* %!)* T (Fbin)
!##&**8,"&* &*( #!"#$"7A
M0!"/<-8(C+& H #!"#$"& "0&)*&+,"& %& #<)D/$-!(8<)* !##&**8,"&* %& "0&)*&+,"& %&
(&-+&* %& T (Fbin) !;-.* !=<8- !;;"89$7 H ? "! ;<*8(8<) "! ;"$* C!$(&A B$(-&+&)(
%8(' 8" +<)(-& "! -<,$*(&**& %& #&( !"/<-8(C+&A
28)!"&+&)(' ;<$- #!"#$"&- "0&)*&+,"& %&* !##&**8,"&* N#<++& +&)(8<))7 %!)* "&
#C!;8(-& OP' )<$* !=<)* ,&*<8) %& %7D)8- "& #!"#$" %$ ;<8)( D:&A 0&*( "& -Q"& %&
"0!"/<-8(C+& RA
1) K!8(' "0!"/<-8(C+& H' "0!"/<-8(C+& I &( "0!"/<-8(C+& R ;-<;<*7* #8@%&**$*
<)( ;-<D(7 "&* (&#C)89$&* %0!##7"7-!(8<) %& "! =7-8D#!(8<) *>+,<"89$& !" #$$
ST->!)( HUVW' T->!)( HUUIX (&""&* 9$& Y
Z >1?$,12 -.23(4()/ !" #$$ %& "0&)*&+,"& %& (&-+&* %& T (Fbin) Y 3&#C@
)89$&+&)( %8(' )<$* )0!##&;(<)* 9$0$) $)89$& -&;-7*&)(!)( ;<$- $) (&-+& %&
T (Fbin)A &"! ;&-+&( "&* %<))7&* #<%7&* %&* +<%."&* (-.* #<+;!#(&*A
Z @13&3(51)(&3 %& "0&)*&+,"& %& (&-+&* %& T (Fbin) Y M&* !"/<-8(C+&* ;-<D(&)(
,8&) "&* ;-<;-87(7* "8)7!8-&* %&* (&-+&*' %7F? ;!-"7 ? "! *&#(8<) RAH &( ? "! *&#(8<)
OAHAH' ;<$- 9$& "& #<%!/& %&* +<%."&* *<8&)( ;"$* #<+;!#(&*A
! "#$%&'() *+ ,-$./$'&01 2)3 343'56)3 7.76)1'$&()3
"$8#) !" #$$ ! "#$%&' ($)(&) *' )+$),-./0#1' 2 '0 )+$),-./0#1' 3 *-/0 40.'
1/5 '6 ($(#' 7-&. 89/0'. *' .':;$/.' &6 .85&)0$0 *8<= ($)(&)8>
?+'@'17)' 5&/9$60 9$ /))&50.'. (-11'60 )' 7-/60 A@' '50 ($)(&)8 7-&. 6-0.' 80&*'
*' ($5 BCD>
9:)6%.) ;; %&"'& ()''*+ (,+ ,'+,-.!,+ (, /01!,+ Rλecir ∪ Rλe
req2 (, !3,4,-5!,
678 ')9+ ":)'+ ;"!;9!* u1 = HRreq
λe∪Rcir
λe(u) = {u11, u12, u13, u14} ":,;
u11 = t(i( (⊥,⊥), i(⊥,⊥)), i(i(⊥,⊥), i(⊥,⊥))), u12 = t(i(i(⊥,⊥), (⊥,⊥)), i(i(⊥,⊥), i(⊥,⊥))),u13 = t(i(i(⊥,⊥), i(⊥,⊥)), i( (⊥,⊥), i(⊥,⊥))), u14 = t(i(i(⊥,⊥), i(⊥,⊥)), i(i(⊥,⊥), (⊥,⊥))).
<, (,94=0-, 5"+ (, /*>*;/=&9/, ,+& (, /,>?"=/, ,4";&,-,'& !3,4,-5!, 5/*;*(,'& ":,;
!3,'+,-.!, (, &,/-,+ (, u18 c. .d ')9+ ;"!;9!)'+
u2 = HRreq
λe∪Rcir
λe(u1) = HR
req
λe∪Rcir
λe(u11 + u12 + u13 + u14) =
H(t(i( (⊥,⊥), i(⊥,⊥)), i(i(⊥,⊥), i(⊥,⊥)))) + H(t(i(i(⊥,⊥), (⊥,⊥)), i(i(⊥,⊥), i(⊥,⊥)))) +H(t(i(i(⊥,⊥), i(⊥,⊥)), i( (⊥,⊥), i(⊥,⊥)))) + H(t(i(i(⊥,⊥), i(⊥,⊥)), i(i(⊥,⊥), (⊥,⊥)))).
@5/0+ !3*!=-='"&=)' (,+ &,/-,+ /,()'("'&+ ,& !" ;"')'=+"&=)'8 ')9+ ;"!;9!)'+
;)--, +9=&
t(i( (⊥,⊥), (⊥,⊥)), i(i(⊥,⊥), i(⊥,⊥))) + t(i(i(⊥,⊥), i(⊥,⊥)), i( (⊥,⊥), (⊥,⊥))) +t(i( (⊥,⊥), i(⊥,⊥)), i( (⊥,⊥), i(⊥,⊥))) + t(i( (⊥,⊥), i(⊥,⊥)), i(i(⊥,⊥), (⊥,⊥))) +t(i(i(⊥,⊥), (⊥,⊥)), i( (⊥,⊥), i(⊥,⊥))) + t(i(i(⊥,⊥), (⊥,⊥)), i(i(⊥,⊥), (⊥,⊥))) +t( ( (⊥,⊥), i(⊥,⊥)), i(i(⊥,⊥), i(⊥,⊥))) + t( (i(⊥,⊥), (⊥,⊥)), i(i(⊥,⊥), i(⊥,⊥))) +t(i(i(⊥,⊥), i(⊥,⊥)), ( (⊥,⊥), i(⊥,⊥))) + t(i(i(⊥,⊥), i(⊥,⊥)), (i(⊥,⊥), (⊥,⊥)))
@5/0+ !" ;"')'=+"&=)'8 ')9+ 5)9:)'+ ).&,'=/ 9', /,5/*+,'&"&=)' 5!9+ ;)-5";& ;=>
(,++)9+ A
t(i(i(⊥,⊥), i(⊥,⊥)), ( (⊥,⊥), i(⊥,⊥)) + (i(⊥,⊥), (⊥,⊥)) + i( (⊥,⊥), (⊥,⊥))) +
t(i( (⊥,⊥), (⊥,⊥)) + ( (⊥,⊥), i(⊥,⊥)) + (i(⊥,⊥), (⊥,⊥)), i(i(⊥,⊥), i(⊥,⊥))) +
t(i( (⊥,⊥), i(⊥,⊥)), i( (⊥,⊥), i(⊥,⊥)) + i(i(⊥,⊥), (⊥,⊥))) +
t(i(i(⊥,⊥), (⊥,⊥)), i( (⊥,⊥), i(⊥,⊥)) + i(i(⊥,⊥), (⊥,⊥)))
<,+ 5"+ (, /*>*;/=&9/, +9=:"'&+ ?)';&=)'',/)'& (, !" -B-, -"'=0/,2 C! ?"9& ')&,/
D9, ("'+ ;, ;"+>! u, v $∈ HRreq
λe∪Rcir
λe(u) -"=+ u, v ∈ FPRreq
λe∪Rcir
λe(u)2
E60&/0/9'1'60F 6-&5 9-G-65 %&' )' 0'.1' A6$) '50 %&$5/:5/1/)$/.' $& 0'.1' /6/0/$)
5$&; )' (#$6,'1'60 *'5 5G1H-)'5 *'5 5-&5:$.H.'5 *' )$ .$(/6' 80/%&'08' 7$. )' 5G1H-)'
t> I+$&0.' 7$.0F )'5 -78.$0/-65 *' (/.(&)$0/-6 7-&. *'5 5-&5:$.H.'5 *' )$ .$(/6' ;-60
)' 141' (-17-.0'1'60>
J6' %&'50/-6 89/*'60' '50 %+&>=! 5)++=.!, (3";;*!*/,/ !, ;"!;9! ,' *:=&"'& !" /*5*&=>
&=)' (,+ )5*/"&=)'+ 5)9/ !" ;=/;9!"&=)' ,' (,+;,'("'& 9', +,9!, ?)=+ !3,'(/)=& ='>
&*/,++* ,& +"&9/,/ !, '),9( ;)9/"'& "9&"'& D9, 5)++=.!,>
"'5 0'(#6/%&'5 6-&5 7'.1'00'60 *' )&00'. (-60.' )+'@7)-5/-6 (-1H/6$0-/.' 7-&.
)'5 1-*K)'5 5G180./%&'5> L$)#'&.'&5'1'60F )$ (-17$(/08 *'5 *-668'5 6-&5 (-6*&/0
= &6 $&0.' 7.-H)K1' %&/ '50 )+'M'0 *'5 *-668'5 /60'.18*/$/.'5> ?'5 7-/605 A@'5 )-:
($&@ N?ODP N-& )$ 50.$08,/' *' 5$0&.$0/-6 Q"/$.*- 3RRSTP 7.85'608'5 *$65 )$ 5'(0/-6
5&/9$60' 5-60 &6' H-66' 5-)&0/-6 7-&. (' 7.-H)K1'>
!"! #$%&'&()%&*+ ,-
./0*1&%2'3 4 !"#$" %& FPHRλe
(s) !'&# %&( )*+"&( %& ,-./0 12&
5167*+8&%&*+ 9 s ∈ T (Fbin)3 H,H⋆ ∈ FNT &0 Rλe $/ &/(&45"& %& )*+"&(
%6789:;(
! res1 ← ∅ <
"! res2 ← s <
#! %)+%:;3 res1 $= res2 <)&13
$! res1 ← res2 <
%! res2+← H⋆Rλe
(res2)
&! res2+← HRλe(res2)
'! =+ %)+:;3
(! 13%*;1+31 res2
!" #$%&'&()%&*+
&00& (�.-/ 0)!.0& "6-,0.4.(!0.-/ %&( !"+-).0=4&( %6>'!"$!0.-/ &/ $0.".(!/0 %&(
0&#=/.?$&( %6!##>">)!0.-/ %&( #!"#$"( !5-$0.((!/0 @ $/ -$0." %& '>).1#!0.-/ (A45-"B
.?$& &C#!#&D
E-$) ?$&"?$&( 0A,&( %& )*+"&( >">4&/0!.)&( @ %)-.0&3 /-$( !""-/( 4-/0)&) "! #!,!#B
.0> %6!##>">)!0.-/ %&( #!"#$"(D F'.%&44&/03 "6&C#!#.0> %& #&00& -,0.4.(!0.-/ %>,&/%
%$ /-45)& %& )*+"&( %& )>B>#).0$)& !,,".#!5"&(3 %& "6.4,-)0!/#& %& #&( )*+"&( !./(.
?$& %$ /.'&!$ %& (A4>0).& %$ 4-%*"& %-//>D
!"!# $%&'(&)*+, &%-. '& /)0&)12*- 3- /&)(0&)*+,
G-0)& 5$0 &(0 %6!##>">)&) "& #!"#$" &/ >'.0!/0 "! )>,>0.0.-/ %&( -,>)!0.-/( ,-$)
"! #.)#$"!0.-/ &/ %&(#&/%!/0 $/& (&$"& H-.( @ "6&/%)-.0 ./0>)&((> &0 (!0$)&) "& /-&$%
#-$)!/0 0!/0 ?$& ,-((.5"&D I&( ,-./0( 12&( "-#!$2 JI8EK J-$ "! (0)!0>+.& %& (!0$)!B
0.-/K (-/0 $/& 5-//& (-"$0.-/D
I! (0)!0>+.& %& (!0$)!0.-/ J!,,&">& >+!"&4&/0 I&( !"#$% &"'(% )!*+,'- )& K
&(0 ,)-,-(>& %!/( L .!)%- MNNOPD 7""& "$00& #-/0)& "6&Q&0 %& "! 0!.""& %&( %-//>&(
./0&)4>%.!.)&(D 7Q�.'&4&/03 #&0 !"+-).0=4& &(0 0)*( &C#!#& ,-$) %&( (0)$#0$)&(
0)*( #-4,!#0&( @ "! RSS 0&"( ?$& TSS L .!)%- MNNOP3 SSS L -$')&$) MNNMP3 ;SS
L -$')&$) MNNUP3 V;SS L9=.&))ABT.&+ MNNWP3 L9=.&))ABT.&+ MNNXPD
9-$0 %6!5-)%3 /-$( ./0)-%$.(-/( $/ /-$'&!$ (A45-"& /-/B0&)4./!" H⋆,-$)
#=!?$& (A45-"& /-/B0&)4./!" HD &( /-$'&!$2 (A45-"&( ,&$'&/0 Y0)& $0.".(>( %!/(
%&( )*+"&( )>B>#).0$)& #-44& $/ (A45-"& /-/B0&)4./!" /-)4!"D Z"( (-/0 ./0&),)>0>(
#-44& Hn,-$) %&( &/0.&)( /-/ />+!0.H( nD
>6=+&%&*+ -- ./+#%0!/1+$"!# 2(% 34&.35% 2( $67( 89
5!"$ Rλe→I ,# (#%(1:;( 2( /<=;(% +6+#$ ,# *>$? =+,*@( A," (%$ 2( ;+ 0!/1(
H (a(x, y)) !B H ∈ FNT ($ a ∈ Fbin9 5(;!# 2(% *+% 2( /<=;(% 2, *>$? 2/!"$- #!,%
+C!#% ;(% #!,C(;;(% /<=;(% %,"C+#$(% D
! "#$%&'() *+ ,-$./$'&01 2)3 343'56)3 7.76)1'$&()3
Rλe→I R′λe→I
H (a(x, y))→ a(x, y) H⋆ (a(x, y))→ a(x, y)
H (a(x, y))→ a(G (x), y) H⋆ (a(x, y))→ a(G⋆ (x), y)
H (a(x, y))→ a(x,D (y)) H⋆ (a(x, y))→ a(x,D⋆ (y))
H (a(x, y))→ a(G (x), y)
H (a(x, y))→ a(x,D (y)) H⋆ (a(x, y))→ a(G⋆ (x), D⋆ (y))
H (a(x, y))→ a(x, y)
H (a(x, y))→ a(x,D (y)) H⋆ (a(x, y))→ a(x,D⋆ (y))
H (a(x, y))→ a(x, y)
H (a(x, y))→ a(G (x), y) H⋆ (a(x, y))→ a(G⋆ (x), y)
H (a(x, y))→ a(x, y)
H (a(x, y))→ a(G (x), y)
H (a(x, y))→ a(x,D (y)) H⋆ (a(x, y))→ a(G⋆ (x), D⋆ (y))
!"# H, G, D ∈ FNT $ H⋆, G⋆, D⋆ ∈ FNT "% a ∈ Fbin
! "#$% &'%() *$( !# %)#&+"'),#%-'& .(+ /012/3 .( %45( 5($% 6%)( #55!-*$7(
+75#)7,(&% 5'$) !8(&+(,9!( .( ):;!(+ H (a(x, y)) → α (% !8(&+(,9!( .( ):;!(+
H (b(x, y)) → β '< α, β ∈ T (Fbin ∪ FNT ,X ) +8-! &8(=-+%( 5#+ .( ):;!( .( %45(
%(!!( *$( H (a(x, y))→ b(x, y) '$ H (b(x, y))→ a(x, y)>
8791&'&01 :! !"#$%&!'"()&# *+$ ,-. ,/$ *+ (01+ 223
/&)( Rλe→II 4# +#$+'56+ *+ !786+$ "0"#( 4# 9:(; 8"49<+ =4) +$( *+ 6" %&!'+
H (a(x, y)) &> H ∈ FNT +( a ∈ Fbin3 /+6&# 6+$ 9"$ *+ !786+$ *4 9:(; *!&)(? #&4$
"@&#$ *+ #&4@+66+$ !786+$ 9&''+ 6+$ $4)@"#(+$ A
Rλe→II R′λe→II
H (a(x, y))→ G(D(a(x, y)))
G (a(x, y))→ a(x, y) H⋆ (a(x, y))→ D⋆ (a(x, y))
H (a(x, y))→ G(D(a(x, y)))
D (a(x, y))→ a(x, y) H⋆ (a(x, y))→ G⋆ (a(x, y))
H (a(x, y))→ G(D(a(x, y)))
G (b(x, y))→ b(x, y) H⋆ (a(x, y))→ G⋆ (c(x, y))
G (a(x, y))→ c(x, y)
G (c(x, y))→ α
D (a(x, y))→ b(x, y) H⋆ (a(x, y))→ D⋆ (b(x, y))
D (c(x, y))→ c(x, y)
D (b(x, y))→ β
"@+9 H, G, D ∈ FNT ? H⋆, G⋆, D⋆ ∈ FNT +( a ∈ Fbin? α, β ∈ T (Fbin ∪ FNT ,X )?
Var(α) = {x, y} +( Var(β) = {x, y}3
!"#$%&'()")$% *&%+ $,(- .,(/,$- 0,$-&*!1)1 2) -3"4,2) H⋆0,"") ($ "#15
'()(1 &$*&'(#$% 6 '()2 )$*1,&% $,%1) -3-%7") *) 1!5!01&%(1) )-% )$ %1#&$ *) 2#$0)1
($ .,&$% 89): ;#1 0,$-!'()$%+ 2)- 17<2)- %1#$-=,1"#%&,$ .)(/)$% $,(- #&*)1 6 2#$0)1
!"! #$%&'&()%&*+ ,-
./0*1&%2'3 !"#$" %& αRλe(u, v) !'&# %&( )*+"&( %& ,-./0 12&
4156*+7&%&*+ 8 u, v ∈ T (Fbin)3 α ∈ T (Fbin ∪ FNT ,X ) &0 Rλe $/ &/(&45"& %&
)*+"&( %6789:;(
! (& <α = ⊥= )/*1(
"! 13%*91+31 ⊥
#! (&+*+ (& <α = x= )/*1(
$! 13%*91+31 u
%! (&+*+ (& <α = y= )/*1(
&! 13%*91+31 v
'! (&+*+ (& <α = H(α′) !'&# H ∈ FNT = )/*1(
(! 13%*91+31 HRλe(α′
Rλe(u, v))
)! (&+*+ (& <α = H⋆(α′) !'&# H⋆ ∈ FNT = )/*1(
*! 13%*91+31 FPHRλe
(α′Rλe
(u, v))
! (&+*+ (& <α = a(α1(x, y), α2(x, y)) !'&# a ∈ Fbin= )/*1(
"! 13%*91+31 a(α1Rλe(u, v), α2Rλe
(u, v))
#! :+ (&
"&( #!"#$"( %& >8? %!/( "&( &/%)-.0( ./0@)&((!/0( ,-$) -,0.4.(&) "& 0&4,( %& #!"A
#$"B C@(-)4!.(3 /-$( &((!D-/( %6./0@+)&) #&( 0)!/(E-)4!0.-/( %!/( /-( !"+-).0F4&(
%6@'!"$!0.-/ #"!((.G$&(B
>6!"+-).0F4& H &(0 "! /-$'&""& '&)(.-/ %& "6!"+-).0F4& I <"6!"+-).0F4& %$ ,-./0
12&= G$. #-4,0& "&( /-$'&!$2 (D45-"&( /-/A0&)4./!$2 H⋆B C!/( "! 5-$#"& %& #&
/-$'&" !"+-).0F4&3 "&( )*+"&( %& "6-,@)!0&$) %& ,-./0 12& (-/0 !,,".G$@&( &/ !,,&"!/0
"! E-/#0.-/ H⋆Rλe
(s) <"6!"+-).0F4& J !'&# %&( (D45-"&( /-/A0&)4./!$2 H⋆=B
>6!"+-).0F4& K &(0 @+!"&4&/0 ($5(0.0$@ ,!) "6!"+-).0F4& LB 7/ )&+!)%!/0 "& /-$A
'&!$ (D45-"& /-/A0&)4./!" H⋆3 #&0 !"+-).0F4& !,,&""& "6!"+-).0F4& H <"6!"+-).0F4&
%$ ,-./0 12&=B
M/0$.0.'&4&/03 (&"-/ "! ,)-,-(.0.-/ L &0 "! #-/(0)$#0.-/ %&( !"+-).0F4&( ".(0@& !$
%&((-$( N
O >.+/&( L %& "6!"+-).0F4& H
O >.+/&( P &0 JQ %& "6!"+-).0F4& L
-/ ,&$0 #-/#"$)& G$& "6@'!"$!0.-/ ,!) "6!"+-).0F4& H &0 ,!) "6!"+-).0F4& I %-//& "&
4R4& )@($"0!0B
7/ E!.03 &/ "!/S!/0 "& #!"#$" H⋆Rλe
(res2)<>.+/& L %& "6!"+-).0F4& H= !'!/0 "& #!"#$"
HRλe(res2)<>.+/& T %& "6!"+-).0F4& H= /-$( ,-$'-/( !00&./%)& "& ,-./0 12& ,"$( '.0&
G$& "6!"+-).0F4& IB
41*$*(&%&*+ !"# Rλe $% &'() *# s ∈ T (Fbin)+ ,"- *."/#* 0*/ Rλe→I ⊆ Rλe
!$ Rλe→II ⊆ Rλe 1*23-"//4%# 0*/ 5!%0"#"!%/ 0* -4 067%"#"!% 88 !$ 0* -4 067%"#"!%
89 4"%/"
(Rλe ∪ (R′λe→I ∪R
′λe→II))
⋆(E′) = R⋆λe(E
′).
! "#$%&'() *+ ,-$./$'&01 2)3 343'56)3 7.76)1'$&()3
!"# E ⊆ T (Fbin) "$ E′ = {H(t)|t ∈ E}%
8()/-) 9 & R⋆λe(E
′) ⊆ (Rλe ∪ (R′λe→I
∪ R′λe→II
))⋆(E′)' #(")$ *!+,"-$ )"./-
Rλe ⊆ (Rλe ∪ (R′λe→I
∪R′λe→II
))
& (Rλe ∪ (R′λe→I
∪R′λe→II
))⋆(E′) ⊆ R⋆λe(E
′). 0/12)1+!/-) "- $2 +$ -$ .( - .3)"
," # ) 4
& (Rλe ∪R′λe→I
)⋆(E′) ⊆ R⋆λe(E
′) 4
& R′λe
⋆H⋆ (a(x,y))→a(x,y)(E
′) ⊆ Rλe⋆H (a(x,y))→a(x,y)(E
′)
& R′λe
⋆H⋆ (a(x,y))→a(G⋆ (x),y)(E
′) ⊆ Rλe⋆H (a(x,y))→a(G (x),y)(E
′) # 2
Hn (a(x, y)) →⋆H (a(x,y))→a(G (x),y) a(Gn (x), y) !"# n )15) 66"-$
72 -,%
& 8+6+. +2"6"-$ 9/12 ." 2")$"%
& (Rλe ∪R′λe→II
)⋆(E′) ⊆ R⋆λe(E
′) 4
& 8+6+. +2"6"-$ 9/12 .") 1$2") # )%
!"# $%%!&# '%%"#()*) +*# &!",*$"- $%.!)'(/0*# 1$) &!()* 2("3* 3* +$# 4567
:;)6%.) <= 8/+$ Rcirλe ,*:-+ , -) .(";"69." 92*#*,"-$% <") 2=7.") ," #+2#1. $+/-
>?%@AB' %%% >?%@CB ," Rcirλe *$ -$ ,") *.*6"-$ +2") D ,2/+$" $39" E >Rcir
λe →IB 9"1!"-$
F$2" /9$+6+)*") 9/12 ." G .#1. ," 9/+-$ :;" #/66" . H I/- )1+! -$" 4
H⋆(i(x, y)) → i(H⋆(x), H⋆(y)) !"#$
H⋆(r(x, y)) → r(H⋆(x), H⋆(y)) !"%$
H⋆(b(x, y)) → b(H⋆(x), H⋆(y)) !"&$
H⋆(t(x, y)) → t(H⋆(x), H⋆(y)) !"'$
J- 99.+K1 -$ .( .7/2+$L6" ," 9/+-$ :;" D . #+2#1. $+/- 9/12 .") 2=7.") ," #+2#1. M
$+/- /9$+6+)*") R′cirλe →I
>N%@B' >N%AB' >N%OB' >N%?B FPHRcirλe
∪Rreq
λe∪R′cir
λe →I
(u) ,/--" 92=)
K1".K1") # .#1.)% < )36*$2+" ,") )/1)M$"26") -/1) 9"26"$ ," 2"M92"-,2" .") 2*)1.$ $)
,") 9/+-$) :;") ./# 1; ,*PD # .#1.*) "- # #L"% Q+- ."6"-$' -/1) $$"+7-/-) ." 9/+-$
:;" 7./R . FPHRcirλe
∪Rreq
λe∪R′cir
λe →I
(u) = t(a(a(⊥,⊥), a(⊥,⊥)), a(a(⊥,⊥), a(⊥,⊥))) /S
a ∈ {i, r}%
8% 9$"( &!(*) :"* %*# $);)*# #!&( 3*# #()"+(")*# ()<# +!01$+(*# .)=+* $" ($;%*$"
3>"&'+'(2 *( ? %$ +$&!&'#$('!&7
!"# )*0$):"!&# $"##' :"* :"*%:"*# +$%+"%# 1!") 3*# )<.%*# 3$&# Rλe→I *(
Rλe→II #!&( '&"('%*# +$) '%# #!&( 32@? +$%+"%2#7 6!")#"',!&# %*# &!",*$"- $%.!)'(/0*#
1!") )23"')* +*# '&+!&,2&'*&(# 3$&# %$ #*+('!& #"',$&(*7
!"! #$%&'&()%&*+ ,
-./*0&%1'2 3 !"#$" %& FPHRλe
(s) !'&# %& ()$'&""&* +,-"&* %& .)/(0 12&
4056*+7&%&*+ 8 s ∈ T (Fbin)3 H,H⋆ ∈ FNT &0 Rλe $( &(*&45"& %& +,-"&*
%6789:;*
! res1 ← ∅ <
"! res2 ← s <
#! %)+%9:2 res1 '= res2 ;)&02
$! res1 ← res2 <
%! res2+← H⋆R′λe→I
∪R′λe→II
(res2)
&! res2+← HRλe ,R′
λe→I,R′
λe→II,Rλe→I ,Rλe→II
(res2)
'! <+ %)+9:2
(! 02%*:0+20 res2
!"!" #$%&'()*+, -./01$21)(&3 &4)(+(5/,
)44& %=>? 4&(0/)((=3 ()$* .+).)*)(* %& ()$'&!$2 !"-)+/0@4&* .)$+ ='/0&+
"&* +&A#!"#$"* /($0/"&* .)$+ %&* +,-"&* %!(* Rλe→I ∪Rλe→II B
C6!"-)+/0@4& D &*0 "! ()$'&""& '&+*/)( %& "6!"-)+/0@4& E F"6!"-)+/0@4& %$ .)/(0
12&G H$/ #)4.0& "6 /(0+)%$#0/)( !$0)4!0/H$& %$ ()$'&!$ 9:;B I!(* "! 5)$#"& %&
"6!"-)+/0@4&3 %&* +,-"&* !'&# "6).=+!0&$+ %& .)/(0 12& *)(0 !.."/H$=&* &( !..&"!(0
$(& J)(#0/)( H⋆R′λe→I
∪R′λe→II
(s) &0 &( !..&"!(0 "6!"-)+/0@4& K HRλe(s) "/4/0= .)$+
"! +,-"& =0!(0 %!(* Rλe→I )$ Rλe→II C6!"-)+/0@4& L %)((& $(& J!M)( %6!.."/H$&+
HRλe(s) *!(* $0/"/*&+ !$#$(& +,-"& %!(* Rλe→I &0 Rλe→II .)$+ "! .+&4/,+& =0!.&
%& "! 0+!(*J)+4!0/)(B
C6!.."/#!0/)( %&* ()$'&!$2 !"-)+/0@4&* %)((& &2!#0&4&(0 "& 4N4& #!"#$" #)44&
"& #!"#$" %& *O*0,4&* $0/"/*!(0 "&* +,-"&* ).0/4/*=&* %& "! *�/)( .+=#=%&(0& .)$+ "&
4N4& .+)5",4& %!(* $( 0&4.* %& #!"#$" H$!*/A/%&(0/H$&B
P(0$/0/'&4&(03 *&")( "! .+).)*/0/)( D &0 "! #)(*0+$#0/)( %&* !"-)+/0@4&* "/*0=&
#/A%&**)$* Q
R C/-(&* S &0 D %& "6!"-)+/0@4& D
R C/-(&* S &0 D %& "6!"-)+/0@4& L
()$* .)$')(* #)(#"$+& H$& "6='!"$!0/)( .!+ "6!"-)+/0@4& D &0 .!+ "6!"-)+/0@4& T
%)((& "& 4N4& +=*$"0!0 %& "6!"-)+/0@4& EB
U)$* !')(* %)(# "! .+).)*/0/)( D3 $( +=*$"0!0 ."$* J)+0 H$& "! .+).)*/0/)( SB
40*$*(&%&*+ 3 !"# Rλe $% &'() *# s ∈ T (Fbin)+ ,"- *."/#* 0*/ Rλe→I ⊆ Rλe
!$ Rλe→II ⊆ Rλe 1*23-"//4%# 0*/ 5!%0"#"!%/ 0* -4 067%"#"!% 88 !$ 0* -4 067%"#"!%
89 4"%/"
(Rλe ∪ (R′
λe→I ∪R′
λe→II) \ (Rλe→I ∪Rλe→II))⋆(E′) = R⋆
λe(E′).
4:*5 E ⊆ T (Fbin) *# E′ = {H(t)|t ∈ E}+
! "#$%&'() *+ ,-$./$'&01 2)3 343'56)3 7.76)1'$&()3
8.90(&'#6) !"#$" %& HRλe ,R′
λe→I,R′
λe→II,Rλe→I ,Rλe→II
(s) !'&# %& ()$'&""&* +,-
."&* %& /)0(1 23&
:(7;012&'&01 <
s ∈ T (Fbin)4 H ∈ FNT
Rλe ,R′
λe→I,R′
λe→II,Rλe→I ,Rλe→II *)(1 %&* &(*&56"&* %& +,."&* 789:;*
! res← ∅
"! 3& ⊥ ∈ s &1
H(⊥)→ β ∈ Rλe \(Rλe→I ∪Rλe→II) ∪(R′
λe→I∪R′
λe→II) $.0(3
#! +&* +← β
$! =1 3&
%! %0/( a(u, v) ∈ s &1
H(a(x, y))→ α ∈ Rλe \(Rλe→I ∪Rλe→II) ∪(R′
λe→I∪R′
λe→II) >$&()
&! +&* +← αRλe∪(R′I∪R′
II)(u, v)
'! =1 %0/(
(! %0/( a(u,⊥) ∈ s &1
H(a(x,⊥))→ x ∈ Rλe \(Rλe→I ∪Rλe→II) ∪(R′
λe→I∪R′
λe→II) >$&()
)! +&* +← u
*! =1 %0/(
! %0/( a(u, v) ∈ s &1 H(a(x, y)) → a(x, y),H(a(x, y)) → H(H ′(a(x, y))) ∈ Rλe
>$&()
"! +&* +← FPH′
Rλe({a(u, v)})
#! =1 %0/(
$! ()'0/(1)( res
:()/-) ?@ !"#$%# &$''()'*# +# ,- ."/./&(*(/) 01
(Rλe ∪ (R′
λe→I ∪R′
λe→II) \ (Rλe→I ∪Rλe→II))⋆(E′) = R⋆
λe(E′).
-%#' E ⊆ T (Fbin) #* E′ = {H(t)|t ∈ E}2
34-."5& ,- ."/./&(*(/) 6 )/$& /7*#)/)&
(Rλe ∪ (R′
λe→I ∪R′
λe→II))⋆(E′) = R⋆
λe(E′).
-%#' E ⊆ T (Fbin) #* E′ = {H(t)|t ∈ E}2
34-$*"# .-"*8 -$ 9/9#)* +/)):8 )/$& ./$%/)& ."/$%#" ;$4$)# "5<,# #) R′
λe→I
#&* -..,('-7,# &#$,#9#)* &( ,- "5<,# '/""#&./)+-)*# #) Rλe→I #&* -..,('-7,#2 3# .,$&
,# ":&$,*-* +4$) .-& +# ":=:'"(*$"# +# ,- "5<,# #) R′
λe→I#&* (+#)*(;$# > '#,$( +# ,- "5<,#
'/""#&./)+-)*# #) Rλe→I 2 ?(9(,-("# ./$" R′
λe→II-%#' Rλe→II 2 @#,- )/$& .#"9#*
+4/7*#)(" ,# ":&$,*-* +4$) .-& +# ":=:'"(*$"# &-)& $*(,(&#" ,- "5<,# #) (Rλe→I∪Rλe→II)2
@#,- 9/)*"# ,- "#%#)+('-*(/)2
A73/.'$'3 )B%7(&6)1'$/B ? 3-)& ,- .-"*(# #A.:"(9#)*-,#8 )/$& -%/)& (9.,:=
9#)*: &$" +#& 9/+5,#& -"7/"#&'#)*&8 #* &$" +#& 9/+5,#& +# ":&#-$A +# !#*"( -B)
+# 9/)*"#" ,4#C'-'(*: +# ,- *#'D)(;$# +# EF!2
!"! #$%&'&()%&*+ ,,
!"#$%&'
()*$$' &)+*&)$' ,- &",%$' ('&./ ,' 0)$0-$
1"23 23 1"23 23
4!#"!'/0'56/ 25 2400 → 25007
89 /'0"5,'/ : 8 /'0"5,'
;</')-+ ,' '6!* 22 → 24 24 → 235= > ?'-!'/ 2→ 20 &*5-6'/
"@ $'/ &",%$'/ )!#"!'/0'56/ /"56 $' .!"6"0"$' ,A4!#*6!)B' 4!#"!'/0'56 C(4 DE $' .!"F
6"0"$' ,' '!0"$)6' C D '6 $' .!"6"0"$' ,AG$'06*"5 4!#"!'/0'56 C2G DE '6 $'/ &",%$'/
,' !</')-+ ,' '6!* /"56 $' .!"#$%&' ,'/ ?*$"/".?'/E $' .!"6"0"$' ,' H$"66',F;*5B '6
$' .!"6"0"$' ,' ;"-5,F;"#*5 I-6'+J
-.')/01.( , 1"-/ !'K'5"5/ )-+ Rλe→III $'/ !%B$'/ ;G3(;H ,' 6L.' MMM ,' $)
N"!&'
O H (a(x, y))→ b(x, y)
O H (a(x, y))→ a(y, x)
O H (a(x, y))→ a(⊥, a(x, y))
)K'0 H ∈ FNT et a, b ∈ FbinJ
G$$'/ 5' /"56 .)/ '50"!' 6!)*6<'/ 0)! $) /)6-!)6*"5 ,'K*'5,!) 6!%/ 0"&.$'+' P '+<F
0-6'! )K'0 ,' 6'$$'/ N"!&'/J Q)5/ $'/ 0)/ .$-/ /*&.$'/E .)! '+'&.$' H(a(x, y)) →
b(x, y) R-)5, ,'/ /"-/F6'!&'/ ,' a, b /"56 ⊥E 5"-/ ."-K"5/ 'S'06-'! -5' 6!)5/N"!F
&)6*"5 /*&.$'J Q' .$-/E ."-! R-'$R-'/ 6L.'/ ,' ;G3(;HE $'/ 0)$0-$/ ,' ."*56/ T+'/
B$"#)-+ /"56 6'!&*5</J
!"#$ %& '(#)!*+, -
!"# $% $&!'()*%+ ",-# !.,"# '*/#%")/ 0%# !01,*()&2%# 34/.!0-!)(," $0!##(5-%#
',-* 0%# #6#)72%# 3% */8/$*()-*% /0/2%")!(*%# 9:;<=># ,- =:;<=>#?@ A%# !-)*%#
90%# $0!##(5-%# %) 0%# ;<=>#? 3,(.%") '!##%* '!* -"% )*!"#B,*2!)(," #(2'0% 9'*,',#/%
!- $&!'()*% '*/$/3%")? !.!") 04/.!0-!)(," '!* ",# !01,*()&2%#@
C% $&!'()*% $,")(%") /1!0%2%") -"% '!*)(% 34,')(2(#!)(," 3%# !01,*()&2%# 34/.!08
-!)(," %" -)(0(#!") 3%# )%$&"(5-%# 34!$$/0/*!)(," 3%# $!0$-0# !D,-)(##!") E -" ,-)(0
3% ./*(F$!)(," #62D,0(5-% %G$!$%@ H.(3%22%")+ 04%G$!$()/ 3/'%"3 3- ",2D*% 3%
*710%# 3% */8/$*()-*% !''0($!D0%#+ 3% 04(2',*)!"$% 3% $%# *710%# !("#( 5-% 3- "(.%!-
3% #62/)*(% 3- 2,370% 3,""/@
!"# 0! '!*)(% %I'/*(2%")!0% !- $&!'()*% J+ ",-# !00,"# $,2'!*%* 3%# !01,*()&2%#
0(#)/# $(83%##-# !F" 3% 2,")*%* 04%G$!$()/ *%2!*5-!D0% 3% 04!01,*()&2% 3% #!)-*!)(,"
',-* 0%# 2,370%# #62/)*(5-%# )%0# 5-% 0% <KL !("#( 5-% 0%# !-)*%# '*,),$,0%# 3% )!(00%
3- 2,370% /",*2%@
!"#$#%&' ()!*#'
!"#$% &&& ' ())*$+!#$,- . *!
/0"$1+!#$,-
!"#$%&' (
!"#$%&'(%!) *'+ ,-./0
!"#$! %# &'()*+ , -
!"# $% &'!()*+% ,- "./# !0."# (+1#%"*1 23456 (+1#%+0!"* $! (/)##!"&% 78%9(+%#:
#)." 7%# #;#*<=%# >."&*).""%$# %* 7%# *%&'")?/%# 78!&&1$1+!*)." 7%# &!$&/$# !@./*)#:
#!"* A /" ./*)$ 7% 01+)B&!*)." #;=@.$)?/% %C&!&%D !"# $% &'!()*+% E- "./# !0."#
=."*+1 &.==%"* "./# (./0."# 10!$/%+ 7%# 23456#D F%# >.+=!$)#=%# %* $%# !$G.:
+)*'=%# 7810!$/!*)." 7% $823456 #."* )$$/#*+1# (!+ $% (+.*.&.$% 4HID
!"# &% &'!()*+%- "./# !$$."# =."*+%+ $! (/)##!"&% 7% $823456 7!"# $! =.71$):
#!*)."D H(+<# $% 4HI- "./# &."*)"/."# A %9($)?/%+ &.==%"* "./# (./0."# #)=/$%+
?/%$?/%# !/*+%# (+.*.&.$%# *%$# ?/% $% (+.*.&.$% 7% I%+&.$!*% JIIK- $% (+.*.&.$% 78:
2$%&*)." H+@.+%#&%"* JF2IKD L.# !=@)*)."# #."* 1G!$%=%"* 781&$!)+&)+ $! +%$!*)."
%"*+% 456 %* 3456- %* %"*+% 3456 %* 23456 #/+ &%# (+.*.&.$%# (./+ $8.@M%&*)> 7%
+1!$)#%+ $% (+%=)%+ (!# 7% $! *+!"#>.+=!*)." !/*.=!*)?/%D
8!/*+% (!+*- $! &!(!&)*1 7% $! #)=/$!*)." 78/" +1#%!/ 7% I%*+) #.)* 7% *;(%
.+7)"!)+% J!((%$1 1G!$%=%"* IN4K #.)* 7% *;(% ')1+!+&')?/% !0%& 7%# 23456 %#*
1G!$%=%"* 7)#&/*1% 7!"# &% &'!()*+%D
".//0123
!" #$%&'() *+,$+()-(./) ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! 01
!"!" #$%&%'%() *) #)$'%(+&) ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ,-
!"!. #$%&%'%() */0()'&1%2 3$4%$)5')2& ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ,6
!2 34)(*56 %( 7(/+8 79: ;7'*-(9:+*.)8/8$.< ! ! ! ! ! ! ! ! ! ! ! 00
!1 34)(*56 %( 7(/+8 =84+*+-=8>5() ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ?
!"! #$%&'() *+,$+()-(./) 01
!" #$%&'() *+,$+()-(./)
!"# $! #%&'()" *+,+-. ")/# !0)"# 123#%"'3 ")'2% #4#'56% 789:; #/2 /#!<% !0%&
$% 12)')&)$% 9=>+ !"# &%''% #%&'()". ")/# &)"'("/)"# ? %@1$(A/%2 &)66%"' ")/#
1)/0)"# #(6/$%2 A/%$A/%# !/'2%# 12)')&)$%# '%$# A/% $% 12)')&)$% B% >%2&)$!'%. $%
12)')&)$% BC7$%&'()" =2D)2%#&%"'+ E)# !6D('()"# #)"' BC3&$!(2&(2 $! 2%$!'()" %"'2%
9:; %' 89:;. %' %"'2% 89:; %' 789:; !0%& $C)DF%&'(G B% 23!$(#%2 $% 12%6(%2 1!#
B% $! '2!"#G)26!'()" !/')6!'(A/%+
>)/2 &H!A/% 12)D$56%. ")/# !$$)"# &)"#'2/(2% 1!# ? 1!# B%# #4#'56%# 23I3&2('/2%
B% $! 6!"(52% #/(0!"'% J
-+ 2$.)/+3-/4$. %53. 678 -'*))493(+
,+ 6+*.):$+;*/4$. %3 678 <(+) 3. =678 K;%&'()" *+-+-L J
M 9)/' BC!D)2B. &H!A/% 25<$% ")26!$% B% 9:; α→ β #%2! &)"0%2'(% 0%2# /"%
25<$% 89:; B% '41% J H(α)→ β !0%& α, β ∈ T (Fbin,X ) +
M >/(# ")/# !0)"# D%#)(" BC/" %"#%6D$% B% 25<$%# B% &(2&/$!'()" 1)/2 A/% B%#
25<$%# H(α)→ β #)(%"' !11$(&!D$%# J
H(a(x, y))→ a(H(x), y),
H(a(x, y))→ a(x,H(y))
M N" %"#%6D$% B% 25<$%# B% 1)("' O@% %#' &)"#'2/(' &(IB%##)/# J
F (x)→ x,
F (x)→ F (H(x))
P+ 6+*.):$+;*/4$. %3 =678 <(+) 3. >=678 J >)/2 &H!A/% 12)D$56%. !125#
$! '2!"#G)26!'()" %" <3"32!$ KQ)(2 &G+ $! #%&'()" *+,+PL. ")/# !$$)"# 123#%"'%2
B!"# &%''% #%&'()" /"% #)$/'()" #(61$(O3% 6!(# 1$/# 30(B%"'% %" 789:;+
*+ ?@/4;4)*/4$. K;%&'()" R+,L J E)/# )D#%20)"# A/% $%# 25<$%# B% &(2&/$!'()" #)"'
3$36%"'!(2%# ? B2)('% B% '41% S. !("#( %$$%# #)"' '2!"#G)263%# <2T&% ? $! 25<$%
#/(0!"'% J
H⋆(a(x, y))→ a(H⋆(x),H⋆(y))
S$ G!/' ")'%2 A/% $%# A/!'2% #4#'56%# B% 23I3&2('/2% #)"' $(#'3# &(IB%##/# KU)"#'2/&I
'()" BC/" 9:; &$!##(A/%. 92!"#G)26!'()" B/ 9:; 0%2# /" 89:;. 92!"#G)26!'()" B/
89:; 0%2# /" 789:;. %' V1'(6(#!'()"L &)66% )DF%&'(G B% &!$&/$%2 $C%"#%6D$% B%#
3'!'# !&&%##(D$%# B/ 6)B5$%+ E)/# B(#&/'%2)"# &)66%"' ")/# 1)/0)"# %@'2!(2% B%#
12)12(3'3# ("'32%##!"'%# ? 1!2'(2 B% $C%"#%6D$% B%# 3'!'# !&&%##(D$%# B!"# $% &H!1('2%
#/(0!"'+
!"!" #$%&%'%() *) #)$'%(+&)
N" 12)')&)$% B% >%2&)$!'% K123#%"'3 1!2 WX%#'%" -YYZ[L &)"'(%"' /" !2D2% B% 12)I
&%##/#. &H!&/" BC%/@ 1)##5B% /"% 3'(A/%''% $)&!$% !4!"' /"% 0!$%/2 1)##(D$% B!"# /"
! "#$%&'() *+ ,-./0&1$'&-2 %$( 34567
!"# !" # $%&'()*' +%+,+-* E0
'%&'()*' ./0,+12',,'& {p, n, u}! 3'& 4-*'25& p 672 n8 9'24'%, :,5' +%,'5950,0'& ;7(('
!"#$%$&' 672 ()*+%$&' 8< *- 4-*'25 u 9'2, :,5' +%,'5950,0' ;7((' $(,)-($ < 12+
+(9*+12' 12/= *- >%< '**' ;?-%@'5- &7+, '% p &7+, '% n!
A2 .0)2,< ,72,'& *'& B'2+**'& .-%& */-5)5' &7%, 950&'%,0'& ;7((' p(⊥,⊥) 72
n(⊥,⊥)< ', *'& -2,5'& 7%, *- 4-*'25 u! 3' ;7(975,'('%, .' ;' 957,7;7*' '&, .'
9-&&'5 = *- 5-;+%' .' */-5)5' *- 4-*'25 p &+ -2 (7+%& 2%' .'& B'2+**'& - *- 4-*'25 p<
72 n< &+ ,72,'& *'& B'2+**'& &7%, 0,+12',0'& 9-5 *- 4-*'25 n! C+ 2% 957;'&&2& %/- 9-&
'%;75' 2%' 4-*'25 .0>%+' (-+& ,72,'& *'& 0,+12',,'& .' &'& '%B-%,& &7%, .0>%+'&< -*75&
;' 957;'&&2& 0,-)*+, &- 4-*'25 = *- .+&D7%;,+7% .'& 4-*'25& .' &'& '%B-%,&!
E-5 ;7%&012'%,< %72& 97247%& 5'950&'%,'5 2%' ;7%>@25-,+7% .2 957,7;7*' .'
E'5;7*-,' ;7((' 2% -5)5' &25 Fbin = {p, n, u,⊥}! F% '%&'()*' +%+,+-* E0 ⊆ T (Fbin)
'&, .0;5+, .-%& *- >@25' !"!
"! !"#$%&'$(!" )*&" +,- './##(0&1 G +,- RPPG
u(n(x, y), n(z, t)) → n(n(x, y), n(z, t))
u(p(x, y), z) → p(p(x, y), z)
u(x, p(y, z)) → p(x, p(y, z))
H! +%/"#2!%3/$(!" )& +,- 41%# &" 5+,- G 5+,- ;755'&97%.-%, RPPλ G
# ./*0'# *)()1)'# 2 ./*0'# +3+(% 0' 45%) *+647' H(u(...))8
H(u(p(x, y), z)) → p(p(x, y), z) H(u(x, p(y, z))) → p(x, p(y, z))
H(u(n(x, y), n(z, t))) → n(n(x, y), n(z, t))
# ./*0'# ,' 4$1460+%$"( 2 ./*0'# 6($96':'(% +3+(% 0' 45%) *+647' H(u(x, y));<'# +6%1'# c.=.d H(p(x, y)) '% H(n(x, y)) (' #"(% !+# ()4'##+$1'# = *)()1'1
$4$8
H(u(x, y))→ u(H(x), y)H(u(x, y))→ u(x,H(y))
# ./*0'# ,' !"$(% ->'
Percolate(x) → x
Percolate(x) → Percolate(H(x))
I! +%/"#2!%3/$(!" )& 5+,- 41%# &" 65+,- G 65+,- ;755'&97%.-%,
RPPλe G
!"! #$%&'() *+,$+()-(./) 01
!"# !" # $%&'% ())%**+,-% )./0+%/0 *%1-%'%/0 2%* 30+41%00%* n5 p %0 ⊥
# !"#$% "&'&(&$%
H(u(x, y))→ n(Neg(x), Neg(y)) Neg(n(x, y)) → n(x, y)H(u(x, y))→ p(Pos(x), y) Pos(p(x, y))→ p(x, y)H(u(x, y))→ p(x, Pos(y))
# !"#$% )$ *+(*,#-.+/' 01##$% %/'. +)$'.+2,$% 3 *$##$% )$ 45 67
H(u(x, y)) → u(H(x), y)H(u(x, y)) → u(x,H(y))
# !"#$% )$ 8/+'. 9:$ 0;/,% .(-),+%/'% )$ *$##$ )$ 45 6 $' .(-+.-'. )$% *-%
8/%%+<#$% )$% %=></#$% -,?)$%%/,% )$ #- (-*+'$7
Percolate(u(x, y)) → u(x, y) Percolate(u(x, y)) → Percolate(H(u(x, y)))Percolate(p(x, y)) → p(x, y) Percolate(n(x, y)) → n(x, y)
6! 23/454)*/4$. 7 8%* &9:-%* 2% )+&)1-(0+./ *./0 .;0+'+*3%* ).''% -( *1+0% 7
H⋆(a(x, y))→ a(H⋆(x),H⋆(y))!
$<= RPP5 *./ >$<= RPP
λ %0 *./ ?>$<= RPPλe ).&&%*;./2(/0 @.1&/+**%/0 -%
'A'% &3*1-0(0 E1 ⊆ T (Fbin) B ;(&0+& 2% -C%/*%',-% +/+0+(- E0 7
E1 = RPP ⋆(E0) = R
PPλ
⋆(Percolate(E0)) = R
PPλe
⋆(Percolate(E0)).
67)8'/*/) (937+45(./*89 : 1' ,.+#+%-'. '/.($ /,.+# 0)&.-+##& -, *@-8+.($ A7B '/,%
%/>>$% *-8-<#$% )$ .(-+.$( #$ 8(/./*/#$ CC -=-'. 25008(/*$%%,% $' 0, 396 %$*/')$D
D.1* *.''%* +/03&%**3* 1/+41%'%/0 B 1/ %/*%',-% 2% ;&.;&+303* )&9'+$%5 (;E
;%-3 E′1! ?/ @(+0 30(/0 1/ *.1* %/*%',-% 2% E15 -C%/*%',-% E′
1 )./0+%/0 *%1-%'%/0
2%* 30+41%00%* n5 p %0 ⊥ FG.+& 1/ 0%- 0%&'% 2(/* -( H:1&% !"I! G.+& )@! 2%* &9:-%*
2C+/J(&+(/)% %/ ?>$<= 2(/* -% )K(;+0&% *1+J(/0!
!"!# $%&'&(&)* +,-)*('.&/ 0%1&%*2(*/'
8% ;&.0.).-% 2C?-%)0+./ L&,.&%*)%/0 FML,21--( "NN OI %*0 1/ ;&.0.).-% 2C3-%)0+./
2C1/ -%(2%&! P- )./0+%/0 1/ %/*%',-% 2% ;&.)%**1* ).''% -% ;&.0.).-% 2% Q%&).-(0%5
23/.03 ;(& -%* @%1+--%*! RK()1/ 2C%1S 23)+2% 2C(,.&2 *C+- %*0 )(/2+2(0 .1 /./! 8%
;&.)%**1* 2C3-%)0+./ *% 23&.1-% %/ 2%1S 30(;%*!
! "#$%&'() !* +,-./&0$'&,1 %$( 23456
!"# !" # $%&'()*' +%+,+-* E”1 .' *- .'/0+1(' 2,-3'
4- 35'(+15' 2,-3' &' 67(37&' .'& %7'/.& +%,25+'/5& 8-+&-%, /%' 5'9/:,' ; *'/5
%7'/.& .<'%8-%, 37/5 &-=7+5 &+ -/ (7+%& /% .<'/0 '&, 6-%.+.-,2! >-%& /% ,'* 6-&? *'
%7'/. +%,25+'/5 .'=+'%, /% 6-%.+.-, ', &'5- 2,+9/',2 p? &+%7% +* &'5- 2,+9/',2 n!
@+(+*-+5'('%, -/ *' 357,767*' A'567*-,'? -/ .2)/,? ,7/,'& *'& 8'/+**'& .-%& *<-5)5'
&7%, 352&'%,2'& 67((' p(⊥,⊥) 7/ n(⊥,⊥)? *'& -/,5'& 7%, *- =-*'/5 u! 4<7)B'6,+8 .'
*- 35'(+15' 2,-3' '&, .' 3573-C'5 ; *- 5-6+%' .' *<-5)5' *- =-*'/5 p &+ -/ (7+%& /%'
.'& 8'/+**'& '&, 2,+9/',2' 3-5 p? ', n? &+ ,7/,'& *'& 8'/+**'& &7%, 2,+9/',2'& 3-5 n! @+ /%
3576'&&/& %<- 3-& '%675' /%' =-*'/5 .2D%+' 9/' ,7/,'& *'& 2,+9/',,'& .' &'& '%8-%,&
&7%, .2D%+'&? -*75& 6' 3576'&&/& 2,-)*+, &- =-*'/5 ; *- .+&B7%6,+7% .'& =-*'/5& .' &'&
'%8-%,&!
4- .'/0+1(' 2,-3' '&, *- 35762./5' .<2*'6,+7% 52'**'! >-%& *' 6-& 7E *- 5-6+%' -
*- =-*'/5 n? %7/& %<-=7%& 3-& .' 6-%.+.-, ', .7%6 +(37&&+)*' .' 67%,+%/'5 *<2*'6,+7%!
>-%& *' 6-& 67%,5-+5'? +* F - -/ (7+%& /% 6-%.+.-,!
G 3-5,+5 .' *<'%&'()*' .' ,'5('& !"#$% ,57/=2 -351& *- 35'(+15' 2,-3' HI7+5 *-
DC/5' !JK? %7/& 3523-57%& 37/5 *- .'/0+1(' 2,-3' '% (-59/-%, *- 5-6+%' 3-5 /%'
2,+9/',,' c HI7+5 E”1 .-%& *- DC/5' !"K!
G+%&+? *- 5-6+%' 6L7+&+, /% 6-%.+.-, '% 8-+&-%, /% 6L7+0 %7%M.2,'5(+%+&,' 3-5(+
&'& '%8-%,& 2,+9/',2& 3-5 p! N% %7'/. +%,25+'/5 9/+ 2,-+, &2*'6,+7%%2 H%7,2 3-5 /%'
2,+9/',,' cK? 6L7+&+, /% .' &'& '%8-%,& 9/+ - .26*-52 */+M(:(' /% 6-%.+.-,!
O! ",10'(78'&,1 -971 456 8/$00&:7) P 456 RLEPP
c(p(x, y), p(z, t)) → c(c(x, y), p(z, t)) c(p(x, y), p(z, t)) → c(p(x, y), c(z, t))c(p(x, y), n(z, t)) → c(c(x, y), n(z, t)) c(n(x, y), p(z, t)) → c(n(x, y), c(z, t))
J! 4($10;,(<$'&,1 -7 456 =)(0 71 3456 P 3456 6755'&37%.-%, RLEPλ P
# &'()*% (!#!+!*%
H(c(p(x, y), p(z, t))) → c(c(x, y), p(z, t)) H(c(p(x, y), p(z, t))) → c(p(x, y), c(z, t))H(c(p(x, y), n(z, t))) → c(c(x, y), n(z, t)) H(c(n(x, y), p(z, t))) → c(n(x, y), c(z, t))
# &'()*% * ,$+,-)./$0# 1 &'()*% -#$2-*3*#/ .4.#/ )* ,5/! (.-,6* H(c(x, y))78*% .-/+*% c.9.d H(p(x, y)) */ H(n(x, y)) #* %0#/ :.% #!,*%%.$+*% 9 (!#!+*+
$,$;
H(c(x, y)) → c(H(x), y)H(c(x, y)) → c(x,H(y))
# &'()*% * :0$#/ "<*
Election(x) → x
Election(x) → Election(H(x))
!"! #$%&'() *+,$+()-(./) 01
!"# !" # $%&'%( )*)+, -+ ./%'0+ 1% c
2! 2+*.)3$+4*/5$. %6 7289 :(+) 6. ;7289 3 ;7289 .4&&%(54+1)+,
RLEPλe 3
# !"#$% "&'&(&$%
H(c(x, y)) → c(Elec(x), Neg(y)) H(c(x, y)) → c(Elec(x), Pos(y))H(c(x, y)) → c(Pos(x), Elec(y)) H(c(x, y)) → c(Neg(x), Elec(y))Elec(p(x, y)) → c(x, y) Neg(n(x, y)) → n(x, y)Pos(p(x, y)) → p(x, y) Elec(⊥) → ⊥Pos(⊥) → ⊥ Neg(⊥) → ⊥
# !"#$% )$ *+(*,#-.+/' 01##$% %/'. +)$'.+2,$% 3 *$##$% ), 45 67
H(c(x, y)) → c(H(x), y)H(c(x, y)) → c(x,H(y))
# !"#$% )$ 8/+'. 9:$ 0;/,% .(-),+%/'% )$ *$##$ ), 45 6 $' .(-+.-'. )$% *-%
8/%%+<#$% )$% %=></#$% -,?)$%%/,% )$ #- (-*+'$7
Election(c(x, y)) → c(x, y) Election(c(x, y)) → Election(H(c(x, y)))Election(b(x, y)) → b(x, y) Election(b(x, y)) → Election(H(b(x, y)))
"! <=/545)*/5$. 3 6%( &789%( 1% .0&.-9),04+ (4+, 45,0'0(:%( .4''% 9) (-0,% 3
H⋆(a(x, y))→ a(H⋆(x),H⋆(y))!
6% $;<RLEP= (4+ >$;<RLEP
λ %, (4+ ?>$;<RLEPλe .4&&%(54+1)+, @4-&+0((%+,
9% 'A'% &:(-9,), E2 B 5)&,0& 1% 9C%+(%'D9% 0+0,0)9 E”1 3
E2 = RLEP ⋆(E”
1) = RLEPλ
⋆(Election(E”
1)) = RLEPλe
⋆(Election(E”
1)).
8>)6'/*/) (?=>+54(./*6? @ 1' ,.+#+%-'. '/.($ /,.+# 0)&.-+##& -, *@-8+.($ A7B '/,%
%/>>$% *-8-<#$% )$ .(-+.$( #$ 8(/./*/#$ C1D -=-'. 24008(/*$%%,% $' 138, 699 %$*/'?
)$%E
E 5)&,0& 1- &:(-9,), 1% 9C%+(%'D9% E2= +4-( (4''%( 0+,:&%((:( 5)& 9C%+(%'D9% 1%
5&45&0:,:( )*)+, -+ ./%'0+ .4+,%+)+, -+0F-%'%+, c 1% 9) &).0+% B 9) @%-099% GH40& 9)
I8-&% !"J! H40& .@! 1%( &789%( 1C0+K)&0)+.% %+ ?>$;< 1)+( 9% ./)50,&% (-0K)+,!
!"#$%&'( )* +,-./%0#&%,1 $#' 23456
!" #$%&'() *& +&,-. +/0 1+2'3&/0-'4%.,.546
!"# $%&&% #%$&'(") "(*# +,-#%"&("# .*%/.*%# $(0!1%# 2#('& 0% &3+% $/!##'.*%
#('& 0% &3+% 4'-,!,$4'.*%5 +(*, #'6*/%, *" ,-#%!* 0% 7%&,' 8(,"- -/-6%"&!',%9 :(*#
6("&,("# -1!/%6%"& $(66%"& "(*# +(*;("# $!/$*/%, 0%# #*$$%##%*,# 0<*" 6!,.*!1%
0(""- !'"#' .*% /<%"#%68/% 0% 6!,.*!1%# !$$%##'8/%# = +!,&', 0<*" 6!,.*!1% '"'&'!/
0(""- +!, "(# #3#&>6%#9
?(*& 0<!8(,0) "(*# 0%;("# -;(.*%, /% $(0!1% 0<*" %"&'%, +(*, #'6*/%, *" "(68,%
0% @%&(" 0!"# *"% +/!$% 0<*" ,-#%!* 0% 7%&,'9
5(7#'89(0 ! "#$%& '#!( )*( +,-(& .! /%$)$(* )*( %*01*( %*)( 2/* 0 3./0 450.&
s(⊥, 0) 3./0 1& s(⊥, s(⊥, 0)) 3./0 2& *% #$!($ '* (/$%*6 7* 8*%%* 1#!$90*& .! 3*/%
($1/)*0 /! *!(*1:)* $!;!$ '<*!%$*0(6 =*3*!'#!%& !./( 305"50.!( /%$)$(*0 '*( %*01*(
'#!( Fbin 2/$ 8.!%$*!% /! (*/) (>1:.)* '* 8.!(%#!%* ⊥ #?*8 '*( (>1:.)*( :$!#$0*(
%*)( 2/* zero, one, two, three...6
A" 0-&!'/) /%# %"&'%,# 0, 1, 2... #("& ,%+,-#%"&-# +!, zero(⊥,⊥), one(⊥,⊥)) %&$9
7(*, +/*# 0% #'6+/'$'&-) "(*# #*++(#("# .*% &(*# /%# $(0!1%# %" AB?CD +%*;%"&
0-#(,6!'# !;(', +/*#'%*,# $("#&!"&%#9 ("$ *"% ,>1/% TestChange(zero(x, y)) →
une(x, y) +%*& E&,% ,%+,-#%"&-% +!, TestChange(0)→ 19
F'"#') 0<*"% +!,&) +(*, 6(0'G%, /% "(68,% 0% @%&("# 0!"# *"% +/!$%) "(*# *&'/H
'#("# 0%*I ,>1/%# +,'6'&';%# Minus, P lus /'#&-%# $'H0%##(*# JK 7(*, 0%# ,-#%!*I 0% 7%&,' 0#%10 J
Plus(0) → 1Minus(1) → 0
K 7(*, 0%# ,-#%!*I 0% 7%&,' 1,1 0#%10 J
Plus(1) → 2...Minus(2) → 1...
<!*&,% +!,&) +(*, #%*/%6%"& &%#&%, *" %"&'%,) "(*# *&'/'#("# #(*;%"& 0%*I ,>1/%#
0% Test /'#&-%# $'H0%##(*# JK 7(*, 0%# ,-#%!*I 0% 7%&,' 0#%10 J
Test0(0) → 0Test1(1) → 1
K 7(*, 0%# ,-#%!*I 0% 7%&,' 1,1 0#%10 J
Test2(2) → 2Test3(3) → 3
:(*# !//("# +,-#%"&%, &,('# $(0!1%# 0<AB?CD +(*, /%# ,-#%!*I 0% 7%&,' 8(,"-#
-/-6%"&!',%# R =< S, T, Pre, Post,m0 >9 L<'0-% +,'"$'+!/% %#&
M9 0% $(0%, *" 6!,.*!1% %" *" &%,6% 0!"# T (Fbin)9 F* 0-8*&) /% 6!,.*!1% '"'&'!/
m0 %#& ,%+,-#%"&- +!, *" &%,6% '"'&'!/ tm0
N9 %"#*'&% 0% $("#&,*',% *" %"#%68/% 0% ,>1/%# Rtλe +(*, $4!.*% &,!"#'&'(" t ∈ T 9
F'"#') *" O,!"$4' 0%# &,!"#'&'("# 0* 6!,.*!1% m = *" !*&,% m′%#& 6(0-/'#-
+!, /! ,-H-$,'&*,% 0* tm !* tm′ %" *&'/'#!"& 0%# ,>1/%# Rtλe 9
!"! #$%&'() *& +&,-. +/0 1+2'3&/0-'4%.,.546 78
92:5-.,;<& 7 !"#$" %& '!()$!*&+ !##&++,-"&+ ReachRTλe(tm0)
+-$354*.,.54 = ./ 0&('& ,/,0,!" tm0 ∈ T (Fbin)1
RTλe $/ &/+&'-"& %& (2*"&+ %345678+ 0&" )$& R
Tλe =
⋃
ti∈T
Rtiλe
! res1 ← ∅ 9
"! res2 ← tm0 9
#! ,'4,>(& res1 &= res2 ?'.-&
$! res1 ← res2 9
%! @5(- α ∈ res2 ?'.-&
&! res2+← (RTλe)
⋆({Transti(α)|ti ∈ T})
'! A4 @5(-
(! A4 ,'4>(&
)! -&,5(-4&- res2
4/:/1 /;$+ #!"#$";/+ "3&/+&'-"& %& '!()$!*&+ !##&++,-"&+ Reach[R,m0] <!(
45678 #;''& %!/+ "3!"*;(,0='& >?
@;$+ #;''&/A;/+ <!( $/ #;%!*& /!BC D ;%!*& EF &0 &/+$,0& %&$G !$0(&+
D ;%!*&+ H &0 IF &/ <!(0!*&!/0 %&+ %;//J&+ ,/0&('J%,!,(&+ &0 &/ <(;:0!/0 "& #!"K
#$" -!+J +$( "& #!#=& %& ',&$G &/ ',&$G?
!"#$% &
;/+,%J(;/+ S1 "3&/+&'-"& %& @ <"!#&+ %3$/ (J+&!$ %& L&0(,? L;$( <"$+ %& ",+,-,",0J
&0 +!/+ <&(0& %& *J/J(!",0J1 /;$+ +$<<;+;/+ )$3&""&+ +;/0 /;''J&+ p0, p1, ..., pN−1?
B$A4.,.54 CD !"#$% &'
(!)* +, -./%#+ "% 0%*-) 1!-,. .2.3%,*#)-% < S, T, Pre, Post,m0 >'
!,/*-+)/!,/ +, *%-3% tm0 "#,/ T (Fbin) 4!33% +,% 2)/*% "%/ 52#4%/ %* 2%+-/
3#-6+#$%/ %, 4!,/%-7#,* 2% ,!3 "% 48#6+% 52#4%
tm0 = p0(v0, p1(v1, ...pN−1(vN−1,⊥)))
*%2 6+% 5!+- *!+* pi ∈ S, i = 0..N − 1 9
: vi = 1 /) pi ∈ m0
: vi = 0 /),!,'
!,/*-+)/!,/ +, %,/%312% "% -;$2%/ "% -.<.4-)*+-% Rtλe 5!+- 48#6+% *-#,/)*)!,
t ∈ T ' =!+/ 4!,/)".-!,/ 6+#*-% 4#/ "% p 9: p &∈ Pre(t) %* p &∈ Post(t) 9
Transt(p(x, y)) → p(x, Transt(y)) !"#$
: p &∈ Pre(t) %* p ∈ Post(t) 9
Transt(p(x, y)) → p(Plus(x), T ranst(y)) !"%$
: p ∈ Pre(t) %* p &∈ Post(t) 9
Transt(p(x, y)) → p(Minus(x), T ranst(y)) !"&$
! "#$%&'() *+ ,-./0&1$'&-2 %$( 34567
p ∈ Pre(t) !" p ∈ Post(t) # $%&' ()*+,%-' .&! /! 01*.&12! !'" '&3)*+!&*! 4
5)*%6
Transt(p(x, y)) → p(Test1(x), T ranst(y)) !"#$
Transt(p(x, y)) → p(Test2(x), T ranst(y)) !"%$
... !"!$
$%&' 1(%-' )21/!0!-" 7!'%+- &-! *82/! 3%&* /1 "!*0+-1+'%- 9! /1 /+'"! 9! 3/1:!' #
Transt(⊥)→ ⊥6 ;+-'+< /! :=1-2!0!-" 9>)"1" 9! m0 4 m1 !- &"+/+'1-" /1 "*1-'+"+%-
t !'" 0%9)/+') 31*
tm1 ∈ (Rtλe)
⋆({Transt(tm0)})
!"# $%%!&# '%%"#()*) +* +!,$-* .$) "& */*0.%*1
38)9%0) :; ?%912! @6 A%+" &- *)'!1& 9! B!"*+ 7%*-) 9& B*%7/80! 4 C 3=+/%'%3=!'6
A%+" S &- !-'!07/! 9! @C 3/1:!' #
S = {Idle0,WaitLeft0, HasLeft0,WaitRight0, HasRight0, Fork0,
Idle1,WaitLeft1, HasLeft1,WaitRight1, HasRight1, Fork1}
A%+" m0 = {Idle0, Fork0, Idle1, Fork1}D- "!*0! tm0 !'" 9%-: *!3*)'!-") :%00! '&+" EF%+* /1 ,2&*! G6H61I #
idle0(1, waitleft0(0, hasleft0(0, waitright0(0, hasright0(0, fork0(1,
idle1(1, waitleft1(0, hasleft1(0, waitright1(0, hasright1(0, fork1(1,⊥))))))))))
A%+" &-! "*1-'+"+%- GoEat0 "!//! .&! Pre(GoEat0) = {Idle0} !" Post(GoEat0) =
{WaitLeft0,WaitRight0} J" &-! 1&"*! GoLeft0 "!//! .&! Pre(GoLeft0) = {Fork1,WaitLeft0}
!" Post(GoLeft0) = {HasLeft0}
J//!' (%-" K"*! 3*%9&+"!' 31* /!' *82/!' 9! *)L):*+"&*! G6@ 4 G6H :+L9!''&'6 ;/%*'
m1 = m0 − Pre(GoEat0) + Post(GoEat0) '!*1 :1/:&/) 31* &- 31' 9! *)L):*+"&*! #
tm1 = idle0(0,waitleft0(1, hasleft0(0,waitright0(1, hasright0(0, fork0(1,
idle1(1, waitleft1(0, hasleft1(0, waitright1(0, hasright1(0, fork1(1,⊥))))))))))
;/%*' m2 = m1 −Pre(GoLeft0) +Post(GoLeft0) '!*1 :1/:&/) 31* &- 31' 9! *)L):*+"&*! #
tm2 = idle0(0, waitleft0(0,hasleft0(1,waitright0(1, hasright0(0, fork0(1,
idle1(1, waitleft1(0, hasleft1(0, waitright1(0, hasright1(0, fork1(0,⊥))))))))))
6/1<0'$'1 )8%/(&9)2'$<8 ; J- &"+/+'1-" -%"*! %&"+/ E9)"1+//) 1& :=13+"*! MI< :!
:%912! !'" :1317/! 9! "*1+"!* /! 3*%7/80! 1N1-" M 3=+/%'%3=!' !- 2.666, 321 '!:%-9!'6
!"! #$%&'() *& +&,-. +/0 1+2'3&/0-'4%.,.546 78
+5.4,% 95-,% *( 35*':& *& -$%&'( *& +&,-. 8 !"#$%& '()!$% * )+,)%-+!./
+5.4,% 9'.;2&% *( 35*':& *& -$%&'( *& +&,-. 8 0+12 +32%.-+,2 41% )%55%
25.1)51.% 6% 6+,,7%2 ,8%25 #(2 )+"#()5%& $8(-(,5(9% 6%2 :;<= 2 ,8%25 6+,) #(2
15!$!27/
!"#$ %& '&(#)* +),' %&-,"%%". *),# +'/#"*$)*# %" 0)1&2" #,(3&*$. 0)1&2" 45
!"#$% &
6, 1/7,$. *),# #,++)#)*# -," $),$"# %"# +%&0"# 1&*# S #)*$ *)88/"# 1" p0 9pN−1 0)88" 1&*# %" )1&2" :5 ;),$ 1!&7)'1. *),# 1(#0,$)*# 1"# '<2%"# +'(8($(3"#
1, 0)1&2" 4 Postp "$ Prep +),' ,*" +%&0" +5 Postp *),# +"'8"$ 1!&=),$"' ,* ="$)*1&*# %& +%&0" p -,&*1 Prep +"'8"$ 1" '"$('"' ,* ="$)*# 1" %& +%&0" p5 >"# 1",?+",3"*$ /2&%"8"*$ 1/+%&0"' 9 %& +%&0" #,(3&*$" #( %& +%&0" 0),'&*$" *!"#$ +&# p5
Postp(p(x, y)) → p(Plus(x), y)Prep(p(x, y)) → p(Minus(x), y)
<$=4.,.54 8> >+6(9% ?/
+!5 1, .72%(1 6% @%5.! 3+.,7 7$7"%,5(!.% < S, T, Pre, Post,m0 >/
>+,25.1!2+,2 1, 5%."% tm0 6(,2 T (Fbin) )+""% 1,% $!25% 6%2 #$()%2 %5 $%1.2
"(.41(9%2 %, )+,2%.-(,5 $% ,+" 6% )A(41% #$()%
tm0 = p0(v0, p1(v1, ...pN−1(vN−1,⊥)))
5%$ 41% #+1. 5+15 pi ∈ S, i = 0..N − 1 B
C vi = 1 2! pi ∈ m0
C vi = 0 2!,+,/
>+,25.1!2+,2 1, %,2%"3$% 6% .D9$%2 6% .7E7).!51.% #+1. )A(41% 5.(,2!5!+, t ∈ T
%, )+""%,F(,5 #(. 1,% .D9$% 6% )+"#+2!5!+, #.!,)!#($% 6% $( '+."%
Transt(x)→ Post0(Pre0(x))
(-%) Pre0 1,% .D9$% 6% )+"#+2!5!+, 6% 2+1.)% Pre(t) %5 Post0 )%$$% 6% 6%25!,(5!+,
Post(t)/
C >A(41% p ∈ Pre(t) %25 !,5%.#.757% #(.
Prei(x)→ Prep(Prei+1(x)), 0 ≤ i <‖ Pre(t) ‖ %5 Pre‖Pre(t)‖(x)→ x/
C >A(41% p ∈ Post(t) %25 !,5%.#.757% #(.
Posti(x)→ Postp(Posti+1(x)), 0 ≤ i <‖ Post(t) ‖ %5 Post‖Post(t)‖(x)→ x
G%2 .D9$%2 #.!"!5!-%2 Postp %5 Prep #+1. 1,% #$()% #/ Postp ,+12 #%."%5 68(E
H+15%. 1, H%5+, 6(,2 $( #$()% p 41(,6 Prep #%."%5 6% .%5!.%. 1, H%5+, 6% $( #$()% p/
I%2 6%1J #%1-%,5 79($%"%,5 2(15%. * $( #$()% 21!-(,5% 2! $( #$()% ()51%$$% ,8%25 #(2
p/
C >A(41% #$()% p B Prep(p(x, y))→ p(Minus(x), y)/
C @+1. 5+12 $%2 p′ &= p B Prep(p′(x, y))→ p′(x, Prep(y))/
C >A(41% #$()% p B Postp(p(x, y))→ p(Plus(x), y)/
! "#$%&'() *+ ,-./0&1$'&-2 %$( 34567
!"# $!"% &'% p′ �= p ( Postp(p′(x, y))→ p′(x, Postp(y)))
*+,%+- &' ./0,1'2',$ 345$0$ 3' m0 6 m1 "$+&+%0,$ &0 $#0,%+$+!, t '%$ 2!35&+%5 70#
tm1 ∈ (Rtλe)
⋆({Transt(tm0)})
! "#$% &'%() *$( %'$% (+% ,-&-)- (% #..!/*$- 0 .#)%/) 1( !# )#2/&(3 4'$+ #!!'&+
/!!$+%)() 2( 2'1#,( .#) $& (5(6.!(3
!"#$%" &' 8!301' 9) :!+$ &' #5%'0" 3' '$#+ 3' &4';'27&' 9<) :!+$ S ", ',%'2=&'
3' >9 7&0.'% (
S = {Idle0,WaitLeft0, HasLeft0,WaitRight0, HasRight0, Fork0,
Idle1,WaitLeft1, HasLeft1,WaitRight1, HasRight1, Fork1}
:!+$ m0 = {Idle0, Fork0, Idle1, Fork1} ?, $'#2' tm0 '%$ 3!,. #'7#5%',$5 .!22'
&' %"+@0,$ AB!+# &0 C1"#' D)E)0F (
idle0(1, waitleft0(0, hasleft0(0, waitright0(0, hasright0(0, fork0(1,
idle1(1, waitleft1(0, hasleft1(0, waitright1(0, hasright1(0, fork1(1,⊥))))))))))
:!+$ ",' $#0,%+$+!, GoEat0 $'&&' G"' Pre(GoEat0) = {Idle0} '$ Post(GoEat0) ={WaitLeft0,WaitRight0} *&!#% m1 = m0 − Pre(GoEat0) + Post(GoEat0) %'#0 .0&."&5 70# ",
70% 3' #5H5.#+$"#' 0@'. ",' #I1&' 3' .!27!%+$+!, 3' %!"#.' Pre(GoEat0)- c.6.d. J&&'@0 K$#' 7#!3"+$' 70# &'% #I1&'% 3' #5H5.#+$"#' +,3+G"5'% .+H3'%%!"%)
RPre(GoEat0)λe =
8
<
:
GoEat0Pre0(x)→ PreIdle0 (x)PreIdle0 (p(x, y)) → p(x, PreIdle0 (y))∀p ∈ S, p %= idle0PreIdle0 (idle0(x, y))→ idle0(Minus(x), y)
9
=
;
.
'$ .'&&' 3' 3'%$+,0$+!, Post(GoEat0)- c.6.d.
RPost(GoEat0)λe =
8
>
>
>
>
<
>
>
>
>
:
GoEat0Post0(x)→ PostWaitLeft0 (PostWaitRight0 (x))PostWaitLeft0 (p(x, y)) → p(x, PostWaitLeft0 (y)) ∀p ∈ S, p %= waitleft0PostWaitLeft0 (waitleft0(x, y)) → waitleft0(Plus(x), y)PostWaitRight0 (p(x, y))→ p(x, PostWaitRight0 (y)) ∀p %= waitright0PostWaitRight0 (waitright0(x, y))→ waitright0(Plus(x), y)
9
>
>
>
>
=
>
>
>
>
;
.
*+,%+- ,!"% !=$',!,%
tm1 = idle0(0,waitleft0(1, hasleft0(0,waitright0(1, hasright0(0, fork0(1,
idle1(1, waitleft1(0, hasleft1(0, waitright1(0, hasright1(0, fork1(1,⊥))))))))))
J$ ",' 0"$#' GoLeft0 $'&&' G"' Pre(GoLeft0) = {Fork1,WaitLeft0} '$ Post(GoLeft0) ={HasLeft0} *&!#% m2 = m1 − Pre(GoLeft0) + Post(GoLeft0) %'#0 .0&."&5 70# ", 70% 3' #5H
5.#+$"#' 0@'. ",' #I1&' 3' .!27!%+$+!, 3' %!"#.' Pre(GoLeft0)- c.6.d. J&&' @0 K$#'7#!3"+$' 70# &'% #I1&'% #5H5.#+$"#' +,3+G"5'% .+H3'%%!"%)
RPre(GoLeft0)λe =
GoLeft0Pre0(x) → PreFork1 (PreWaitLeft0 (x))et ainsi de suite.
ff
.
'$ .'&&' 3' 3'%$+,0$+!, Post(GoLeft0)- c.6.d.
RPost(GoLeft0)λe =
GoLeft0Post0(x)→ PostHasLeft0 (x)et ainsi de suite.
ff
.
!"! #$%&'() *& +&,-. +/0 1+2'3&/0-'4%.,.546 78
idle
wait lef t
hasleft
wai t r ight
hasright
fork
1
1
1
1
1
1
1
0
0
0
0
1
idle
wait lef t
hasleft
wai t r ight
hasright
fork
0
0
0
0
0
0
1
0
0
0
0
1
p
p p
0 0 0 1
p
p p
? ? ? ?
p
p
p p
1 0 0 0
p
p p
0 1 1 0
p
p
b) Codage 3
a) Codages 1 et 2
...
...
Bin = 1011
Bin = 0110
Bin = 0101
Bin = 0000
!"# !" # $%&'() &*+, -./)'+ &) 0)1-2 3 4 5627%/%56)/ ), T (Fbin)
!"#$%&%"'( ")*+ ),'%")"+
tm2 = idle0(0, waitleft0(0,hasleft0(1,waitright0(1, hasright0(0, fork0(1,
idle1(1, waitleft1(0, hasleft1(0, waitright1(0, hasright1(0, fork1(0,⊥))))))))))
#$%(2,',% &)9$-.:&4,'() ; -" *'!$!+#"' ")'.% )*'!$ /01'#!$$1 #* 23#4!'.% 56( 2%
2)0#7% %+' 2#4#,$% 0% '.#!'%. $% 4.),$8&% #9#"' : 43!$)+)43%+ %" 63, 632 +%2)"0%+;
+5.4,% <5-,% *( 35*'=& *& -$%&'( *& +&,-. " <)*+ ),+%.=)"+ >*% $% 4#.'#7%
0%+ .87$%+ 4)*. $%+ )41.#'!)"+ 4.!&!'!=%+ 0%+ 4$#2%+ %+' ,!%" ).7#"!+1 0#"+ 2% 2)0#7%;
?.@2% A $# &!+% %" 2#23% 0%+ .87$%+ Pre %' Post( ")*+ 4.)B')"+ ,!%" 0% $C#=#"'#7%
0%+ - DEF+;
+5.4,% <'.>2&% *( 35*'=& *& -$%&'( *& +&,-. " <)*+ ),+%.=)"+ 17#$%&%"'
>*% $% 4#.'#7% 0%+ .87$%+ 4)*. $%+ '.#"+!'!)"+ "C%G!+'% 4#+ 0#"+ 2% 2)0#7%;
8%+/ 5-./),1%,/ 7) 9%&'() /+2:',1; 7) 9%&'() <! =',/ 9) 9%&'(); 7) 5'-1'() &)/
->(7)/ 5%+- 7)/ 1-',/212%,/ )/1 %-(',2/. ?2)+@ A+) 7) 9%&'() 5-.9.&),1!
! "#$%&'() *+ ,-./0&1$'&-2 %$( 34567
!"#$% &
! "#$!%& '!(()')*' +!, %)!%,' -,' (-./,' ".*' S 0)*% ", p0 1 pN−1 /)22, ".*'
-, /)".3, 4 ,% 56 789*",:, ", /;.+!, (-./, ,'% !%9-9'#, ()!< "#%,<29*,< !*, !*9+!,
()'9%9)* =!*, >,!9--,? ".*' -8.<$<,6 @)!' "#A*9'')*' Bin(p)|k /)22, -, /;9B<, 1 -.
()'9%9)* k ", Bin(p)6 C.< ,:,2(-,& .0,/ Bin(p) = 01..0& *)!' .0)*' Bin(p)|0 = 0&
Bin(p)|1 = 1 ,% Bin(p)|(h− 1) = 06 C.< /)*'#+!,*%& -. >,!9--, 0 ,'% <#',<0#, ()!< -.
(-./, p0 =Bin(0) = 00..0?&666 +!.*" -. >,!9--, N − 1 =.0,/ Bin(N − 1) = 11..1? ,'%
<#',<0#, ()!< -. (-./, pN−16 @)!' *8.0)*' ")*/ (-!' $,')9* ", /)*',<0,< -, *)2 ",
/;.+!, (-./,& /, +!9 <#"!9% -. %.9--, "! /)".3,6
8/92&'&-2 :; !"#$% &'
(!)* +, -./%#+ "% 0%*-) 1!-,. .2.3%,*#)-% < S, T, Pre, Post,m0 >' !,/*-+4
)/!,/ +, *%-3% t "#,/ T (Fbin) "% 5#+*%+- h = log2(N) *%2 6+% 7!+- 85#6+% i 9 /)
marquage = pi: #2!-/ i ∈ PosF (marquage): #;%8 marquage ∈ {0, 1} %* 0 ≤ i < N '
5#6+% *-#,/)*)!, t ∈ T "!)* <*-% 7-!"+)*% 7#- 2%/ -=$2%/ "% -.4.8-)*+-% 8!34
3%,>#,* 7#- +,% -=$2% 7-),8)7#2% 8!33% "#,/ 2% 8!"#$% ? 9
Transt(x)→ Post0(Pre0(x))
*%22% 6+% Pre0 7!+- 2@%,/%312% "% /!+-8% Pre(t) %* Post0 7!+- 2@%,/%312% "% "%/*)4
,#*)!, Post(t)'
A 5#6+% p ∈ Pre(t) %/* ),*%-7-.*.% 7#-
Prei(x)→ PrehBin(p)(Prei+1(x)), 0 ≤ i <‖ Pre(t) ‖'
A 5#6+% p ∈ Post(t) %/* ),*%-7-.*.% 7#-
Posti(x)→ PosthBin(p)(Posti+1(x)), 0 ≤ i <‖ Post(t) ‖'
B%/ -=$2%/ "% 8)-8+2#*)!, /!,* $.,.-.%/ %* #772)6+.%/ C 7#-*)- "% 2# -#8),% 9
A 5#6+% Preh: 7!+- 0 < k ≤ h
A PrekBin(p)(p(x, y)→ p(Prek−1
Bin(p)(x), y) /) Bin(p)|k = 0
A PrekBin(p)(p(x, y)→ p(x, Prek−1
Bin(p)(y)) /),!,'
D,E,: 6+#," ,!+/ #--);!,/ C 2# 7!/)*)!, "% 2# 72#8% ),")6+.% 9
Pre0(p(x, y))→ p(Minus(x), y)
Pre1(p(x, y))→ p(x,Minus(y))
A 5#6+% Posth: 7!+- 0 < k ≤ h
A PostkBin(p)(p(x, y)→ p(Postk−1
Bin(p)(x), y) /) Bin(p)|k = 0
A PostkBin(p)(p(x, y)→ p(x, Postk−1
Bin(p)(y)) /),!,'
D,E,: 2%/ !7.-#*)!,/ 7!+- #F!+*%- !+ -%*)-%- +, F%*!, "@+,% 72#8% 6+#," ,!+/
#--);!,/ C 2# 7!/)*)!, ),")6+.% /!,* 9
Post0(p(x, y))→ p(Plus(x), y)
Post1(p(x, y))→ p(x, P lus(y))
G),/): 2% 85#,$%3%,* "@.*#* "% m0 C m1 +*)2)/#,* 2# *-#,/)*)!, t %/* 3!".2)/. 7#-
tm1 ∈ (Rtλe)
⋆({Transt(tm0)})
D,' <E3-,' ", /9</!-.%9)* ,:93,*% ',!-,2,*% ", ",'/,*"<, F!'+!81 -. >,!9--, /)<G
<,'()*".*%, ", -. (-./, 9*"9+!#, ,% 2)"9A,< ')* 2.<+!.3,6 D, /,%%, 2.*9E<,& *)!'
()!0)*' (.<%.3,< ",' <E3-,' (<929%90,'6
!"! #$%&'() *& +&,-. +/0 1+2'3&/0-'4%.,.546 78
!"# $%%!&# '%%"#()*) +* +!,$-* .$) "& */*0.%*1
9)&:;2& " !"#$% &' (!)* +% ,-.%#/ "% 0%*,) "% +1%2%34+% 56' (!)* S /7 %7.%38+%"% 95 4+#:%. ;
S = {Idle0,WaitLeft0, HasLeft0,WaitRight0, HasRight0, Fork0,
Idle1,WaitLeft1, HasLeft1,WaitRight1, HasRight1, Fork1}
<!/. #=!7. 8%.!)7 "1/7 #,8,% "% >#/*%/, h = 4 4!/, :!"%, +1%7.%38+% S? [email protected]. 7!/.
/*)+).!7. 95 4,%3)A,%. B%/)++%. "% :%* #,8,% C/) .!7* ,-.%,=-%. 4!/, +%. 95 4+#:%. "%
S? "!7: +%. 6 "%,7)A,%. 7% .!7* 4#. /*)+).-.' (!)* m0 = {Idle0, Fork0, Idle1, Fork1}'
D7 *%,3% tm0 %.* "!7: ,%4,-.%7*- :!33% %7 E$/,% F'G'8'
(!)* /7% *,#7.)*)!7 GoEat0 *%++% C/% Pre(GoEat0) = {Idle0} %* Post(GoEat0) ={WaitLeft0,WaitRight0} H+!,. m1 = m0 − Pre(GoEat0) + Post(GoEat0) .%,# :#+:/+- 4#, /7
4#. "% ,-I-:,)*/,% #=%: /7% ,A$+% "% :!34!.)*)!7 "% .!/,:% Pre(GoEat0)? [email protected]. J++%=# K*,% 4,!"/)*% 4#, +%. ,A$+%. "% ,-I-:,)*/,% )7")C/-%. :)I"%..!/. ;
RλePre(GoEat0) =
8
>
>
>
>
<
>
>
>
>
:
GoEat0Pre0(x)→ Pre0000(x)Pre0000(p(x, y)) → p(Pre000(x), y)Pre000(p(x, y))→ p(Pre00(x), y)Pre00(p(x, y))→ p(Pre0(x), y)Pre0(p(x, y)) → p(Minus(x), y)
9
>
>
>
>
=
>
>
>
>
;
.
%* :%++%. "% "%.*)7#*)!7 Post(GoEat0)? [email protected].
RλePost(GoEat0) =
8
>
>
>
>
>
>
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
>
>
>
>
>
>
:
GoEat0Post0(x) → Post0001(Post0011(x))
Post0001(p(x, y)) → p(Post001(x), y)Post001(p(x, y)) → p(Post01(x), y)Post01(p(x, y))→ p(Post1(x), y)
Post0011(p(x, y)) → p(Post011(x), y)Post011(p(x, y)) → p(Post11(x), y)Post11(p(x, y))→ p(x, Post1(y))
9
>
>
>
>
>
>
>
>
>
>
>
>
>
>
=
>
>
>
>
>
>
>
>
>
>
>
>
>
>
;
.
L# ,A$+% Post1(p(x, y))→ p(x, P lus(y)) #=%: +%. ,A$+%. 4,)3)*)=%. .!7* 4#,*#$-%.'
#$%(2,',% &);$-.:&4,'() J7 /*)+).#7* 7!*,% !/*)+ M"-*#)++- #/ :>#4)*,% NO? :%
:!"#$% %.* :#4#8+% "% *,#)*%, +% 4,!8+A3% #P#7* N 4>)+!.!4>%. %7 454, 978 .%:!7"%.'
+5.4,% <5-,% *( 35*'=& *& -$%&'( *& +&,-. > Q#7. :% :!"#$%? 7!/. 7% ").*)7I
$/!7. 4#. +% 7!3 "%. 4+#:%.? 7!/. 71#=!7. 8%.!)7 C/% "1/7 .P38!+% p @ +# 4!.)*)!7
"/ 7!3 "% *!/*%. +%. 4+#:%.' %+# ,-"/)* +% 7!38,% "% ,A$+%. "% ,-I-:,)*/,% 4!/, +#
*,#)*%,' H/*,%3%7* ")*? 7!/. ,-"/).!7. +# *#)++% "/ :!"#$%' H)7.)? 7!/. #=!7. 8%.!)7
"1/7 #,8,% #P#7* /7% >#/*%/, log2(N) 4!/, :!"%, +# +).*% "%. 4+#:%. 7/3-,!*-%.'
! "#$%&'() !* +,-./&0$'&,1 %$( 23456
7,&1'0 3$&8/)0 -9 :,-$;) -) (.0)$9 -) 7)'(& < !""# "#$%&!$$' (&)*#++,+-
.!,/ (!*#/ ,$ #$+#"01# *# .12(#+ S- $!,+ 23!$+ 0#+!&$ *4,$ 2/0/# *# 52,%#,/ h =
log2(‖ S ‖)6 7,2$* N =‖ S ‖ $4#+% .2+ ,$ #$%&#/ *# 12 8!/"# 2n- $!,+ /&+9,!$+
*423!&/ ,$ :/2$* $!"0/# *# 8#,&11#+ $!$ ,%&1&+'#+- 9,& /'*,&+#$% 12 .#/8!/"2$(# *,
(!*2:#6 ;$# +!1,%&!$ /2&+!$$201# #+% *# *'(!".!+#/ N #$ +!""# *# .,&++2$(#+ *#
<6 =$+,&%#- $!,+ (!$+%/,&+!$+ ,$ (!*2:# *# /'+#2, *# >#%/& ? .!,/ (52(,$6 =$@$-
$!,+ 2:/':#!$+ (#+ (!*2:#+6
A42,%/# .2/%- +!&% ,$ (!*2:# (!$+%&%,' *# (!".!+2$%+ B2..#1'+ 1#+ +!,+)/'+#2,CD
&*#$%&9,#+6 =E#(%&3#"#$%- 1#+ +!,+)/'+#2,C &*#$%&9,#+ +!$% +!,3#$% (!*'+ *2$+ 1#+
#$*/!&%+ %!%21#"#$% *&E'/#$%+ *, %#/"#6 # (!*2:# $# ./!@%# *!$( .2+ *# 14232$%2:#
*#+ =FGHI+ *2$+ (# (2+)1J6
7(,%,0&'&,1 = I!&% ,$ /'+#2, *# >#%/& 0!/$' '1'"#$%2&/# R =<
S, T, Pre, Post,m0 >6 K1 #C&+%# *#+ (!*2:#+ *# /'+#2,C *# >#%/& L- < #% ? %#1+
9,# 1#+ "2/9,2:#+ 2((#++&01#+ *# (# /'+#2, +!$% (21(,1'+ .2/ 12 (1M%,/# %/2$+&%&3# R⋆λe
+,/ tm0 *# (#+ (!*2:#+6
7()9>) ?? @7()9>) 09::&1:')A
N, "!"#$% *!$$'- $!,+ .!,3!$+ ./!,3#/ 9,4,$# /O:1# #$ Rλe #+% 2..1&(201# +,/ tm++& 12 %/2$+&%&!$ (!//#+.!$*2$%# #$ T #+% 8/2$(5&++201# +,/ m6 =E#(%&3#"#$%- $!,+
.!,3!$+ ./!,3#/ 9,4&1 #C&+%# ,$# 8!$(%&!$ f *# 12 (1M%,/# %/2$+&%&3# R⋆λe +,/ tm0 J
14#$+#"01# *#+ "2/9,2:#+ 2((#++&01#+ *# (# /'+#2, >#%/&6
!"# $%%!&# '(")*+, )+# #!"#-,'#+$". *)+&(*/"+# 0!", $1'%*!,+, %$ 2$0$2*(' )+
1!)'%*#+, )+# ,'#+$". )+ 3+(,* )$&# %$ #+2(*!& #"*4$&(+5
!" #$%&'() *& +&,-. /.$-'-0/.1(&%
6& ,'#+$" )+ 3+(,* 7*',$,27*/"+ 89*&+, :;;;< 0+"( =(,+ )'>&* 0$, "&+ 2!10!#*(*!&
)+# ,'#+$". )+ 3+(,* ?!,&'# '%'1+&($*,+# !,)*&$*,+# @$00+%' #!"# ,'#+$". )$&# 2+((+
#+2(*!&A5
B.C1&'&,1 ?! H'+#2,C *# >#%/& 5&'/2/(5&9,#+6 ;$ /'+#2, *# >#%/& 5&'/2/(5&9,# #+%
*'@$& (!""# ,$ 9,2*/,.1#% < 2R, Tsync, P resync, Postsync > !P
Q 2R ,$ #$+#"01# *# +!,+ /'+#2,C *# >#%/& R =<
Slocal, Tlocal, P relocal, Postlocal,m0local >6
Q Tsync #+% ,$ #$+#"01# @$& *# %/2$+&%&!$+6
Q Presync #% Postsync +!$% *#,C 2..1&(2%&!$+ *# Tsync *2$+ 2R9,& J %!,%# %/2$)
+&%&!$ tsync *# Tsync 2++!(&#$% 1#+ *#,C '%2%+ Pre(tsync) #% Post(tsync) 9,& +!$%
/#+.#(%&3#"#$% 14!/&:&$# #% 1# 0,% *# 12 %/2$+&%&!$ tsync6
B% C$"( &!(+, /"+ %+# #!"# ,'#+$". 0+"4+&( =(,+ *)+&(*/"+#5 3$, 2!&#'/"+&(D %+
2!10!,(+1+&( )+# #!"# ,'#+$". #!&( &$(",+%%+1+&( #*1*%$*,+#5 !"# &!"# *&(',+##!&#
$". 2!)$E+# $F$&( "&*/"+1+&( )+# #!"# ,'#+$". *)+&(*/"+# (+%# /"+ %+ 0,!?%G1+ )+#
37*%!#!07+# !" %+ 0,!(!2!%+ )+ H%!((+)-I*&E5 JK$"(,+ 0$,(D &!"# '(")*!&# 'E$%+1+&(
!"! #$%&'() *& +&,-. /.$-'-0/.1(&% 23
! "#$%$&$ ! '! ($)*'+($,-* .)%!/ 0)- &$*%-!*% )* !*1!2, ! '! "#$&!11)1 -'!*%-0)!1
!% )*! #!11$)#&! "3#%345!6 7!%%! #!11$)#&! "!)% 8%#! &$*1-'5#5! &$22! )* 1$)1 #51!3)
'-95#!*% '!1 3)%#!16
456%,-(0,.56 *&% ,&-7&%
:3*1 3 "#!2-;#! 5%3"!< &$*1%#)-1$*1 )* %!#2! "$)# &$'!# ! *-=!3) '! 1$)1+
#51!3) -'!*%-0)! > local(u, v) 3=!& u !% v 1$*% !1 &$'34!1 ') &$'34! -'!*%-0)!6
:3*1 3 '!)/-;2! 5%3"!< &$*1%#)-1$*1 )* &$'34! -'!*%-0)! *-=!3) ? !* &$2"$13*%
'!)/ 1$)1+#51!3)/ -'!*%-0)!1 %! 1 0)!
sys(local(u, v), local(u, v))
@ "3#%-# '!1 &$'34! -'!*%-0)! '! *-=!3) ?< *$)1 "$)=$*1 &$*1%#)-#! ! &$'34!
-'!*%-0)! '! *-=!3) A '! 3 282! 23*-;#!< !% 3-*1- '! 1)-%!6
B)3*' - !/-1%! )* 1$)1 #51!3) local′(u′, v′) '-95#!*% 3)/ 3)%#!1 Cc.D.d.local(u, v)E<*$)1 $#43*-1$*1 &$22! ! 1)-=3*% >
globalsys(sys′(u′, v′), system)
3=!& ! &F%5 '#$-% system !1% ! &$'34! -'!*%-0)! '! *-=!3) *6
#&7'-1(&% 2 !"#$ % &' ()*+,!-$ ./ *0.1&$2 #3'4/$ ,$!*$ 5$/, 6,!$ #07(,!/-, $7
.+5&-'7, .$( (0/(8,$!*$( -.$7,-4/$( .$ 7-9$'/ 5&/( :'(;
.3-*%!*3*% - *$)1 #!1%! D &$*1%#)-#! '!1 #;4 !1 "$)# 0)G! !1 ")-11!*% 8%#! 3"+
" -0)5!1 1)# %$)% ! 1H1%;2! &$*1%#)-% "3# ! " -34! #5&)#1-I6
456%,-(0,.56 *&% -89:&% *& -$;$0-.,(-&
.3-*%!*3*% &$*1-'5#$*1 )*! %#3*1-%-$* !9!&%)5! 1!) !2!*% "$)# !1 " 3&!1 $+
&3 !1< *$)1 $*1 )* !*1!2, ! '! #;4 !1 '! &-#&) 3%-$* 0)- '!1&!*' '! 3 #3&-*!
J)10)G3) *-=!3) '! &K30)! 1$)1+#51!3)6 @ &! *-=!3) D< *$)1 #5+5&#-=$*1 >
Local(local(x, y)) → γ !"#$
$L γ !1% )*! &$2"$1-%-$* '! 0)! 0)!1 #;4 !1 "#-2-%-=!1 Postp 3*' Prep "$)#
0)! 0)!1 " 3&!16
:G3)%#! "3#%< - !/-1%! '!1 %#3*1-%-$*1 " )1 &$2" !/!1 C !1 %#3*1-%-$*1 1H*&K#$*!1E
c.D.d )*! %#3*1-%-$* !9!&%)5! "$)# '!1 " 3&!1 '3*1 3) 2$-*1 '!)/ 1$)1+#51!3)/ '-I+
I5#!*%1< "3# !/!2" ! '!)/ =$-1-*16 M$)1 $*1 '!1 -89:&% *& %<60/-56.%',.56
&6,-& *&() =5.%.6% CH &$2"#-1 '!)/ 1$)1 #51!3)/ I !% I + 1 3-*1- 0)! '!)/ 1$)1
#51!3)/ 0 !% N − 1E !* '!1&!*'3*% '! 3 #3&-*! J)10)G3) *-=!3) '! &K30)! 1$)1+
#51!3) >
H(sys(x, y))→ SyncII+1(sys(x, y)) !"%$
H(sys(x, y)) → Sync0N−1(sys(x, y)) !"&$
! "#$%&'() *+ ,-./0&1$'&-2 %$( 34567
!"# $%!&# '(#!)& *( +,-.( *( /)+/".$0)!& 1!"+ *(#/(&*+( *" &)%($" 2 $" &)%($"
234 *" /!*$-( 5
SyncII+1(sys(x, y))→ sys(SyncII+1(x), y) !"#$%
SyncII+1(sys(x, y))→ sys(x, SyncII+1(y)) !"##%
6")# &!"# .$&7!&# .(# +,-.(# *( #8&/9+!&)#$0)!& : /9$;"( &)%($" *" /!*$-( 5
SyncII+1(sys(x, y)) → sys(SyncI(x), SyncI+1(y)) !"#&%
Sync0N−1(sys(x, y)) → sys(Sync0(x), SyncN−1(y)) !"#'%
SyncI(sys(x, y)) → sys(x, SyncI(y)) !"#(%
SyncI+1(sys(x, y)) → sys(SyncI+1(x), y) !"#)%
Sync0(sys(x, y)) → sys(Sync0(x), y) !"#!%
SyncN−1(sys(x, y))→ sys(x, SyncN−1(y)) !"#*%
<)&$.(=(&0> $" &)%($" *?"& #!"#3+@#($" *" /!*$-( 5
Sync0(local(x, y))→ α0 !"#+%
SyncN−1(local(x, y))→ αN−1 !"#,%
SyncI(local(x, y)) → βI !"&$%
SyncI+1(local(x, y)) → βI+1 !"&#%
!A α, β #!&0 *(# /!=1!#)0)!&# *( ;"(.;"(# +,-.(# 1+)=)0)%(# Postp $&* Prep 1!"+
;"(.;"(# 1.$/(#B
38)9%0) :; !"#$%&' (') *+$")",*')- ./ 01+23 4/ )"4) !5)'14 161/2 7 ,$18')
{Idle,WaitLeft,HasLeft,WaitRight,HasRight, Fork} 19'8 )'4$'&'/2 : ;'2"/)(1/) $') ,$18') Idle '2 Fork ')2 8"(5 8+<('))"4) =
tlocal = local(idle(1, waitleft(0, hasleft(0, waitright(0, hasright(0, fork(1,⊥))))))
>+/)+3 4/ 8"(1?' (' /+9'14 @ ,"4! : ,*+$")",*') ')2 4/' 8"&,")+2+"/ (' ('4A )"4)<
!5)'14A +('/2+B4') ')2 8"(5 8"&&' )4+2 =
tsys1 = sys(tlocal, tlocal)
>+/)+3 4/ 8"(1?' /+9'14 : ,"4! C ,*+$")",*') ')2 8"&,")5 (' $1 &D&' &1/+%!' =
tsys2 = sys(tsys1 , tsys1 )
C%)*(==(&0> $1+,# .( 1+!'.,=( *(# 69).!#!19(#> .( /!*$-( 9)@+$+/9);"( (& /!=3
1!#$&0 *(# #!"#3+@#($"D )*(&0);"(# /!&%)(&0 ')(& (&/!+( $%(/ .( 1+!0!/!.( *( E.!00(*
F)&- (0 .( 1+!0!/!.( *( F!"&* F!')& G"0(DB H( *(+&)(+ 1+!0!/!.( $ "& /!=1!+0(3
=(&0 "& 1(" *)I@+(&0 *(# $"0+(# /$+ ). *!)0 #8&/9+!&)#(+ 0!"# .(# #!"# +@#($"D $%(/
"&( 1.$/( 1$+0$-@(B
!0+( )=1.@=(&0$0)!& #"+ /(# /!*$-(# *( +@#($"D *( 6(0+) =!&0+( ;"( .$ 0+$
J!+=$0)!& 6(0+)3K<LFE $"0!=$0);"( J!"+&)0 .( =M=( &!='+( *( /!&N-"+$0)!&# 1$+
+$11!+0 $"D $"0+(# $11+!/9(#B
!"! #$%&'() *& +&,-. /.$-'-0/.1(&% 22
#&3'-1(&% 45 !"#$ %& '()*+%,$ !(#&!-+./ *0%))*+#%1+23 '$- !4,*$- )!251$ 6+$3 *0%7
8%31%,$ '$ *% 1$#93+:&$ #%#9$ ; *% <==>
67,.3.%',.89 !"#$ "%$&'(")$ *#& +&$ ',-+&$ .& /0'/#+120") 3456 &2 3455 $")2
7+78&)210'&$ 9 .'"02& .& 2:;& <= 10)$0 &++&$ $")2 '&8;+1/7&$ ;1' +1 ',-+& $#0(1)2&
Sync⋆II+1(sys(x, y))→ sys(Sync⋆II+1(x), Sync⋆II+1(y))
>1' /")$7*#&)2= +&$ 1+-"'02?8&$ .@";2080$120") .& +1 $&/20") A4B $")2 #20+0$1%+&$ .1)$
+& /".1-& ?07'1'/?0*#&4
#&3'-1(&% 44 !"#$ %& '()*+%,$ !(#&!-+./ *0%))*+#%1+23 '$- !4,*$- )!251$ 6+$3 *0%7
8%31%,$ '$ *% 1$#93+:&$ ?@A B2& *0%*,2!+19C$ '$ -%1&!%1+23D>
#$%(:,',% &)7$-.3&9,'() ; E3 &1+*+-%31 321!$ 2&1+* B'(1%+**( %& #9%)+1!$ FD/ +* $-1
)2--+6*$ '$ 1!%+1$! *$ )!26*4C$ %G%31 $38+!23 HI C+**+%!'- )9+*2-2)9$- $3 135, 407
-$#23'$->
#&3'-1(&% 4< J$ #2'%,$ $-1 $K#%#$ :&%3' *$ 32C6!$ '$ -2&- !(-$%&L +'$31+:&$-
M $-1 &3$ )&+--%3#$ '$ N> O+323 #2CC$ *$ #2'%,$ '$ !(-$%&L '$ A$1!+ H/ +* !+-:&$ '0G
%82+! '$ 32C6!$&-$- '$ .$&+**$- 323 &1+*+-($- :&+ !('&+-$31 *% )$!.2!C%3#$ '& #2'%,$>
P3$ -2*&1+23 !%+-233%6*$ $-1 (,%*$C$31 '$ '(#2C)2-$! N $3 -2CC$ '$ )&+--%3#$-
'$ N> E3-&+1$/ 32&- #23-1!&+-23- &3 #2'%,$ 9+(!%!#9+:&$ )2&! #9%#&3> E353/ 32&-
)2&823- -G3#9!23+-$! .%#+*$C$31 #$- #2'%,$->
#&3'-1(&% 4" Q* .%&1 321$! :&0+* $L+-1$ $3#2!$ &3 %&1!$ 1G)$ '$ !(-$%& 9+(!%!#9+:&$
1$* :&$ *$ )%!%C41!$ M $-1 *$ 32C6!$ '$ R$123 '$ #9%:&$ )*%#$/ #2CC$ '$&L $L$C)*$-
-&+8%31- S *$ -G-14C$ '$ T%36%3 $1 *$ -G-14C$ '$ @UO> Q#+ *% 321+23 '$ V-2&-7
!(-$%&V '$8+$31 +3&1+*$/ '23# %#1&$**$C$31 +* 30$L+-1$ )%- $3#2!$ '$ -2*&1+23 $K#%#$
)2&! 2)1+C+-$! #$- #2'%,$->
!"#$ %& '(#)!*+, -
!"# $% $&!'()*%+ ",-# !.,"# /,")*0 1! '-(##!"$% 2% 1345678 2!"# 1! /,29
01(#!)(," 2% :-%1:-%# '*,),$,1%# ',-* /,")*%* 1! *%1!)(," %")*% 678 %) 5678+ %)
%")*% 5678 %) 45678 2!"# ,;<%$)(= 2% *0!1(#%* 1% '*%/(%* '!# 2% 1! )*!"#=,*/!)(,"
!-),/!)(:-%>
?,-# !.,"# 0@!1%/%") '*0#%")0 )*,(# $,2!@%# ',-* #(/-1%* -" *0#%!- 2% A%)*(
;,*"0 ,*2("!(*%+ %) /,")*0 $,//%") ",-# ',-.,"# $!1$-1%* 130)!) 2%# #-$$%##%-*#
2!"# $%# $,2!@%#> B '!*)(* 2% $%) ,'0*!)%-*+ ",-# ',-.,"# $!1$-1%* 1%# 0)!)# !$$%#9
#(;1%#+ 130)!) '*0$02%") C1% $!1$-1 %" !**(D*%E+ 1% $!1$-1 2- $&%/("+ %)$> F% #,") 2%#
$!1$-1# '*(/()(=# ',-* 1! 1,@(:-% F6G> 3!-)*% '!*)+ 1% $!1$-1 2% 130)!) 2%# #-$$%##%-*#
%#) -)(1(#0 0@!1%/%") ',-* 1! 1,@(:-% G6G>
?,-# !.,"# 0@!1%/%") '*0#%")0 -"% #)*-$)-*% !2!')0% 0)!") -" )H'% 2% *0#%!-
&(0*!*$&(:-% !.%$ 2%# #,-# *0#%!-I (2%")(:-%# )%1 :-% 1% '*,;1D/% 2%# A&(1,#,'&%#+ 1%
'*,),$,1% 2% 81,))%2 7("@ %) 1% '*,),$,1% 2% 7,-"2 7,;(" J-)%I> G! )*!"#=,*/!)(,"
!-),/!)(:-% 23-" *0#%!- 2% A%)*( &(0*!*$&(:-% .%*# -" 45678 ",-# $,"2-() K -"
!-)*% '*,;1D/% :-( %#) 2% 20)%$)%* !-),/!)(:-%/%") 2%# #,-#9*0#%!-I (2%")(:-%# K
'!*)(* 23-" *0#%!- 2% A%)*(> % '1-#+ ",-# #,//%# K 1! *%$&%*$&% 23-"% #,1-)(,"
%L$!$% ',-* #(/-1%* 1% )H'% 2% *0#%!-I &(0*!*$&(:-%# )%1# :-% 1% #H#)D/% 2% M!";!"
%) 1% #H#)D/% 2% 5J8> G%# )%1# *0#%!-I #,") '!*!/0)*0# '!* -" ",/;*% 2% <%),"#
2% $&!:-% '1!$%> F%1! "% ",-# 2,""% '!# 2%# ,'0*!)(,"# '*(/()(.%# $,//% 1% )H'%
2% *0#%!- &(0*!*$&(:-% !.%$ 1% '1(!@% *0$-*#(= 2%# #,-# *0#%!-I (2%")(:-%#> G% '1(!@%
*0$-*#(= "3%#) '!# !''1($!;1% ($(+ 1%# )%$&"(:-%# 2% G5A# "% #,") 2,"$ '!# %L$!$%#>
!"#$%&' (
!"#$%&'#() *&" +,-./
!"#$! %# &'()*+ , -
!"# $% &'!()*+% ,- "./# !0."# (+1#%"*1 $234567 (+1#%+0!"* $! (/)##!"&% 82%9(+%#:
#)." 8%# #;#*<=%# >."&*).""%$# %* 8%# *%&'")?/%# 82!&&1$1+!*)." 8%# &!$&/$# !@./*)#:
#!"* A /" ./*)$ 8% 01+)B&!*)." #;=@.$)?/% %C&!&%D !"# $% &'!()*+% ! "#$% &'#"%
()*+,-$. /#00("1 "#$% *#$'#"% .'&+$(2 +(% 34567%8
9&"% +( /:&*,12( ;! "#$% &'#"% 0#"12. +& *$,%%&"/( <(% 34567% <&"% +& 0#<=
.+,%&1,#" *&2 -$(+-$(% *2#1#/#+(%8 9&"% /( /:&*,12(! "#$% /#"1,"$#"% > 0#"12(2 +($2
/&*&/,1. > '.2,?(2 /(21&,"(% *2#*2,.1.% /#""$(% *#$2 /(% 0#<@+(%8
5#$1 <A&B#2<! "#$% <,%/$1#"% $" *2#B+@0( <A&"&+C%( <A&//(%%,B,+,1.8 D *&21,2
<( /( *2#B+@0(! "#$% *#$'#"% 2.%#$<2( <A&$12(% *2#B+@0(% 1(+% -$( +( /&+/$+ <(%
*2#*2,.1.% <( %E2(1. #$ +A,"1(2=B+#/&F( *&2 345678
G( /:&*,12( /#"1,("1 .F&+(0("1 <(% <.0#"%12&1,#"% <( +& %*./,?/&1,#" <(% H#2=
0$+(% <( +& +#F,-$( G5I #$ <( +& +#F,-$( I5I *#$2 +(% 12&<$,2( (" 345678
!""#$%&
!" #$%&'() *+%,,)((-.-&-/0 1234 &+-$/)45.&2,%6) ! ! ! ! ! ! ! ! ! "78
!9 #$%&'() *+%,,)((-.-&-/0 1234 &)( -$:%4-%$/( ! ! ! ! ! ! ! ! ! ! "7;
!"!# $%&'(%&) *+,-'&) ./0+/&)1&23) ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! #45
!"!" $%&'(%&) *+,-'&) ,& /6)&.%7 ,& 8&3/9 ! ! ! ! ! ! ! ! ! ! ! ! ! ###
!8 <26-=3)( /)>124)&&)( ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ""8
!:!# ;+<9(%& 3&*=+/&''& '926.9/& ;>; ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ##:
!:!" ;+<9(%& 3&*=+/&''& ./0+/&)1&23& ?>; ! ! ! ! ! ! ! ! ! ! ! ! ! ##@
!"! #$%&'() *+%,,)((-.-&-/0 1234 &+-$/)45.&2,%6) "78
!" #$%&'() *+%,,)((-.-&-/0 1234 &+-$/)45.&2,%6)
!"# $%#&"'()!*# $+,-!)$ $" &,.&". $(# ,&&(##%-.(# /,) .(# 01234#5 6 /,)'%)
$( &( &,.&".7 *!"# /!"8!*# )9#!"$)( $(# ,"')(# /)!-.:;(# '(.# <"( .( &,.&". $(#
/)!/)%9'9# $( #=)('9 !" .+%*'()>-.!&,?( /,) 012345 @+(*#(;-.( $(# 9','# ,&&(##%-.(#
(#' "'%.%#9 9?,.(;(*' /!") $(# &,.&".# $( /)!/)%9'9 /!") ., .!?%<"( A2@ !" @2@5
0',*' $!**9 "* 9',' %*%'%,. init7 *!"# /!"8!*# !-'(*%) '!"# .(# 9','# ,&&(##%-.(#
(* &,.&".,*' .+(*#(;-.( $( '();(# &.!# $(#&(*$,*' $( .+9',' %*%'%,. init R⋆λe(F (init))5
6/):# .( &,.&". $(# 9','# ,&&(##%-.(# R⋆λe $%#&"'9 &%>$(##"#5 !"# ,8!*# -(#!%*
$+"* ,"')( &,.&". /)%;%'%B C .( &,.&". ,))%:)( R−1λe 5 6&'"(..(;(*'7 *!"# *( '),%'!*# /,#
.( &,# ?9*9),.5 A(/(*$,*'7 /!") .(# ;!$:.(# $( )9#(,"D $( E(')%7 .( &,.&". ,))%:)(
(#' &,.&".,-.( (* &F,*?(,*' ., $%)(&'%!* $(# '),*#%'%!*# $(# )9#(,"D $( E(')% $!**9#
GE!#' $(8%(*' E)( (' E)( $(8%(*' E!#'H5
90:$-/-2$ " !"#$" !%%&'%(
)*&(+, u $+ ,(%-( .!+/ Fbin (, RTλe $+ (+/(-0"( .( %'1"(/ .23456)/ ,(" 7$( R
Tλe =
⋃
ti∈T
Rtiλe8 )2&" (9&/,( v ,(" 7$( v = R
tλe(Transt(u)) !:(# t ∈ T ; !&+/& "2<,!, =%<#<.(+,
.$ ,(%-( v (/, .<>+& #*--( /$&:!+, ?
R−1λe (v) = u
I,%*'(*,*' *!"# /!"8!*# "'%.%#() $("D &,.&".# /)%;%'%B# /!") &,.&".() $(# 9','#
-.!<",*'# &!;;( $,*# ., /)!/!#%'%!* #"%8,*'( C
;4212(-/-2$ < )*&(+, init $+ ,(%-( &+&,&!" .!+/ Fbin (, Rλe $+ (+/(-0"( .( %'1"(/
.23456)8 @2&+,(%A0"*#!1( .2$+ (+/(-0"( .( ,(%-(/ .( Fbin B!==("< deadlockC (/,
#!"#$"< /$% Rλe D =!%,&% .( init #*--( /$&:!+, ?
deadlock = R⋆λe(init) \ R
−1λe (R
⋆λe(init))
;4)3=) "> ?;4)3=) (3,,-$,/) @
deadlock = R⋆λe(init)\R
−1λe (R
⋆λe(init))
E deadlock ⊆ R⋆λe(init)\R
−1λe (R
⋆λe(init)) ? F+,$&,&:(-(+, /("*+ deadlock ⊆
R⋆λe(init) (, deadlock ∩R
−1λe (R
⋆λe(init)) = ∅
E R⋆λe(init)\R
−1λe (R
⋆λe(init)) ⊆ deadlock ? 3G(#,&:(-(+, =!%#( 7$( /2&" (9&/,( t
!##(//&0"( (, &" +2(/, =!/ .!+/ R−1λe (R
⋆λe(init)); !"*%/ &" .*&, H,%( $+ <,!, 0"*A
7$!+,8
A!*#%$9)!*# .( /)!-.:;( $(# EF%.!#!/F(# $%#&"'9 $,*# .( &F,/%')( /)9&9$(*'5
AB)C1&) >< I%*0"'-( D .($9 =J&"*/*=J(/8
@( #*.!1( &.(+,&7$( +&:(!$ K =*$% L =J&"*/*=J(/ (/, $+( #*-=*/&,&*+ .( .($9 /*$/A
%</(!$9 &.(+,&7$(/ #*--( "( /$&:!+, ?
!" #$%&'()* +, -.)'/0%('12 &%) 34567
tsys1 = sys(tlocal, tlocal)
! "# $ "$ %&$'(" ()(#* + ,-(.'$ ,(%/0 -'$1"'--'$ -'$ ,-(.'$ Idle '* Fork . #*0'##'#*
"# 2'* # '$* . 3& .043'$$ "$ 5
tlocal = local(idle(1,waitleft(0, hasleft(0, waitright(0, hasright(0, fork(1,⊥))))))
6'$ ,70- $ ,7'$8 $90-$ (:0$$'#* * "$ 3' ;(< #$ #(=>'$ '* 03'#*01"'$8 %0$1"'#* ; %*
3' $' %'*% ">'% '# $0*"(*0 # 390#*'%4?- .(:'@ A# 'B'*8 0- $"C* 1"' .7(."# $(0$0$$' $(
; "%.7'**' 3' :(".7' '*8 1"9'#$"0*'8 .7(."# (**'#3' 1"' $( ; "%.7'**' 3' 3% 0*' $'
-0?D%' , "% 1"9("."# 39'#*%' '"E #' ,"0$$' /(#:'%@
F "$ .(-."- #$ -'$ &*(*$ (..'$$0?-'$ '#$"0*' # "$ .(-."- #$ -9&*(* ,%&.&3'#* 3' .'$
&*(*$@ G0#(-'/'#*8 # "$ , "> #$ 3&*'.*'% 3'"E 0#*'%4?- .(:'$ . ##"$ .043'$$ "$ 5
t0local = local(idle(0,waitleft(1, hasleft(0, waitright(0,hasright(1, fork(0,⊥))))))
t1local = local(idle(0,waitleft(1, hasleft(0, waitright(0,hasright(1, fork(0,⊥))))))
A* -9&*(* $)/&*%01"' 5
t0local = local(idle(0, waitleft(0,hasleft(1,waitright(1, hasright(0, fork(0,⊥))))))
t1local = local(idle(0, waitleft(0,hasleft(1,waitright(1, hasright(0, fork(0,⊥))))))
!"# $!%&'!%# (") *) +,*+"* -) *./%&)'01*!+,2) ')#&) 3,/#,1*) ,4)+ -)# $!-5*)# -)
2',%-) &,/**) -,%# *, 6,'&/) )768'/$)%&,*) 9:!/' +3; *) +<,6/&') =>;
6.89:(%(8 *;&.)'<*2(%9; = A# "*0-0$(#* # *%' "*0- H3&*(0--& (" .7(,0*%' IJ8 0-
'$* , $$0?-' 3' 3&*'.*'% 3'"E &*(*$ ?- 1"(#*$ 3" ,% ?-D/' 3' 32.000 ,70- $ ,7'$ '#
/ 0#$ 39"#' 7'"%'@
!" #$%&'() *+%,,)((-.-&-/0 1234 &)( -$5%4-%$/(
?)# /%4,'/,%&# Φ #!%& *)# 6'!6'/8&8# -) #@')&8 *)# 6*"# +!$$"%)# )& *)# 6*"# "&/*)#;
!"# -8A%/'!%# +/0-)##!"# -)# /%4,'/,%+)# 6'/$/&/4)# 6!"' "% &)'$) -,%# Fbin; !"#
"&/*/#!%# +)# /%4,'/,%+)# 6'/$/&/4)# 6!"' 48'/A)' -)# 6'!6'/8&8# /%&8')##,%&)# 6!"'
%!# 8&"-)# -) +,#;
>./2'('12 = K 0* t "# *'%/' 3' T (Fbin)@ F "$ 30% #$ 1"' .' *'%/' t $(*0$;(0* Φ
$0 Φ '$* "#' 3'$ ; %/'$ .043'$$ "$ 5
Φ ::= TRUE | ⊥ | a(Φ1,Φ2) | Φ1 ∧ Φ2 | Φ1 ∨ Φ2 | α
(>'. a ∈ Fbin '* α ∈ T (Fbin,X )@
B* )7/#&) -)# CDE# +*,##/(")# 6!"' 48'/A)' "%) &)**) 6'!6'/8&8;
!"! #$%&'() *+%,,)((-.-&-/0 1234 &)( -$5%4-%$/( 678
92$:),/34) 6 !"# t $% #&'(& )& T (Fbin)* +!$, )"'!%, -$& .& #&'(& t ,/#",0/"# 1/
2'!2'"3#3 φ )34%"& )/%, 1/ )34%"#"!% 56 ,7"1 &8",#& $% 9: (!)31",/%# $%& "%;/'"/%.&
#&1 -$& (RInv)⋆({t}) %7&,# 2/, ;")& )/%, $%& )&, 0!'(&, ."<)&,,!$, =
5* Invall /;&. %7"(2!'#& -$&1 #&'(& t
RInvall ={
x→ x}
.
>* Inv⊥ /;&. t = ⊥
RInv⊥ ={
⊥ → ⊥}
.
?* InvaInvl,Invr /;&. t = a(tl, tr)@ a ∈ Fbin@ Invl(tl) &# Invr(tr) ,!%# ;'/"&,
a(x, y)→ a(x, y)
/;&. Pos(tl) = 00,Pos(tr) = 01 &#
(RInvl)⋆(tl) ∩ (RInvr)⋆(tr) (= ∅
A* (Inv1 ∨ Inv2)(t) /;&. Inv1(t) !$ Inv2(t) &,# ;'/"&, =
(RInv1∨Inv2)⋆(t) = (RInv1)⋆(t) ∪ (RInv2)⋆(t)
B* (Inv1 ∧ Inv2)(t) /;&. Inv1(t) &# Inv2(t) ,!%# ;'/"&, =
(RInv1∧Inv2)⋆(t) = (RInv1)⋆(t) ∩ (RInv2)⋆(t)
C* InvαRInvα =
{
α→ α, α ∈ T (Fbin,X )}
.
!" #$%#$&'(' )" *+$"(' &!('$"**,!(" φ - .'$&/"$ *0$ 1" #$%2134" 567 "*( 809&1
":&*(" (%0;%0$* 0! 0!&80" ;"(%! <&$<01,!( ),!* 19,$2$" )" =&'$,$<=&"> 6&!*&? !%0* #%0@
.%!* A%$401"$ <"((" #$%#$&'(' <%44" 0! #$')&<,( .'$&/"$ *0$ <=,80" <%!/B0$,(&%!
,<<"**&21">
;<)=1&) "> D!%,")3'!%, 1&, ,E(F!1&, ⊥ &# ∅ '&,2&.#";&(&%# .!((& 1&, ;/1&$',
9:GH &# IJK H@ !"# OneToken 1& 2'3)"./# )34%" .!((& 1& ,$";/%# 2!$' $% #&'(&
α ∈ T (Fbin) =
OneToken(α) =
⊥ si ∀p, p′Pos(α),
α(p) = α(p′) = t⇒ p = p′
∅ sinon.
L/' .!%,3-$&%#@
TAP satisfait φ⇔ ∀α ∈ R⋆({u}), OneToken = ⊥
!" #$%&'()* +, -.)'/0%('12 &%) 34567
!"#$%$&" '(")
TAP satisfait φ⇔ ∀α ∈ R⋆({u}), OneToken (= ∅
*+$, u $-" .$ "$#%$ t(i(i(⊥,⊥), i(⊥,⊥)), i(i(⊥,⊥), i(⊥,⊥)))/
01!- +$&1&- '$ '23&(# .* 4#14#(2"2 φ 5 (. $6(-"$ "1!71!#- !& !&(8!$ 7$"1& ,(#,!.*&"
'*&- .9*#:#$ '$ ;(2#*#,;($/ <1&-('2#1&- ¬φ 8!( -(=&(3$ 8!$ (. $6(-"$ 4$!">?"#$ !&$
,1&3=!#*"(1& &$ ,1&"$&*&" 4*- 7$"1& 1! *! %1(&- '$!6 7$"1&-/
01!- %1&"#1&- 2=*.$%$&" ,1%%$&" &1!- 41!+1&- 4#1!+$# ,$""$ 4#14#(2"2 φ $&
@AB/ 01!- ,1%%$&C1&- 4*# .$- D$!(..$- $& %1&"*&" 4*- E 4*- '$ :*- $& ;*!") &1!-
#$%4.*C1&- ,;*8!$ &1$!' 4*#
F conflict -( "1!- ,$- $&D*&"- -1&" onlyt 1! *! %1(&- !& $-" conflict)
F not -( "1!- ,$- $&D*&"- -1&" not)
F onlyt -(&1&/
G9$&-$%:.$ '$ #H=.$- '$ @AB ROneToken$-" .(-"2$- ,(>'$--!- 5
a(⊥,⊥) → not(⊥,⊥) with a ∈ {i, r, b} !"#$
t(⊥,⊥) → onlyt(⊥,⊥) !"%$
a(conflict(x, y), z) → conflict(conflict(x, y), z) !"&$
a(x, conflict(y, z)) → conflict(x, conflict(y, z)) !"'$
a(onlyt(x, y), onlyt(z, w)) → conflict(onlyt(x, y), onlyt(z, w)) !"($
t(onlyt(x, y), onlyt(z, w)) → conflict(onlyt(x, y), onlyt(z, w)) !")$
a(not(x, y), not(z, w)) → not(not(x, y), not(z, w)) !"!$
t(not(x, y), not(z, w)) → onlyt(not(x, y), not(z, w)) !"*$
a(not(x, y), onlyt(z, w)) → onlyt(not(x, y), onlyt(z, w)) !"+$
a(onlyt(x, y), not(z, w)) → onlyt(onlyt(x, y), not(z, w)) !"#,$
t(not(x, y), onlyt(z, w)) → conflict(not(x, y), onlyt(z, w)) !"##$
t(onlyt(x, y), not(z, w)) → conflict(onlyt(x, y), not(z, w)) !"#%$
I1!# +2#(3$# 8!$ .$ TAP satisfait φ) ,$.* -!J" '$ "$-"$# -9(. $6(-"$ !&$ ,1&3=>
!#*"(1& 2"(8!$"2 onlyt E .* #*,(&$ 5
TAP satisfait φ⇔
(InvOneToken) ∧ InvonlytInvall,Invall⇔
(ROneToken)⋆({u}) ∩ (InvonlytInvall,Invall)⋆({u}) (= ∅
1K InvonlytInvall,Invall$-" .9$&-$%:.$ '$ #H=.$- 41!# 3."#$# !&$ ,1&3=!#*"(1& 2"(8!$"2
onlyt E .* #*,(&$ 5
onlyt(x, y)→ onlyt(x, y)
*+$, Pos(tl) = 00,Pos(tr) = 01 $" (RInvall)⋆(tl) ∩ (R
Invall)⋆(tr) (= ∅/
!"# !$#%&'!(# )"% (!"# *%'!(# "+,-,#%& %( ./0 1-2##,)"% *%# 32&)"%"&# )",
234(%(+ #!"'%(+ 5 "(% %67-!#,!( *89+2+# *" #:#+43%; <%7%(*2(+= -%# #:#+43%# >!(1?
+,!((%-# *8,('2&,2(1% 7&9#%(+9# 1,?*%##!"# 7%"'%(+ 9',+%& *%# 32&)"%"&# %( "+,-,#2(+
!"! #$%&'() *+%,,)((-.-&-/0 1234 &)( -$5%4-%$/( 67
!" #$#%&!'()#*+,- .!" "/"&0(!" 1$#2&)$##!3" 4)#5*')*#2! 6')()&)1" "$#& 78#)"
2$((! "+)& 9
82$9),/34) " !"# t $% #&'(& )& T (Fbin)* +!$, )"'!%, -$& .& #&'(& t ,/#",0/"# φ
)12%"& )/%, 3/ )12%"#"!% 45 ,6"3 &7",#& $% 89 (!)13",/%# $%& "%:/'"/%.& #&3 -$&
(RInvλe )
⋆({Inv(t)}) %6&,# ;/, :")& )/%, $%& )&, 0!'(&, ."<)&,,!$, =
4* Invall /:&. %6"(;!'#& -$&3 #&'(& t
RInvallλe =
{
Inv(a(x, y))→ a(x, y) ∀a ∈ FbinInv(⊥)→ ⊥
}
.
>* Inv⊥ /:&. t = ⊥
RInv⊥λe =
{
Inv(⊥)→ ⊥}
.
?* InvaInvl,Invr /:&. t = a(tl, tr)@ a ∈ Fbin@ Invl(tl) &# Invr(tr) ,!%# :'/"&,*
RInvaInvl,Invrλe =
{
Inv(a(x, y))→ a(Invl(x), Invr(y))}
.
A* Inv1 ∨ Inv2 /:&. Inv1(t) !$ Inv2(t) &,# :'/"&,*
RInv1∨Inv2λe =
Inv(a(x, y))→ Inv1(a(x, y)) ∀a ∈ FbinInv(a(x, y))→ Inv2(a(x, y)),
Inv(⊥)→ Inv1(⊥),
Inv(⊥)→ Inv2(⊥)
.
B* Inv1 ∧ Inv2 /:&. Inv1(t) &# Inv2(t) ,!%# :'/"&,*
RInv1∧Inv2λe =
{
Inv(a(x, y))→ Inv1(Inv2(a(x, y))) ∀a ∈ Fbin,
Inv(⊥)→ Inv1(Inv2(⊥))
}
.
C* Invα +!$, /:!%, D&,!"% )& 36EF89 Rα→βcheck )!%%1 )/%, 3/ ,&.#"!% A*> -$" %!$,
;&'(&# )& :1'"2&' ," $% #&'(& t)& T (Fbin) .!%.!')& /:&. α*
:# 1*)&; #$+" 6$+5$#" 57')8!' ") 34!#"!(<3! ! (*+5*)" 7&*&" Bad = ¬φ =
¬Inv(R⋆(init)) "$)& &$+=$+'" 1*+,- ># 7&*& ¬φ 5'*) !"& 2$(6&7 2$((! +# !,!(63!-
:0;$-/-2$ 6< !"# t $% #&'(& )& T (Fbin)* G%& "%:/'"/%.& %1H/#":& ¬Inv(t) &,#
:'/"& ,&$3&(&%# ," ¬Inv &,# $%& )&, 0!'(&, ."<)&,,!$, =
4* ¬Invall = ∅
>* ¬Inv⊥ /:&. t = a(t1, t2)
?* ¬InvaInvl,InvrI /:&. t = b(tl, tr)@ b (= a | b ∈ Fbin@ !$
I /:&. t = a(tl, tr) Invl(tl) ∨ Invr(tr) &,# 0/$7*
A* ¬(Inv1 ∨ Inv2) /:&. ¬Inv1(t) ∧ ¬Inv2(t) &,# :'/"&*
B* ¬(Inv1 ∧ Inv2) /:&. ¬Inv1(t) ∨ ¬Inv2(t) &,# :'/"&*
!" #$%&'()* +, -.)'/0%('12 &%) 34567
! ¬Invα "#$% Rα→βcheck &'( t $&) #*+$
! "#$%& $' ()$*%( +!( #+%,( *-'+%$-! .-+, /#'/+'(, ¬Inv (! +%$'$*#!% '( /#'/+' 0(*
1%#%* #//(**$2'(* R⋆3 ''( (*% .,1*(!%1( 0#!* '# /-!4(/%+,( *+$5#!%( 6
#128*0(9)* : ,-.")*/0 +1*0#"(*"0%$!
¬Inv(α) = R⋆(init)\Inv(α)
"#$% α '0 )$(2$ %3/&!
7(5(!-!* 8 !-%,( 1%+0( 0( /#* 9:;3
3;*<&=* :! 4156789 &'*#"0) R¬φλe &:-%*;$ 3" :(/:(*-)- ¬φ!
No t(r(x, y)) → r(No t(x), No t(y)) !"#$%
No t(b(x, y)) → b(No t(x), No t(y)) !"#&%
No t(i(x, y)) → i(No t(x), No t(y)) !"#'%
No t(⊥) → ⊥ !"#(%
Two t(r(x, y)) → r(Two t(x), y) !"#!%
Two t(r(x, y)) → r(x, Two t(y)) !"#)%
Two t(b(x, y)) → b(Two t(x), y) !"#*%
Two t(b(x, y)) → b(x, Two t(y)) !"+,%
Two t(i(x, y)) → i(Two t(x), y) !"+#%
Two t(i(x, y)) → i(x, Two t(y)) !"++%
Two t(t(x, y)) → i(x, Two t(y)) !"+$%
9-2"0)*<'$2$0) +*)= 3$ &>2?/3$ No@t ∈ FNT $&) *0)(/+'*) :/'( #-(*;$( <'1'0
)$(2$ 0$ %/0)*$0) :"& 3$ &>2?/3$ t! 5A$%)*#$2$0)= *3 01> " "'%'0$ (B.3$ +$ 3" C/(2$
No@t(t(x, y))→ β! 50 %/0&-<'$0%$= :/'( '0 )$(2$ α ∈ T (Fbin) :/'( 3$<'$3 *3 $D*&)$
p ∈ Pos(α) )$3 <'$ α(p) = t= (R¬φλe )
∗({No@t(α)}) = ∅!
4$ &>2?/3$ Two@t ∈ FNT $&) *0)(/+'*) :/'( #-(*;$( <'$ 3$ &>2?/3$ t "::"("E)
+"0& '0 )$(2$ "' 2/*0& '0$ C/*&! F*0&*= :/'( '0 )$(2$ α ∈ T (Fbin)=
G (R¬φλe )
⋆({Two@t(Two@t(α))}) = ∅ &*.0*;$ <'$ 3$ &>2?/3$ t "::"("E) +"0& α
'0$ &$'3$ C/*& /' H"2"*& I
G (R¬φλe )
⋆({Two@t(Two@t(α))}) (= ∅ &*.0*;$ <'$ 3$ &>2?/3$ t "::"("E) +"0& α
"' 2/*0& +$'D C/*&!
J"( %/0&-<'$0)=
3$ 7FJ &")*&C"*) φ ⇔(R¬φ
λe )⋆((RTAPλe )⋆({Two t(Two t(t′0))})) ∧ (R¬φ
λe )⋆((RTAPλe )⋆({No t(t′0)})) = ∅
<'"0+ t′0 = Arbiter(t0) $) t0 $&) 3" %/0;.'(")*/0 *0*)*"3$ +' 7FJ!
!"! #$%&'() *+%,,)((-.-&-/0 1234 &)( -$5%4-%$/( 678
!"# $%&#'()!(# "(' *")%' #!+"),!( '( -*+-"+*() R⋆λe({InvOnlyT (α)}) -,./'##!"# 0
InvOnlyT (a(x, y)) → a(InvOnlyT (x), InvNoT (y)) !"#$%
InvOnlyT (a(x, y)) → a(InvNoT (x), InvOnlyT (y)) !"#&%
InvOnlyT (t(x, y)) → t(InvNoT (x), InvNoT (y)) !"#$%
InvNoT (a(x, y)) → a(InvNoT (x), InvNoT (y)) !"#!%
InvNoT (⊥) → ⊥ with a ∈ {i, r, b} !"#&%
!"#$%&%" '()!*+,'-%&#( . ! "#$%$&'!# !(#)* ("#$% + ! ,-#'$% ,'!& %* ./'0$#)*
123 !("& &(44*& .'0'5%*& ,* 6-)$7*) 8"* φ *&# 6)'$ 0(") %* 9:; #)'$#'!# 24000)(<
.*&&*")& *! 2, 933 &*.(!,*&=
!"!# $%&'(%&) *+,-'&) ./0+/&)1&23)
/010202 3!*+45&%+6- 78+-9&*+&-%" 7# ::
!"#$%&'!"# () *'!+!,!() %) -)',!(.+) )" #),+$!" /01010 2!3# "!3# $"+&')##!"#
3"$43)5)"+ .36 *'!*'$&+&# %) E1 43$ ,!"+$)"")"+ #)3()5)"+ %)# &+$43)++)# n7 p )+
⊥ 89!$' (. :;3') /0<=0
E1 = ((RPP )
⋆(E0) ∩ T ({n, p,⊥})).
>" )"#)5?() %) '@;()# %ABCDEF (RFilternp⊥λe )⋆({Inv(α)}) ,!'')#*!"%."+ )#+
%&,'$+ ,$G%)##!3# H
Inv(⊥) → ⊥ !"#'%
Inv(p(x, y))→ p(Inv(x), Inv(y)) !"()%
Inv(n(x, y)) → n(Inv(x), Inv(y)) !"(*%
C$".()5)"+7 "!3# ,!"#$%&'!"# (. *'!*'$&+& +)(() 43) #. '.,$") )#+ &+$43)+&) *.'
p H
E′1 = (E1 ∩ {p(t1, t2)|t1, t2 ∈ T}).
I$"#$7 (RPλe)
⋆({P (α)}) .J), P (p(x, y))→ p(x, y)0
B" ,!",(3#$!"7 E′1 #= ∅ #$;"$:) 43A.3 5!$"# 3") %)# K)3$(()# *!##@%) 3") J.()3'
p L E′1 = ∅ #$;"$:) 43) +!3+)# ()# K)3$(()# "A!"+ 43A3") J.()3' n0
!"#$%&%" '()!*+,'-%&#( 2; ! "#$%$&'!# !(#)* ("#$% +,-#'$%%- '" ./'0$#)* 123 $%
*&# 0(&&$5%* ,* #)'$#*) %* 0)(#(.(%* ,* ;*).(%'#* '>'!# 25000)(.*&&"& *! 4($!& ,?"!*
,*4$* ,* &*.(!,*=
! "#$%&'() *+ ,-(&./$'&01 %$( 23456
*+7+ +7 ,-(&./$'&01 89&1:$(&$1'; 8< %(0'0/0=) 892=)/'&01 >(?0();/)1'
!"#$%&'!"# () *'!+!,!() %-.(),+$!" /'0!')#,)"+ )" #),+$!" 123242
56"# (6 %)78$9:) &+6*); "!7# )##6<)'!"# %) %)#,)"%') )" #&(),+$!""6"+ () ,6"=
%$%6+ >&+$?7)+& *6' p@ #$ *!##$0()2
/ *6'+$' %7 '(+6+ %) E2; "!7# "!7# $"+&')##!"# 678 *'!*'$&+&# 6<6"+ 7" ,A):$"
?7$ ,!"+$)"+ 7"$?7):)"+ c %) (6 '6,$") B (6 C)7$(()2
D" +)( .EFGH (RPathcλe )⋆({Inv(α)}) )#+ %&,'$+ ,$=%)##!7# I
O(a(x, y)) → a(x, y) avec a ∈ {n, p} !"#$%
Inv(⊥) → ⊥ !"##%
O(⊥) → ⊥ !"#&%
Inv(c(x, y)) → c(O(x), Inv(y)) !"#'%
Inv(c(x, y)) → c(Inv(x), O(y)) !"#(%
J-)"#):0() E′2 )#+ $((7#+'& )" KL7') 12M2
N!7# :!"+'!"# &L6():)"+ ,!::)"+ "!7# *!7O!"# :!%&($#)' ,) .EFGH )" FGH I
N!7# *!7O!"# %&"!:0')' () "!:0') %) C)7$(()# &+$?7)+&)# *6' c2
N!7# *'!*!#!"# 7") 67+') #!(7+$!" I N!7# *!7O!"# +'!7O)' ,) '(+6+ K"6( )"
(6"P6"+ (RPathcλe )⋆({Inv(α)}) #7' E”1 >Q!$' (6 KL7') 12M@2
5-;<='$'; )@%-(&A)1'$<@ ! "#$"%$#!& (RPathcλe )⋆({Inv(α)}) '%( E”1 )#(
!*&(+ *%&,$ -./&#,$$/ #% "0#),&(+ 123 ,$ +'& )*'',4$+ .+ &(#,&+( $+ )(*&*"*$+ .5 $+"&,*!
6(4*(+'"+!& #7#!& 2450)(*"+''%' +! 8*,!' .5%!+ '+"*!.+9
FGH RPathoI
c(⊥,⊥)→ o(⊥,⊥) !"#!%
c(o(x, y), z)→ o(o(x, y), z) !"#)%
o(x, o(y, z))→ o(x, o(y, z)) !"#*%
D" +)( .EFGH RPathoλe )#+ %&,'$+ ,$=%)##!7# I
R :;<$+' </!/(/+'
O(o(x, y))→ o(x, y) !"&+%
O(⊥)→ ⊥ !"&,%
H(c(x, y)) → o(O(x), y) !"&$%
H(c(x, y)) → o(x,O(y)) !"&#%
R :;<$+' .+ ",("%$#&,*!
H(c(x, y)) → c(H(x), y) !"&&%
H(c(x, y)) → c(x,H(y)) !"&'%
!"! #$%&'() *+%,,)((-.-&-/0 1234 &)( -$5%4-%$/( 666
!"#$% &$ '()*+ ,-$
Patho(x) → x !"#$%
Patho(x) → Patho(H(x)) !"#!%
!"#$ %"&&'()"($ * +,-./- 0# 1".."& '( -'&+2,),(. %3,4#' 5./4#'..' c +,- o
'. &"(.'- ("'#0 +,- ("'#06 7, +-"+-/5.5 ,8,(. #(' 5./4#'..' o * 2, -,%/(' '$. %'22'
4#/ ("#$ /(.5-'$$'6
E2 = (RPatho)⋆(E”1) = (R
Pathoλe )⋆(Inv(E”1)).
9/($/: ("#$ +-5$'(."($ 2'$ $8$.;&'$ %/<0'$$"#$ +"#- %3'-%3'- #(' +-"+-/5.5 ,8,(.
#(' 5./4#'..' o * 2, -,%/(' =
E”2 = (E′
2 ∩ {o(t1, t2)|t1, t2 ∈ T}).
9/($/: (ROλe)
⋆({O(α)}) ,>'% O(o(x, y)) → o(x, y)"
?'..' +-"+-/5.5 '$. '@,%.'&'(. %"&&' '( AB#-' C6D $,#E 4#' 2' %3'&/( 0'$
5./4#'..'$ c '$. -'&+2,%5 +,- %'2#/ 0'$ 5./4#'..'$ o6
70(3&/%/( )8104-9)$/%38 6" .* /0#/1#0*+ (RPathoλe )⋆({Inv(α)}) %12 E”1 '02
*(+2$ (1+)# 3&4+0)##4 01 /50')+2$ 678 )# $%+ '(%%)9#$ &$ +20)+$2 #$ '2(+(/(#$ &:.#$/+)(*
;29(2$%/$*+ 0<0*+ 2400'2(/$%%1% $* '#1% &$ &$1- =)*1+$%>
!"!" #$%&'$%( )*+,&%( +% -.(%/$0 +% 1%2-3
F,($ 2'$ &"0;2'$ 0' -5$',#@ 0' G'.-/: /2 E,#. 1/'( &,H.-/$'- 2'$ -;B2'$ 0' %/-%#2,<
./"( +"#- ,22'- * 2I'(0-"/. 0' 2, +2,%' /(0/4#5' '. .'$.'- 2' &,-4#,B' $"#3,/.56
!"!"!6 :04-;,%/-2$ *+-$5%4-%$/( *)( <=-&2(21=)(
J'$. 0I/(>,-/,(%' +"#- >5-/A'- 4#I/2 8 , ,#%#(' %"(AB#-,./"( .'22' 4#'
0'#@ >"/$/($ +-'(('(. 2, &K&' E"#-%3'..'6 7' +-"12;&' 05$"-&,/$ 0'>/'(. 0'
2,(%'- #( $8$.;&' 0' -5<5%-/.#-' RConflitλe +"#- .'$.'- 2'$ 0'#@ >"/$/($6 L/
(RConflitλe )
⋆({Conflit(α)}) E"#-(/. #( -5$#2.,. ("( >/0': ,2"-$ /2 '@/$.' #(' .'22' %"(<
AB#-,./"(6
Conflit(system(x, y))→ Sync1N(system(x, y)) !"#&%
Conflit(system(x, y))→ SyncIIPP (system(x, y)) !"#'%
SyncIIPP (system(x, y))→ system(SyncIIPP (x), y) !"()%
SyncIIPP (system(x, y))→ system(x, SyncIIPP (y)) !"(*%
SyncIIPP (system(x, y)) → system(TestI(x), T estIPP (y)) !"(+%
Sync1N(system(x, y)) → system(TestIPP (x), T estI(y)) !"(,%
TestI(system(x, y)) → system(x, TestI(y)) !"(#%
TestIPP (system(x, y)) → system(TestIPP (x), y) !"((%
TestI(local(x, y)) → local(x, LocalTestI(y)) !"($%
TestIPP (local(x, y)) → local(x, LocalTestIPP (y)) !"(!%
LocalTestI(x)→ Testhasright(x) !"(&%
LocalTestIPP (x) → Testhasleft(x) !"('%
! "#$%&'() *+ ,-(&./$'&01 %$( 23456
!"# $!%$% &'()($(*% &+"' ,-'(.!' %( /! &0(1+%+&0! 2 3(!4 &'(% "4! *+"'/0!$$! 524%
1! )2(4 62"/0! 75'+($!8 9
Testhasleft(hasleft(x, y)) → hasleft(Test(x), y) !"#$%
Testhasleft(fork(x, y))→ fork(x, Testhasleft(y)) !"#&%
Testhasleft(hasright(x, y)) → hasright(x, Testhasleft(y)) !"#'%
Testhasleft(waitright(x, y))→ waitright(x, Testhasleft(y)) !"#(%
Testhasleft(waitleft(x, y)) → waitleft(x, Testhasleft(y)) !"#)%
Testhasleft(idle(x, y)) → idle(x, Testhasleft(y)) !"#*%
Testhasleft(fork(x, y))→ fork(x, Testhasleft(y)) !"##%
Testhasright(hasright(x, y)) → hasright(Test(x), y) !"#!%
Testhasright(fork(x, y)) → fork(x, Testhasright(y)) !"#+%
Testhasright(hasleft(x, y)) → hasleft(x, Testhasright(y)) !"#,%
Testhasright(waitright(x, y))→ waitright(x, Testhasright(y)) !"!$%
Testhasright(waitleft(x, y))→ waitleft(x, Testhasright(y)) !"!&%
Testhasright(idle(x, y)) → idle(x, Testhasright(y)) !"!'%
Testhasright(fork(x, y)) → fork(x, Testhasright(y)) !"!(%
Test(1) → 1 !"!)%
5-789'$'7 ):%-(&;)1'$8: < ! "#$%$&'!# !(#)* ("#$% +,-#'$%%- '" ./'0$#)* 123
!("& &(44*& .'0'5%*& ,* 6-)$7*) 8"* 0')4$ 2350/$%(&(0/*&3 $% !9: ' '"."!* .(!;
7<")'#$(! #*%%* 8"* ,*"= 6($&$!& 8"$ 0)*!!*!# %' 4>4* ?(")./*##* *! "!* ,$@'$!* ,*
4$!"#*&A
*+!+!+! ,-(&./$'&01 =)7 &1>$(&$1'7 =8 %(0'0/09) =) 5081=?50@&1 A8'):
:+"% (4$-'!%%+4% -621!)!4$ ; "4 $!%$ 5<(4,2'(24/! ϕ &+"' ,-'(.!' ="<(1 > 2 "4!
%!"1! '!%%+"'/! 524% 1! )+5?1!@ A(4%(B 4+"% !%%2>+4% "4 $!%$ 5! 4-62$(+4 5<(4,2'(24/!
¬ϕ &+"' /0!'/0!' "4! /+4.6"'2$(+4 524% 12 ="!11! 9
C %+($ 2" )+(4% 5!"# &'+/!%%"% ="( "$(1(%!4$ /!$$! '!%%+"'/! !4 )D)! $!)&%
H(x)→ TestR(TestR(x)) !"!*%
TestR(systemGlobal(x, y)) → systemGlobal(x, TestR(y)) !"!#%
TestR(system(x, y)) → system(TestR(x), y) !"!!%
TestR(system(x, y)) → system(x, TestR(y)) !"!+%
TestR(local(x, y))→ local(x, TestR(y)) !"!,%
TestR(r(x, y)) → r(Test(x), y) !"+$%
TestR(psend(x, y))→ psend(x, TestR(y)) !"+&%
TestR(pload(x, y))→ pload(x, TestR(y)) !"+'%
TestR(pok(x, y)) → pok(x, TestR(y)) !"+(%
TestR(buffull(x, y))→ buffull(x, TestR(y)) !"+)%
TestR(bufidle(x, y)) → bufidle(x, TestR(y)) !"+*%
TestR(pask(x, y)) → pask(x, TestR(y)) !"+#%
TestR(pwait(x, y))→ pwait(x, TestR(y)) !"+!%
Test(1) → 0 !"++%
!"! #$%&'()* +),-$.)//)* 00"
!"#$ %&'&( )% $"&'*+,
H(x) → TestZ(x) !"#$%
TestZ(systemGlobal(x, y)) → systemGlobal(TestZ(x), T estZ(y)) !"$&%
TestZ(system(x, y)) → system(TestZ(x), T estZ(y)) !"$'%
TestZ(res(x, y))→ res(Test0(x), y) !"$(%
TestZ(local(x, y))→ local(x, TestZ(y)) !"$)%
TestZ(r(x, y)) → r(Test(x), y) !"$*%
TestZ(psend(x, y))→ psend(x, TestZ(y)) !"$+%
TestZ(pload(x, y))→ pload(x, TestZ(y)) !"$,%
TestZ(pok(x, y)) → pok(x, TestZ(y)) !"$!%
TestZ(buffull(x, y))→ buffull(x, TestZ(y)) !"$#%
TestZ(bufidle(x, y)) → bufidle(x, TestZ(y)) !"$$%
TestZ(pask(x, y)) → pask(x, TestZ(y)) !"'&&%
TestZ(pwait(x, y))→ pwait(x, TestZ(y)) !"'&'%
Test0(0) → 0 !"'&(%
1),2.'()* 03 !"# !$#%&'!(# )"% *%# #+#,-.%# */01%##"# #!(, "(% '%&#/!( 1%
Two2t %, No2t 1" 345 6!"& 7% 6&!,!*!7% 1% 8!"(1 8!$/( 9",%:; <(,"/,/'%.%(,=
7%# &->7%# 1% */&*"7?,/!( 1% TestR @TestZA #!(, )"?#/.%(, /1%(,/)"%# B *%77%# 1%
Two2t @No2tA;
14*(/+2+* )5-4.&,)6+2(5 03 C( ",/7/#?(, (!,&% !",/7 @1D,?/77D ?" *E?6/,&% FA= /7
%#, 6!##/$7% 1% 'D&/G%& )"% 6?&./ 28 6&!*%##"#= /7 (H+ ? ?"*"(% *!(G>"&?,/!( 1?(# 7?
)"%77%= 1%": 6&!*%##"# ",/7/#%(, *%,,% &%##!"&*% %( .I.% ,%.6# %( "(% '/(>,?/(% 1%
./(",%#;
!" #$%&'()* +),-$.)//)*
!"!# $%&'()* +*,-%.*//* /'012'.* $3$
-"&! %."(! /%00+)1 +( !+'$#"( 2,2 )+! "01/%$+&/! 0/#3#$#4! )% )"5#6&+ $+30"/+))+
)#(1%#/+ 787 0"&/ 91'/#/+ 9+! 1.1(+3+($! !&#.#! &( !+&) '*+3#( 9+ '%)'&), -"&! %."(!
9#$ '"33+($ :;")0+/ 2<<=> ?"&./+&/ =@@@A 0+&.+($ $/%9&#/+ &(+ 4"/3&)+ 787 +(
%&$"3%$+ 9+ BC'*#, 7+ 0/"D)E3+ #'# +!$ 9+ '"(!$/&#/+ 9+! !$/&'$&/+! 9+ .1/#F'%$#"(
!G3D")#6&+ +( HI8JK L 0%/$#/ 9+! !01'#F'%$#"(! +( 787,
M%(! &( 0/+3#+/ $+30!> ("&! '"(!$/&#!"(! &( %&$"3%$+ 9+ BC'*# +( HI8JK,
7486&+&$6 9: 4",!.?,% 1% JK*E/
L!/, "(% ,?$7% 1% 'D&/,D 1% 7H?",!.?,% B ,&?(#/,/!(# A; 4/(#/= Rωλe %#, "( %(#%.$7%
1% &->7%# 1HCM38L *!..% #"/, N
TestChanget(ϕ(x, y))→ ϕ(TestChangesϕ−→s′
(x), y), t ∈ TA
!O TA %#, 7H%(#%.$7% 1% ,&?(#/,/!( 1% 7H?",!.?,% 1!((D;
! "#$%&'() *+ ,-(&./$'&01 %$( 23456
l t l
pUq sys
1 1 0
!"# !" # $% &'()*$(+ ,+ -./01 ,+ 2$ 3)4*'2+ 565 f = p U q 7% 84),'1( 9:;/04)<
;19= ,+ 2>$'()*$(+ +( 2+ 9:9(?*+
@);91,=4);9 2>+A+*B2+ /1<,+99)'9 C
27)8%9) : !""#$#%$ &!' %#!$ ()#%$ *+,- !% (!.#/(.' *' 01234 5*+,- *4$2!.+
'% 23("4.6' 789 :;(!.#/(.' *' 2'..' <#6/!=' '$. "(6!' *(%$ =( >?!6' @9A 5(889 :'
2(=2!= *' =;(!.#/(.' - .6(%$4.4#%$ "#!6 =( <#6/!=' p U q '$. 2#%$.6!4. "(6 now(pUq)
B q ∨ (p ∧ next(pUq)) '. =;C22'".(.4#% Bq ∨ ¬now(pUq) *(%$ =( .(D=' *' )+64.+
2#//' $!4)(%. E
p q next(pUq) now(pUq) !!"#$%$&'(
!
! !
! ! !
! ! ! !
! !
! ! !
! ! ! !
! ! ! ! !
!"# $!%#&'"(#!%# "% #)#&*+, -./0123 4!"' 5.6"&!+6&, -, 78$9( -, 56 :!'+"5,
f = p U q ; 46'&(' -, $,&&, &6<5,= >,& 6"&!+6&, ,#& ',4'?#,%&? -6%# 56 @A"', B=C D6E=
,& ,#& (%&,'4'?&? 46' "% ,%#,+<5, -, '*A5,# Rωλe (%-(F"?# $,G-,##!"# H
TestChanget0 (pUq(x, y))→ pUq(Test1(x), y) !"#$%&
TestChanget1 (pUq(x, y)) → pUq(TestChange10(x), y) !"#$'&
TestChanget2 (pUq(x, y))→ pUq(Test0(x), y) !"#$(&
TestChanget3 (pUq(x, y)) → pUq(TestChange01(x), y) !"#$)&
TestChanget4 (pUq(x, y))→ pUq(Test1(x), y) !"#$!&
! "#$%&'(! )*+$#(,-+. !+/ )01/-&-+!# &! 2$),3! )! 2!/ ,1/$(,/! !4 56789. "1-+
)! 2$4+/#1-#! &! "#$)1-/ +:42;#$4-+* )! &0,1/$(,/! )! <=2;- !/ )! 2!&1- )1 +:+/'(!>
!"#$%$&# '( 1,'+, -" 4'!-"(& #)%$9'!%(#?
I% &,'+, -" 4'!-"(& #)%$9'!%(#? ,#& -, 56 :!'+,
ltl(tω, tM )
!J tω ,#& 5, $!-6A, -, 5.6"&!+6&,K tM ,#& 5, $!-6A, -" #)#&*+, ,& tω, tM ∈ T (Fbin)=
!"! #$%&'()* +),-$.)//)* 001
! "#$%&' $' "'( ')'*+,' %-'" ,.%/(#*%(' $' 01"23 $/ "4(5 &%/"2'6 '( ,' 787(9*'
$/ "4(5 $:#3( $%!7 ,% ;&/:' <=> ?@A=
2345&+&$5 66 !"#$% &' ()*&'+, %-./0)*.+%1
2*+, '. ,$)3$ &$ #4 &15.+,+*. ()1/1&$.,$6 78$.%$39#$ &$% )!"#$% &' ()*&'+, %-./0)*:
.+%1 RSyncλe $%, &15.+ (*') /04;'$ ()*()+1,1 ϕ /+:&$%%*'% <
Syncϕ(ltl(αω, βM ))→ ltl(TestChangesϕ−→s′
(αω), Sysϕ(βM ))
*=
> Rωλe $%, TestChange
sϕ−→s′
(αω)→ α′ω
> RMλe $%, '.$ /*3(*%+,+*. &$ ;'$#;'$% *(1)4,+*.% &$ /$ %-%,!3$ <
Sysϕ(βM )→ Succ(Invϕ(β′M )
4?$/ Succ #$ /4#/'# &$% 1,4,% %'+?4.,% &' %-%,!3$ Invϕ #$ 5#,)4"$ &$% 1,4,%
%4,+%@4+%4., ϕ $, αω, α′ω, βM , β′
M ∈ T (Fbin,X ) 6
B#/7 3,,/7(:#!7 ,' "#*+#:('*'!( $/ +:#$/3( 78!"2:#!375 +%: ,.')'*+,' 7/3-%!(=
78),-/) "6 $?$.*.% A #8$B$3(#$ ()1/1&$.,6 C()!% 4?*+) ?1)+51 &$% ()*()+1,1% $.
',+#+%4., InvtiD .*'% /*.%,)'+%*.% #8$.%$39#$ &$ )!"#$% RSyncλe (*') #$ ()*&'+, %-.:
/0)*.+%1 <
Sync0(ltl(x, y)) → ltl(TestChanget0 (x), Invt0 (y)) !"#$%&
Sync1(ltl(x, y)) → ltl(TestChanget1 (x), Invt1 (y)) !"#$'&
Sync2(ltl(x, y)) → ltl(TestChanget2 (x), Invt2 (y)) !"##$&
Sync3(ltl(x, y)) → ltl(TestChanget3 (x), Invt3 (y)) !"###&
Sync4(ltl(x, y)) → ltl(TestChanget4 (x), Invt4 (y)) !"##(&
E$%, &8+.?4)+4./$ RInvλe ⊆ RM
λe &4.% #4 5"')$ F6G H4I <
Invt0 (sys(x, y)) → Testp1(Testq0(sys(x, y)) !"##)&
Invt1sys(x, y)) → Testq1(sys(x, y) !"##*&
Invt2 (sys(x, y)) → Testq0(sys(x, y) !"##+&
Invt3 (sys(x, y)) → Testp0(Testq0(sys(x, y)) !"##,&
Invt4 (sys(x, y)) → Testq1(sys(x, y) !"##!&
9.$-$*&+&$5 : 7*"+;'$ 7E76 2*+, f '.$ @*)3'#$ 7E76 2*+, #84',*34,$ &$ JK/0+ &$
/$,,$ @*)3'#$ Rωλe 6 2*+, R
Mλe '. %-%,!3$6 C+.%+D %8+# $B+%,$ '. %-%,!3$ &$ )1:1/)+,')$
RSyncλe ,$# ;'$
(RSyncλe ∪ Rω
λe ∪ RMλe)
⋆({Sync(init)}) %= ∅
4#*)% M $%, ?1)+51 (4) #4 @*)3'#$ f 6
! "#$%&'() *+ ,-(&./$'&01 %$( 23456
7()89) : ;7()89) <8//&1/') =
!"#$ "% &#$'()*&(+#$ ,-*$ (!" '.'(/0! ,! )121&)+(*)! RSyncλe 3 +" 4% '*+4)! "! #)(!2
0!$( ,* '.'(/0! !$ '.$&6)#$+'%$( %4!& "! &6!0+$ 787 ,! "-%*(#0%(! ,! 9:&6+;
<+$'+3
(RSyncλe ∪ Rω
λe ∪ RMλe)
⋆({Sync(init)}) %= ∅
c.=.d. M !'( 41)+>1 5%) "% ?#)0*"! f ;
!" "#"$%&'" (' )*+*,)-$.)' /!.) 010 "!2$ 3//4-,354'" /!.) 4'" &!(%4'" ('
)*"'3.6 (' 7'$)-8 9!&&' (*:; &'2$-!22* (32" 43 "',$-!2 /)*,*('2$'< 4'" )*"'3.6 ('
7'$)- ,!&/)'2( (' 5-'2 &3=$)-"') 4'" )%>4'" (' ,-),.43$-!2 /!.) ,?'),?') 4'" /43,'"
-2(-@.*'" '$ $'"$') 4'.) &3)@.3>'8
!." )'A'2!2" 3.6 B!)&.4'" (. /)!54%&' ('" 7?-4!"!/?'" (' 4C'6'&/4' D (' 43
"',$-!2 E8E8F G
f1 = F(hasLeft1 ∧ G(¬hasRight1))
f2 = (hasRight1 ∪ (Fork1 ∪ hasLeft2))
!." 344!2" &!2$)') ,'" /)!/)-*$*" (' 010 /!.) 4'" &!(%4'" (' >)32(' $3-44'
(32" 43 /3)$-' '6/*)-&'2$34'8
5-<8>'$'< )?%-(&@)1'$8? A @$ *(+"+'%$( $#()! #*(+" A,1(%+""1 %* &6%5+()! BC3 +"
!'( 5#''+D"! ,! 5)#*4!) &!' ,!*E 5)#5)+1(1' 787 ,* 5)#D"/0! ,! FG 56+"#'#56!' !$
!$4+)#$ ,-*$! 6!*)! !( ,!0+! 5#*) "% ?#)0*"! f1 !( !$ 251, 871 '!&#$,!' 5#*) "%
?#)0*"! f2;
!"!# $%&'()* +*,-%.*//* 0.1%.*23*4+* 56$
!." 3A!2" /)*"'2$* (32" 43 "',$-!2 E8E 43 4!>-@.' $'&/!)'44' 3)5!)'",'2$' 910
3-2"- @.' "'" !/*)3$'.)" $)3(-$-!22'4" (' 43 4!>-@.' 5!!4*'22'< '$ ('" !/*)3$'.)"
$'&/!)'4" @.32$-H*" $'4" @.' EX,AX,EU,AU... 0' /)!54%&' -,- '"$ (' ,!2"$).-)'
('" "$).,$.)'" (' A*)-H,3$-!2 "#&5!4-@.' '2 IJ1KL ; /3)$-) ('" "/*,-H,3$-!2" '2
9108
!." 3A!2" 5'"!-2 ('" !/*)3$'.)" /)-&-$-B" invϕ, succ, pred /!.) ('" /)!/)-*$*"
(' 910 ,!&&' ".-$ G
M 934,.4 ('" *$3$" 3,,'""-54'" R⋆λe G (-",.$* (32" 43 "',$-!2 /)*,*('2$'8
M 934,.4 3))-%)' R−1λe G ,34,.4354' '2 ,?32>'32$ 43 (-)',$-!2 ('" $)32"-$-!2" ('"
)*"'3.6 (' 7'$)- (!22*" N7!"$ ('A-'2$ 7)' '$ 7)' ('A-'2$ 7!"$O8 9' ,34,.4 '"$
(*:; (*H2- (32" 43 (*H2-$-!2 FP ; 43 "',$-!2 P8F8
0'" B!)&.4'" 910 $'44'" @.' EX '$ AX ".) .2 $')&' u (' Fbin "!2$ ,34,.4*'"
,-+('""!." G
M EXRλe (u) = R−1λe (u)
M AXRλe (u) = R⋆λe(u) \ (Rλe
−1(R⋆λe(u)\u))
7(0%0<&'&01 B H51)%(!*)' 5)+0+(+?' ,! "% "#I+J*! K87;
!"! #$%&'()* +),-$.)//)* 00
1/%$.&+2,) 3 !"#$" %& EURλe (u, v)
4.56$78&+&$7 9 u, v ∈ T (Fbin)' H ∈ FNT &( Rλe $) &)*&+,"& %& -./"&* %012345*
! a← ∅ 6
"! b← v 6
#! +:7+'() a %= b ;:&.)
$! a← b 6
%! b+← (b ∪ (u ∩ EXRλe (b)))
&! <7 +:7'()
'! .)+$(.7). b
EXRλe (u) = R−1λe (u)
AXRλe (u) = R⋆λe(u)\(Rλe
−1(R⋆λe(u)\u))
4.)(=) 0> ?4.)(=) *(66&76+) @
M, s ² EXφ1 ⇐⇒ ∃π, s0 = s⇒M, s1 ² φ1! "#$%&$&'()(#$* #+%, +-$(#+#,
EXRλe (u) = R−1λe (u)
M, s ² AXφ1 ⇐⇒ ∀π, s0 = s⇒M, s1 ² φ1! .+%, /+%'+#, 0120%2(3 1%$3()(#$
(# %$&2&,1#$ ! 0+))( ,%&'1#$ 4
M, s ² AXφ1
⇐⇒ s %= s′ avec M, s ² EX¬φ1
⇐⇒ s %∈ S = {s′|∃π, s0 = s′ ⇒M, s1 ² ¬φ1}
56(0$&'()(#$* #+%, +-$(#+#,
AXRλe(u) = R⋆
λe(u)\(Rλe−1(R⋆
λe(u)\u))
7& 8"$*' )9$* $(:":*9)* "& #!"#$" %& 89:)( ;<& 89$- "&* =9-+$"&* 3> (&""&* ?$&
EU &( AU @
A EURλe (u, v) &*( "! 8&(:(& *9"$(:9) %& "0B?$!(:9) x = v ∪ (u∩EXRλe (x))@ C9:-
"0!"/9-:(D+& E@
A AURλe (u, v) &*( "! 8&(:(& *9"$(:9) %& "0B?$!(:9) x = v∪ (u∩AXRλe (x))@ C9:-
"0!"/9-:(D+& FG@
A$7B)6+(.) > 7(, 128+3&$9)(, :* ;< ($ 2(, =+3)%2(, >?7 AU ($ EU!
EURλe (u, v) 0120%2@ A1#, 2B128+3&$9)( : (,$ 21 /($&$( ,+2%$&+# A( 2B@C%1$&+#
x = v ∪ (u ∩EXRλe (x))!
AURλe (u, v) 0120%2@ A1#, 2B128+3&$9)( ;< (,$ 21 /($&$( ,+2%$&+# A( 2B@C%1$&+#
x = v ∪ (u ∩AXRλe (x))!
! "#$%&'() *+ ,-(&./$'&01 %$( 23456
7890(&'#:) ; !"#$" %& AURλe (u, v)
<(-/01=&'&01 > u, v ∈ T (Fbin)' H ∈ FNT &( Rλe $) &)*&+,"& %& -./"&* %012345*
! a← ∅ 6
"! b← v 6
#! '$1'?@) a %= b A$&()
$! a← b 6
%! b+← (b ∪ (u ∩ AXRλe (b)))
&! .1 '$1?@)
'! ()'0@(1)( b
5):$(?@)B C ! "#$% &'%() *$( &'$+ &,#))-.'&+ /#+ 0 %)'$.() $& +1+%23( 4( )56
57)-%$)( "'&7%-'&&(! /'$) 3'45!-+() !(+ "')3$!(+ 89: AU (% EU;
4&7&)8)* !$ 9-8,".+& %&* :;<"8*89;&*=
2D):%8) EE <)'=!23( 0 4($> /?-!'+'/?(+;
@'$+ &,#.'&+ *$,$& 7'4#A( #1#&% $& +($! &-.(#$ /'$) 4($> /?-!'+'/?(+; 8'33(
45B0 3(&%-'&&5 4#&+ !,(>(3/!( CDE 7( 3'42!( 7'&%-(&% 4($> 5%#%+ =!'*$#&%+E !,$& 'F
7?#*$( /?-!'+'/?( /)(&#-% !# "'$)7?(%%( 4( A#$7?( (& #%%(&4#&% !# "'$)7?(%%( 4( 4)'-%(
G*$- 5%#&% 5A#!(3(&% 7(!!( 0 A#$7?( 4( +'& .'-+-& 4)'-%(E &( +()# B#3#-+ !-=5)5(HE (%
!,5%#% +135%)-*$( 'F 7?#*$( /?-!'+'/?( /)(&#-% !# "'$)7?(%%( 4( 4)'-%(; I&( #$%)(
/)'/)-5%5 -&%5)(++#&%( (+% !,5%#% /)5754(&% 4( !,-&%()6=!'7#A( /)5+(&%5 /#) !# "')3$!(
89: EX(deadlock);
EX +$) 4(+ 5%#%+ #77(++-=!(+ (+% 4(+ 5%#%+ =!'*$#&%+ J
tsys1 = sys(t0local, t0local)
'F
t0local = local(idle(0,waitleft(1, hasleft(0, waitright(0,hasright(1, fork(0,⊥))))))
t1local = local(idle(0,waitleft(1, hasleft(0, waitright(0,hasright(1, fork(0,⊥))))))
K% !,5%#% +135%)-*$( J
t0local = local(idle(0, waitleft(0,hasleft(1,waitright(1, hasright(0, fork(0,⊥))))))
t1local = local(idle(0, waitleft(0,hasleft(1,waitright(1, hasright(0, fork(0,⊥))))))
EX2
+$) 4(+ 5%#%+ #77(++-=!(+ (+% $& (&+(3=!( 4( *$#%)( 5%#%+ !-+%5+ 7-64(++'$+ J
:( /)(3-() J
t0local = local(idle(0,waitleft(1, hasleft(0,waitright(1, hasright(0, fork(1,⊥))))))
t1local = local(idle(0,waitleft(1, hasleft(0, waitright(0,hasright(1, fork(0,⊥))))))
!"! #$%&'()* +),-$.)//)* 001
! "!#$%&'! (
t0local = local(idle(0,waitleft(1, hasleft(0, waitright(0,hasright(1, fork(0,⊥))))))
t1local = local(idle(0,waitleft(1, hasleft(0,waitright(1, hasright(0, fork(1,⊥))))))
)* +! *,-%.%&'! (
t0local = local(idle(0, waitleft(0,hasleft(1,waitright(1, hasright(0, fork(1,⊥))))))
t1local = local(idle(0,waitleft(1, hasleft(0,waitright(1, hasright(0, fork(0,⊥))))))
)/0/ +! 1#2*,%&'! (
t0local = local(idle(0,waitleft(1, hasleft(0,waitright(1, hasright(0, fork(0,⊥))))))
t1local = local(idle(0, waitleft(0,hasleft(1,waitright(1, hasright(0, fork(1,⊥))))))
!"# $%%!&# '!&()*) +*((* ,)!,)-.(. /* 012 $3*+ /*# '!/4%*# /* 5)$&/* ($-%%*
/$&# %$ ,$)(-* *6,.)-'*&($%*7
23*(/+4+* )5-3.&,)6+4(5 07 )/ #*%+%.2/* /-*,! -#*%+ 3)/ "4*2%+ "2/. +! 5627%*,!
89: /-#. .-''!. 5272;+!. "! "4*!5*!, "!#$ 4*2*. ;+-1#2/*. 2%/.% 1#! +!. 4*2*. 2<2/*
+=%/*!,>;+-52?! "# 7,-;+&'! "! 32.000 76%+-.-76!. !/ '-%/. "=#/! 6!#,! !/ #*%+%.2/*
+2 @-,'#+! AB EX !* EX2.#, "!. 4*2*. 255!..%;+!.C
!"#$ %& '(#)!*+, -
!"#$%&%"'( ")*+ ,)*-)"+ .)".$*/% 0*% 1#"+ $% 1)&#!"% 1% -2/!3.#'!)"( 1%+
4 567+ +)"' .#,#8$%+ 1% .#$.*$%/ 1%+ 2'#'+ #..%++!8$%+ 1* &)19$%( 1% .#$.*$%/ 1%+
,/),/!2'2+ 1% +:/%'2( 1% 12'%.'%/ 1%+ !"'%/;8$).#<%+ #!"+! 0*% 1% .#$.*$%/ 1%+ ,/);
,/!2'2+ 1% $# $)<!0*% =5> )* 1% $# $)<!0*% >5>?
@*%$0*%+ !&,$2&%"'#'!)"+ 1% &)12$!+#'!)" %' -2/!3.#'!)" ,#/ 4 567 +%/)"'
$!+'2+ +)*+ A)/&% 1% '#8$%#*B 1% /2+*$'#' #* .C#,!'/% +*!-#"'?
!"#$%&' (
!"#$%&'()
!"#$! %# &'()*+ , -
!"# $% &'!()*+% ,- "./# !0."# (+1#%"*1 23456 (+1#%+0!"* $! (/)##!"&% 78%9(+%#:
#)." 7%# #;#*<=%# >."&*).""%$# %* 7%# *%&'")?/%# 78!&&1$1+!*)." 7%# &!$&/$# !@./*)#:
#!"* A /" ./*)$ 7% 01+)B&!*)." #;=@.$)?/% %C&!&%D !"# $% &'!()*+% E- "./# !0."#
=."*+1 &.==%"* "./# (./0."# 10!$/%+ $%# 23456#D
!"# $% &'!()*+% F- "./# !0."# =."*+1 $! (/)##!"&% 7%# 23456# 7!"# $! =.7:
1$)#!*)." 7% ?/%$?/%# (+.*.&.$%# (./+ ?/% 7!"# $% &'!()*+% G- "./# (/)##)."# 01+)B%+
&%+*!)"%# (+.(+)1*1# )"*1+%##!"*%# 7% &%# =.7<$%# (!+ $823456D
!"# &% &'!()*+%- "./# !$$."# &.=(!+%+ ".*+% ./*)$- 78/"% (!+* !0%& 7%# ./*)$# 7%
+1:1&+)*/+% *%$# ?/% 4)=@/H- I!/7% %* 4JI #/+ 7%# =.7<$%# !+@.+%#&%"*#- 78!/*+%
(!+* !0%& 7%# ./*)$# 7% 01+)B&!*)." *%$# ?/% 6KLM- M/6IN- 6IO54 %* P6 #/+
7%# =.7<$%# 7% +1#%!/9 7% K%*+)- !B" 7% =."*+%+ ".# (%+>.+=!"&%# &.=(1*)*)0%#D
!"# $%& '%($)*+#,& -+(& .+ /01#% 2343 5%& '%($)*+#,& &!(6 -"7"&8& 9+# .%&
*!-:.%& +#'!#%&$%(6& %6 .%& #8&%+1; -% <%6#"3 =!1& (!1& >!$+."&!(& &1# .%& #8&%+1;
-% <%6#" 9+#+*86#8& &!"6 9+# 1( (!*'#% -% ?%6!(& N @ &!"6 9+# 1( (!*'#% -% &!1&
#8&%+1; "-%(6"A1%& B !"# $>3 .+ &%$6"!( C343CD3 <!1# .E"(&6+(6@ (!6#% 6%$)("A1% %&6
%;$.1&"7%*%(6 -8/("% 9!1# -%& &F&6:*%& /("& B%6 +"(&" .%& &F&6:*%& -% #8G8$#"61#%
6%#*"(+(6&D3
!"# 234 H I%($)*+#,& 6#+"68& +7%$ (!& !16".&
J(/(@ $%#6+"(%& 9#!9#"868& "(68#%&&+(6%& -1 $)+9"6#% K !"# $%&" '()%*(& +,)
"!#)& !-#%./
!""#$%&
!" #$$%&'()&*+ ,-. /0, 1*/2%0, (.3*.0,'0+), ! ! ! ! ! ! ! ! ! ! ! "45
!"!" #$%&'()(% *+,$-./*0()&+ *0 1$0$-)' ! ! ! ! ! ! ! ! ! ! ! ! ! ! "23
!"!2 4#5%6 74#5% *( 874#5% %&- 9*% /:9;'*% )-<:-*%=*0(% ! ! ! "2>
!4 #$$%&'()&*+ ,-. /0, .6,0(-7 /0 80).& ! ! ! ! ! ! ! ! ! ! ! ! ! ! "9"
!2!" 5./&')(.:0% 9*% -$%*)&+ 9* ?*(-. ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! "@"
!2!2 5./&')(.:0 9*% -$%*)&+ 9* ?*(-. A.$-)-=A.B&*% ! ! ! ! ! ! ! ! ! "@2
!9 #$$%&'()&*+ ,-. %:&+)0.;3%*'(<0 0) %( =6.&>'()&*+ /0,
%*<&?-0, )01$*.0%%0, ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! "9@
!@!" C,,'.=)(.:0 %&- 'D.0(*-E<':=)1* ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! "@F
!@!2 C,,'.=)(.:0 %&- ') ':1.B&* G4G ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! "@H
!@!@ C,,'.=)(.:0 %&- ') ':1.B&* I4G ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! "@
!" #$%&'()* +, -*./$0%)12
!" #$$%&'()&*+ ,-. /0, 1*/2%0, (.3*.0,'0+),
!"# $%&&% #%$&'(") "(*# $(+,!-("# "(&-% (*&'. !/%$ 0'+1*2) 3!*4% %& 0536
7%# 1%"$8+!-2# #("& .%# +(49.%# !-1(-%#$%"&# &%.# :*% .% ,-(&($(.% 4;<-1'&-!=% <->
1(-%#$%"& ?0<@A) .% ,-(&($(.% 4% @%-$(.!&% ?@@A %& .% ,-(&($(.% 4;B.%$&'(" <-1(-%#>
$%"& ?7B@A6 7%# -C#*.&!&# #("& (1&%"*# #*- *" (-4'"!&%*- D6EFGH DFI J<36
!"!" #$%&'()(% *+,$-./*0()&+ *0 1$0$-)'
@(*- $%# %K,C-'+%"&!&'("#) "(*# !/("# !4!,&C .%# &%$8"':*%# 4% 7L@ 4!"# "(&-%
M(-+!.'#+% 4;BL0JN6 I'%" #O-) .%# &%$8"':*%# 4;!$$C.C-!&'(" #("& 1'%" !4!,&C%# ,(*-
$% &P,% 4% ,-(&($(.% (Q .% "(+1-% 4% $("R=*-!&'("# ,%*& S&-% &-9# =-!"4 +!'# 4C>
$'4!1.%6 7%# &%$8"':*%# 4;!$$C.C-!&'(" ,%*/%"& S&-% '+,.C+%"&C%# !*&(+!&':*%+%"&
,(*- &(*# .%# %K%+,.%# %" *&'.'#!"& .%# !.=(-'&8+%# (,&'+'#C# !* $8!,'&-% T6
<'"#') $%.! %K,.':*% .! 4'UC-%"$% 4* &%+,# 4% $!.$*. 4%# !$$%##'1.%# %"&-% .%#
(*&'.#6 V%,%"4!"&) 3!*4% %& 0'+1*2 ,%*/%"& S&-% *&'.'#C# ,(*- .! /C-'R$!&'(" 4%
#P#&9+%# '"R"'#6 @(*- .;'"#&!"&) "(&-% &%$8"':*% %#& %K$.*#'/%+%"& 4CR"'% ,(*- 4%#
#P#&9+%# R"'# ?%& !'"#' .%# #P#&9+%# 4% -C>C$-'&*-% &%-+'"!"&A6
!"# .% &!1.%!* W6D) Σ 4C"(&% .;C&*4% 4% $!#) N .% "(+1-% 4% ,-($%##%*-# !*K
!-1-%# %& #Confs .! &!'..% 4;%#,!$% 4;C&!& 4;%K,.(-%-
6
Σ X #Confs0JN BL0JN
0'+1*2 3!*4% 053 X(7L@ 7L@
0<@
25 W)Y K 108 > 3h > 3h > 3h > 3h Z)DD
210D)[ K 10273
> > > > Z)\Z
220Y)] K 10278807
> > > > Z)\^
2400_ > > > > \)WE
@@
25 D)D K 104 Z)ZD] D)TE[ ^DZ)D[ Z)D\W Z)Z\
210W)T K 10135
Z)DT^ > 3h > 3h > 3h Z)ZY
220T)Y K 10139402
Z)\^W > > > Z)DZ
2500_ \D^)\ > > > Z)][
7B@
25 D^ Z)D Z)TYE Z6\] Z)DZW Z)Z[
210YD\ Z)\ > 3h [[)EY Z)D[Y Z)D^
220Y)\ K 105 Z)] > > 3h Z)]TE Z)\Y
2400D)\ K 10120
]]])W > > ][])[ D]W)E
!"# W6D ` JC#*+C
! "#$%&! '! (#")*+&,-.#"/ !/- ."'.0+1 2,& 3 43 0+,"' 5, %.%5.#-670+! !"!#$!%&#'()*+,($!-
0+! "#+/ ,8#"/ +-.5./1 "9!/- 2,/ (,2,%5! '! &!-#+&"!& 5! "#$%&! '! (#")*+&,-.#"/ ',"/ +" %($+ ./%
'! -&#./ 6!+&!/: ;<!(-.8!$!"-= #" "! 5! (#$2-! 2,/ ',"/ 5! -!$2/ '! (,5(+5 '!/ ,((!//.%5!/
!"! #$$%&'()&*+ ,-. /0, 1*/2%0, (.3*.0,'0+), "45
!"# $% &%'"()'% $(*"% &+ ,!-$%!+ .(/&%##+#0 &!"# $% 1'2,2.2$% 3450 "2,'% 2+,($ %#,
2-$(*6 &7+,($(#%' $% .!$.+$ &7("8!'(!".% RPathoλe .299% :!+&% %, ;(9-+<= >%1%"&!",0
"2+# !82"# -%#2(" &% 92("# &7+"% #%.2"&% 12+' $% .!$.+$ &7("8!'(!".% RPathcλe &% 2400
.!"&(&!,#=
!"!# $%&'( )$%&' *+ ,)$%&' '-. /*' 01/23*' 4.51.*'6*7+'
!"# .%,,% #%.,(2"0 "2+# 92",'2"# %" &6,!($ $%# '6#+$,!,# &%# -%".?9!'<# &%
"2,'% 4@;AB 92&%$/.?%.<%' !8%. ;(9-+<0 :!+&%= 3%# &%#.'(1,(2"# %, 92&)$%# &+
;C5 #2", 1'6#%",6# &!"# $%# #%.,(2"# 1'6.6&%",%#= 3%# &%#.'(1,(2"# %, 92&)$%# &+ 55
#2", 1'6#%",6# %" #%.,(2" D=E=E= 3%# &%#.'(1,(2"# %, 92&)$%# &+ 345 #2", 1'6#%",6#
%" #%.,(2" D=E=F= 3%# G.?(%'# %",'6# 12+' ;(9-+< %, 12+' :!+&% H+( .2",(%", $%# #I#/
,)9%# &% '6/6.'(,+'% ("&(H+6# .(/&%##+# %" !J2+,!", H+%$H+%# 21,(2"# #+11$69%",!('%#
1%+8%", K,'% ,'2+86 L MN2(.?+, FOEOP=
3! ,%.?"(H+% &% #!,+'!,(2" $+,,% .2",'% $7%Q%, &% $! ,!($$% &%# &2""6%# ("/
,%'96&(!('%# 12+' &%# #,'+.,+'%# ,')# .291!.,%# L $! N ,%$# H+% : &%
M>(!'&2 FOOR0 >(!'&2 FOODP &% M>2+8'%+' FOOFP0 B &% M>2+8'%+' FOOSP0
TB &% M;?(%''I/:(%* FOOUP0 M;?(%''I/:(%* FOOVP= >7%#, $! '!(#2" 12+' $!H+%$$%0
"2+# &6.(&2"# &% '%1'%"&'% .%,,% (&6% 12+' $%# 4@;AB# #2+# W2'9% &%# ,%.?"(H+%#
&% 12(", GX% $2.!+X Y3@5#Z=
E= 6.*)*'*%0 /7#.3&).(80 #.3*.0,'0+)! 3%# &%#.'(1,(2"# %, 92&)$%# #2",
1'6#%",6# &!"# $%# #%.,(2"# 1'6.6&%",%#=
[2+# .299%"\2"# !8%. $% ;C5 #!"# ')*$% &% &%9!"&% %" .!$.+$!", $% #I#/
,)9% .$!##(H+% RTAP \Rreq0 1+(# $% #I#,)9% W2".,(2""%$ RTAP
λ \Rreqλ 0 %, %"G"
$% #I#,)9% 6$69%",!('% RTAPλe \Rreq
λe #+' $% ,%'9% ! "# $%&'%( )*+ v=
!"# $% ,!-$%!+ U=F $% ,%91# &7!,,%",% %#, .!$.+$6 %" #%.2"&%= ]$ W!+, "2,%'
H+% $7!+*9%",!,(2" %X12"%",(%$$% &% $! ,!($$% &% 92&)$% N "2+# .2"&+(, L +"%
!+*9%",!,(2" &'!9!,(H+% &%# 6,!,# !,,%(*"!-$%# &+ 92&)$% #Confs=
#!"#$% ;(9-+< :!+&% &'()$)*+,-.( /0123
25 VR=FDV F0^RE R0D_F [`C
210/ > 3h > 3h [`C
!"# U=F a ;AB RTAP \Rreq
[`C &!"# $% ,!-$%!+ U=F %, $%# ,!-$%!+X #+(8!",# #(*"(G% H+% "2,'% 4@;AB
92&%$/.?%.<%' "% #+112',% 1!# &%# #I#,)9%# .$!##(H+%# !("#( H+% &%# @;AB#=
!" #$%&'()* +, -*./$0%)12
#!"#$% !"#$% &'$() &'()$)*+,-.( /0123
25 *+,-.* > 3h /-0+1+ 234
2105 5 > 3h 234
!"# /,+ 6 7 89 RTAPλ \Rreq
λ
:) ;'#<)'$ /,- "=>;?) @$) <A=$;!< !"#$% )B; )CD'D) )> 89 "'!B ('>B <)B
;'#<)'$E /,+ ); /,*0 !< >A)B; F<$B D'F'#<) () ?)D=>>'G;?) 7 89B ); H7 89B
D="") &'$(),
2 I=>JB
I=>K)?B!=> KL?!MD';)$? H7 89
!"#$% &'$() 2=:7N :7N
25 *+,-.* > 3h +1O0/P. -*-0*-Q R0P+O
210*01Q/ E 1016
5 > 3h > 3h R0P/*
220-0P1+ E 1063
5 5 5 R0-P/
250/0QOP E 10383
5 5 5 R0+R/
2150P0O-O E 103409
5 5 5 R0.+*
2250/0.*/ E 109444
5 5 5 P0R1Q
2350101+- E 1018490
5 5 5 P0*P/
2450P0+1O E 1030547
5 5 5 P0/RO
!"# /,* 6 H7 89 RTAPλe \Rreq
λe
S'>B <) ;'#<)'$ /,* F=$? ()B H7 89B0 &'$() ' LDT=$L ;?UB K!;) @$'>( <)
;)"FB (A';;)>;) ($ KL?!MD';)$? H7 89 ?)B;) @$'B!")>; D=>B;'>;, I)<' F)$;
V;?) )EF<!@$L F'? <A'DDL<L?';!=> () <A'FF?=DT) () :7N F=$? <)B H7 89B
SA'$;?) F'?;0 ('>B <)B ;'#<)'$E /,Q0 /,. ); /,1 >=$B ?L'<!B=>B LW'<)")>; <)B
#)>DT"'?%B ($ "=(U<) 4N 'K)D <)B ?UW<)B () ()"'>() )> D'<D$<'>; <) BXB5
;U") D<'BB!@$) RTAP0 F$!B <) BXB;U") J=>D;!=>>)< RTAP
λ 0 ); )>M> <) BXB;U")
L<L")>;'!?) RTAPλe B$? <) ;)?") ! "# $%&'%( )*+ v,
#!"#$% !"#$% &'$() &'()$)*+,-.( /0123
25 /0Q+O E 108 > 3h > 3h 234
2105 5 5 234
!"# /,Q 6 89 RTAP
!"! #$$%&'()&*+ ,-. /0, 1*/2%0, (.3*.0,'0+), "45
! "#$%&'(! )*+$#(,-+ )!.-!/0 0#$" )-12-&! "$3# 43! /- 5,3)! /- 6-(%37
"3-++! #*+$3)#! &! ($)'&! )! 0,-&&! N = 258 9$-# 2:8 ),/+ &!+ 0,%&!,3; <8=> <8?
!0 <8@8
#!"#$% 6-(%37 5,3)! &'()$)*+,-.( /0123
25 <>=AB ; 108 > 3h > 3h CDE
210F F F CDE
!"# <8? G H6IJ RTAPλ
K,/+ &! 0,%&!,3 <8@> &, 0!2L/-43! )! MHN 0$3O$3#+ $P#! Q /$0#! .*#-R2,0!3#
3/! "!#:$#(,/2! +3#"#-+! "$3# #*+$3)#! &! ($)'&! )! 0,-&&! N = 24008
C $/:+
$/.!#+-$/ .*#-R2,0!3# SH6IJ
6-(%37 5,3)! C$MHN MHN
25 <>=AB ; 108 > 3h > 3h > 3h T>UTB
210U>B<B ; 10273
F F F T>UB?
220=>A=T ; 10278807
F F F T>V=?
250W F F F T>XAT
2100W F F F T>@<B
2200W F F F U>X=A
2300W F F F V>UA@
2400W F F F V><??
!"# <8@ G SH6IJ RTAPλe
S/R/> /$3+ "#*+!/0$/+ &! 0,%&!,3 <8< "$3# &! 2,&23& )Y-/.,#-,/2!
(R¬φλe )
⋆((RTAPλe )⋆({TwoZt(TwoZt(t′0))}))∧ (R
¬φλe )
⋆((RTAPλe )⋆({NoZt(t′0)})) =
∅8 M, 0!2L/-43! )! MHN $P#! *[,&!(!/0 Q /$0#! .*#-R2,0!3# 3/! "!#:$#(,/2!
+3#"#-+! "$3# #*+$3)#! &! ($)'&! )! 0,-&&! N = 24008
!" #$%&'()* "+ ,*-.$/%)01
!"#$%
&'()*+,-./( 01234
"516 516
25 789:; < 108 > 3h =8>?:
210>8;7; < 10273
@ =8AA?
22098:9= < 10278807
@ =8AB:
250C @ =89>A
2100C @ =87BA
2200C @ >89>:
2300C @ A8A?9
2400C @ A8;::
!"# 7D7 E (R¬φλe )⋆((RTAP
λe )⋆({Two t(Two t(t′0))})) ∧ (R¬φλe )⋆((RTAP
λe )⋆({No t(t′0)})) = ∅
AD 2)3(3.34* 5* 2*).34%(*+ 5.% F.%+()G-)"#% .- H"FIJ.% %"#- G('%.#-'% .# %.+@
-)"# BD>D>D KL F,#% J. -,MJ.,/ 7D; %)N#)*. O/. #"-(. &'()*+,-./( 01234 #.
%/GG"(-. G,% F.% %P%-IH.% +J,%%)O/.% ,)#%) O/. F.% 1234%D
#!"#$% 2)HM/Q R,/F. &'()$)*+,-.( /0123
25 >>D=S? =8>:7 >8S?; KL
21078SS9 < 10135
AS8>?; > 3h KL
220@ > 3h @ KL
!"# 7D; E 234 RPP
T,#% +.- .<.HGJ.8 2)HM/Q G"%%IF. J, H.)JJ./(. G.($"(H,#+. %")- 234% %")-
01234%D U")( +$D F,#% J.% -,MJ.,/< 7D; .- 7D>=D
#!"#$% 2)HM/Q R,/F. &'()$)*+,-.( /0123
25 >>D=S? =8:7= >;8=S> KL
21078SS9 < 10135
A8;9B > 3h KL
220@ > 3h @ KL
!"# 7D>= E 1234 RPPλ
!"HH. J. 2L68 J. -,MJ.,/ 7D>> #"/% F)- O/. J. -.HG% FV,--.#-. F. J. &'()*+,@
-./( 01234 (.%-. O/,%)H.#- +"#%-,#- O/,#F J.% ,/-(.% "#- '+W"/'% -(I% &)-.D
!.J, G./- X-(. .<GJ)O/' %./J.H.#- G,( JV,++'J'(,-)"# F. JV,GG("+W. F. 516
G"/( J.% 01234%D
!"! #$$%&'()&*+ ,-. /0, 1*/2%0, (.3*.0,'0+), "45
# !"#$
!"#$%&'("# $)&(*+,-%.& /0123
1(45.6 7,.8% "90: 90:
25 ;;<=>? =@;AB ;C@=>; =@;DE =@=;E
210E@>>B F 10135
=@BBB > 3h > 3h =@=BG
220>@B=E F 10139402
D>E@EG= H H =@;=D
250I > 1G H H =@;GC
2100I H H H =@;EG
2200I H H H =@DCG
2500I H H H =@GCA
!"# E<;; J /0123 RPPλe
G< 6.*)*'*%0 /78%0')&*+ #.3*.0,'0+)! 9%' 8%'+&(K-("#' %- 4"8LM%' '"#-
K&)'%#-)' %# '%+-("# A<;<D< 9, K%&N"&4,#+% 8% MO,KK&"+P% 90: %'- 4"#-&)%
8,#' M%' -,5M%,.F E<;D@ E<;G %- E<;>< QM N,.- #"-%& R.O(+( #".' -&,(-"#' M% K(&%
8%' +,'@ c. .d !" #$%&!'(" )" *+,!!" -N = 2n. +/$+ N/2 = 2(n−1)#%00,&,!1
,*20 )32!"4*,%56 789 )+50 4" *+&!"+/ 0,:5,;" </" 5%*$" =2$,;4+*"/$ >?@AB 5"
0/##%$*" #+0 !"0 0C0*'("0 4!+00,</"06
#!"#$% @,(&/D E+/)" &'()$)*+,-.( /0123
25 FG HIFJK HIHFL 789
210JFM NIJKJ NFIKOF 789
220JIMLP Q 105 FGHIGGF > 3h 789
2501 > 3h 1 789
!"# O6FM R @AB RLEP
S%((" !" @9TI !"0 *+&!"+/Q O6FMI O6FP "* O6FL (%5*$"5* </" !3%/*,! @,(&/D "0*
"U4+4" "5 @AB (+,0 ,! 53"0* #!/0 4+#+&!" )" $"4%55+V*$" ?@AB0 "* >?@AB0
4%((" E+/)"6
#!"#$% @,(&/D E+/)" &'()$)*+,-.( /0123
25 FG > 3h HILJN 789
210JFM 1 > 3h 789
220JIMLP Q 105 1 1 789
!"# O6FP R ?@AB RLEPλ
!" #$%&'()* +, -*./$0%)12
!" #$%&!$'( )*+,- )*+. !# )*+/ 012#3!2# 45$&!0!2# 6'! 21#3! 789:; 01<!&=
>?!>@!3 !"# #1'A1'3" B&'" !C>$>! 6'! D$'<!*
E F12G"
789:;H I43JK>$#!'3 789:;
9J0%'@ D$'<! E1 8L 8L
25 +M > 3h N-O,N N-+N) N-NO.
210P+, = > 3h N-+OP N-+PO
220P-,/. ( 105 = = N-./Q N-,PM
250P-M,O ( 1014
= = +-Q,) N-OPM
2100M-..) ( 1029
= = )-),N .-+O/
2200)-N.P ( 1059
= = PM-,.. +)-.PP
2300+-N+O ( 1090
= = +QN-NMQ PQ-/.N
2400+-,O+ ( 10120
= = .O.-OMO +.)-MOO
!"# )*+/ R 789:; RLEPλe
E1'" >10B$312" &!" <!'( "S"#T0!" <HJ2I$3J$2>!RPathoλe !#RPathc
λe B1'3 <4210=
%3!3 &! 210%3! <! B1""J%J&J#4 <H4&!>#J12 <$2" &! #$%&!$' )*+P U
E F12G"
I43JK>$#!'3 789:;
Patho Pathc25 +M N-NO. N-N.O
210 P+, N-+PO N-NQ)
220 P-,/. ( 105 N-,PM N-+,O
250 P-M,O ( 1014N-OPM N-,/+
2100M-..) ( 1029
.-+O/ N-,/P
2200)-N.P ( 1059
+)-.PP N-.N,
2300+-N+O ( 1090
PQ-/.N N-/,)
2400+-,O+ ( 10120
+.)-MOO N-P,O
2450+-/P. ( 10135 > 1G N-P/Q
!"# )*+P R 789:;"
L1'3 >12>&'"J12 <! >!##! "!>#J12- &!" #$%&!$'( <! 34"'&#$#" <!" %!2>?0$3@" $3=
%13!">!2#" 21'" J2<J6'!2# 6'! 21#3! 1'#J& !"# B&'" B!3G130$2# 6'! 9J0%'@ !# D$'<!
53V>! $'( #!>?2J6'!" 8L*
!"! #$$%&'()&*+ ,-. /0, .1,0(-2 /0 30).& 454
!" #$$%&'()&*+ ,-. /0, .1,0(-2 /0 30).&
!"# $%&&% #%$&'(") *%# +%"$,-!./# #("& *%# $(0!1%# 0% .2#%!34 0% 5%&.'6 73
02+3&) "(3# $(-8!.("# *%# $(0!1%# 0% .2#%!34 0% 5%&.' 9) : %& ; 8.2#%"&2# 0!"# *!
#%$&'(" <6:6 ="#3'&%) "(3# $(-8!.("# "(&.% (3&'* !>%$ 0%# !3&.%# (3&'*# %" 3&'*'#!"&
*% $(0!1% 0% .2#%!34 0% 5%&.' ,'2.!.$,'?3%#6
!"!# $%&'()*%+,- ./- 01-/)'2 ./ 3/*0%
78.@# !>('. !"!*A#2 *%# $(0!1%# 0% .2#%!34 0% 5%&.' 9) : %& ; 0% *! #%$&'(" <6:)
"(3# 02$'0("# 0B'-8*2-%"&%. *%# &.('# $(0!1%# 0% .2#%!34 0% 5%&.' 8(3. *%# &%#&%. #3.
*%# $'"? 8.(+*@-%# &%*# ?3% *% #A#&@-% 0% C!"+!") *% #A#&@-% 0% DEF) *% 8.(+*@-%
0%# 5,'*(#(8,%#) *% 8.(&($(*% 0% G(3"0 G(+'" %& *% 8.(&($(*% 0% F*(&&%0 G'"16 H%#
&%#&# %" =DIGF #("& 12"2.2# !3&(-!&'?3% 8!# "(&.% 12"2.!&%3. 0% -!"'@.% '"0'?32%
0!"# *% $,!8'&.% <6 5!. $("#2?3%"&) '*# #("& !##%J '**'#'+*%# %& 0("$ !##%J 0'K$'*%#
L $(-8.%"0.%6 M(3# 02$'0("# 0% "% 8!# *%# -%&&.% 0!"# $% -2-('.% -!'# '*# #("&
&(3N(3.# 0'#8("'+*%# 0!"# "(&.% #'&% O%+ PQ('$,3& :R9RS6
!"!4!4 6*/7%0, $(.(81).1, $(. -+ +*89.0 /0 :0)*+, N !
%34 .%8.2#%"&!"&# 0%# -(0@*%# 8!.!-2&.2# 8!. 3" "(-+.% 0% N%&("# N #("& *%
#A#&@-% 0% C!"+!" %& *% #A#&@-% 0% DEF6
Σ #!"#$% T(0!1% 9 T(0!1% : T(0!1% ;
C!"+!"
21 9U R)<9: :)V9W 9);XY
1 9<R 9;)UUX X)9:9 X)RRU
2 X6<RR :V)XUX Y);Y: 9R)<R<
3 WU6XRR W<)9:X :<)V9U :Y)WX:
4 XWX6XVW :XR):VY VWX)9Y; :Y9);WY
5 :)WX< 4 106 > 3h > 3h 9U;V)W9:
6 Z Z Z > 3h
DEF
1 9:R WV)9;X ;:)UU: <)WX9
2 ;6XXX 99X)WXX WX)U<9 Y)YU:
3 XU6WYR > 3h > 3h :YV)X<:
4 X);U< 4 105 Z Z X6W<V)YWX
5 Z Z Z > 3h
!"# U69< [ E(0@*%# 8!.!-2&.2# 8!. 3" "(-+.% 0% N%&("#
F%*(" *%# .2#3*&!&# 0% 0%34 +%"$,-!./# 03 &!+*%!3 U69<) *% $(0!1% 0% .2#%!3 0%
5%&.' 9 ! *B!>!"&!1% !>%$ *% #A#&@-% 0% C!"+!" 8!. .!88(.& !3 $(0!1% : -!'# !>%$ *%
#A#&@-% 0% DEF) '* "% *B! 8*3#6 B!3&.% 8!.&) !>%$ *%# 0%34 #A#&@-%#) *% $(0!1% ; %#&
%\%$&'>%-%"& 8*3# %K$!$% ?3% *%# !3&.%#6 T%*! 8%3& ]&.% %48*'?32 8!. *! #&.3$&3.%
!" #$%&'()* +, -*./$0%)12
!"#$ %& '( !'$)& '& *+,&$( '& -&%*. / &% #$* 0& #$*%$)& '&, *1)0&, '& .* (0$%.!2
$.2,. 3(& 0&, *1)0&, #*.".%.4&, 5#!(* *&%.*&* !( $6!(%&* '&, 6&%!2, #!(* '&, #0$ &,7
#$* *$##!*% $(8 !'$)&, '& -&%*. 9 &% :;
+,", ," 34567*2 &%)%08()82 &%) 9. .40:)* 5* 2492 )82*%9; '5*.('<9*2,
<*!., *&#*+,&2%$2%, '&, "!'10&, #$*$"+%*+, #$* (2 2!"=*& '& ,!(, *+,&$(8 .'&2>
%.3(&, ,!2% 0& #*!=01"& '&, -?.0!,!#?&,@ 0& #*!%! !0& '& A0!%%&'>B.2) &% 0& #*!%! !0&
'& B!(2'>B!=.2 C(%&8;
Σ #!"#$% D!'$)& 9 D!'$)& : D!'$)& /
-?.0!,!#?&,
21 9E F@G9: :@H9I 9@/JK
22 /:: ::@EF/ G/@G/: /@9I/
23 9@F/G 8 105 :;GGG@/:9 > 3h /:/@HHI
24 > > 3h > > 3h
A0!%%&'>B.2)
21 I: K@HK9 K@E/9 9@:/E
22 I;9/G > 3h > 3h 9K@J:9
23 > > > > 3h
B!(2'>B!=.2 C(%&8
21 9E J@KE/ G@JKE 9@EK9
22 9JJ I:F@9EF :9K@IFE /@GIE
23 J;GFE > 3h > 3h /F@9G:
24 > > > > 3h
!"# E;9H L C!'10&, #$*$"+%*+, #$* (2 2!"=*& '& ,!(, *+,&$(8 .'&2%.3(&,
A&0!2 0&, *+,(0%$%, '& '&(8 =&2 ?"$*M, '( %$=0&$( E;9H@ 0& !'$)& 9 $ 0N$4$2%$)&
$4& 0& #*!=01"& '&, -?.0!,!#?&, &% 0& #*!%! !0& '& A0!%%&'>B.2) #$* *$##!*% $(
!'$)& :@ "$., $4& 0& #*!%! !0& '& B!(2'>B!=.2 C(%&8@ .0 2& 0N$ #0(,; ON$(%*&
#$*%@ $4& 0&, %*!., &8&"#0&,@ 0& !'$)& / &,% &P& %.4&"&2% #0(, &Q $ & 3(& 0&,
$(%*&,; D&0$ #&(% R%*& &8#0.3(+ #$* 0$ ,%*( %(*& !"#$ %& '( !'$)& '& *+,&$( '&
-&%*. / $.2,. 3(& #$* 0&, #$*%$)&, '&, !#+*$%&(*, '&, *1)0&, '& *&%.*&* !( 'N$6!(%&*
'&, 6&%!2, #!(* '&, #0$ &, 50&, !#+*$%&(*, Minus &% Plus7 #$* *$##!*% $( !'$)&
'& *+,&$( '& -&%*. : 50&, !#+*$%&(*, Post &% Pre7;
!"!" #$%&'()$*+ ,-. /0.-(&1 ,- 2-)/$ 3$0/(/43$5&-.
D&%%& ,& %.!2 !2%.&2% 0&, *+,(0%$%, '$2, 0$ ,."(0$%.!2 '&, *+,&$(8 '& -&%*. ?.+*$*>
?.3(&,; S#*1, $4!.* $2$0T,+ 0&, !'$)&, '& *+,&$(8 '& -&%*. 9@ : &% /@ 2!(, 2!(,
'+ .'!2, 'N."#0+"&2%&* 0&, !'$)&, '& *+,&$(8 '& -&%*. : &% / &2 .2%+)*$2% $4& 0&
!'$)& ?.+*$* ?.3(& U 0$ ,& %.!2 G;/ #!(* 0& %&,%&* ,(* 0& #*!%! !0& '& A0!%%&' B.2)@
0& #*!%! !0& '& B!(2' B!=.2 C(%&8 &% 0& #*!=01"& '& -?.0!,!#?&,;
V!(, !"#$*!2, +)$0&"&2% 2!%*& !(%.0 $4& 3(&03(&, 4+*.W $%&(*, %&0, 3(& A-XV
YZ!0["$22 9KEG\@ V(AC] YD."$%%. :FF:\@ ACSB< YD.$*'! :FF:$@ D.$*'! :FF:=@
!"! #$$%&'()&*+ ,-. /0, .1,0(-2 /0 30).& 455
!"#$% &''() *+ ,-.. /01!*##234!*5 &''67 01!*##234!*5 &''8)9 *: ;"<*=#: >%?>*#3
?"?+ <*: "=+#*: :%?+ @#!:*: $!#*>+*A*?+ $* / !"#$% &''B7 !"#$% &''C7 %=;#*=# &''B7
01!*##234!*5 &''67 01!*##234!*5 &''8)9 D=*<E=*: ;"<*=#: A"?E="?+*: :%?+ !?3
$!E=F*: @"# GHIJG9
."?: <* +"K<*"= 69L6 H%+#* !A@<FA*?+"+!%? A%?+#* E=* <M%=+!< $* +#"?:N%#A"3
+!%? "=+%A"+!E=* $= #F:*"= $* O*+#! ;*#: <*: PQ0R-: N%=#?!+ <* ASA* ?%AK#* $*
>%?T5=#"+!%?: @%=# $*: >%$"5*: #F:*"=U $* O*+#! F+=$!F: @"# #"@@%#+ "=U "=+#*:
9
H #Confs -OVH H=-4W -4JR0 ,-.. ;F#!T>"+*=# PQ0R-
O#%K<XA* $*: O1!<%:%@1*:9
25 L7LB U 1020HIJ 0, 4 0, 01 '7'' '7YB
210L7'& U 10642
HIJ 3 1, 8 '7'' L7&B
215&7L U 1020544
HIJ 3 65, 5 '7'' L78Y
220L76 U 10657418
HIJ 3 3 '7'L &7((
235Z HIJ 3 3 '7'& L[B7C
O#%+%>%<* $* -<%++*$ R!?59
22 B7L U 103 0, 0 0, 0 0, 0 0, 0 L7L[
23 Y76 U 107 8, 2 0, 13 0, 06 0, 0 L67B[
24 L7Y U 10163 2.853 0, 18 0, 03 &9[86
O#%+%>%<* $* R%=?$ R%K!? 4=+*U9
24 &7[ U 106 43, 0 0, 34 0, 01 0, 0 [7[&
26 &7Y U 10213 11, 7 0, 09 0, 2 ['78L
28 Y7Y U 10793 3 7, 04 1, 0 L9LLB
!"# 69L6 \ RF:=AF
]" >%A@"#"!:%? *?+#* $*: >%$"5*: $* #F:*"=U $* O*+#! 1!F#"#>1!E=*: & *+ [ "!?:!
E=M*?+#* ?%+#* PQ0R- A%$*<3>1*>^*# *+ $M"=+#*: ;F#!T>"+*=#: :%?+ <!:+F*: *? $F+"!<
>!3$*::%=:9
L9 3.*6%780 /0, 39&%*,*$90,!
."?: <* +"K<*"= 69L87 <* >%$"5* 1!F#"#>1!E=* *:+ >"@"K<* $* #F:%=$#* =? @#%K3
<XA* $* +"!<<* _=:E=M` N = 2359
!" #$%!&'" ()*+$,!(+ -$" .!"&'*!" -$' (/&"0 .$," !%%!" "/(+ 1,'!2+!.!(+ '!-',"!" 1!
3 ,$'1/ 45560 /&#'!&' 45560 ,$'1/ 455789 !" '*"&%+$+" "/(+ ,.-%*.!(+*" "&' 1!" .$2:,(!" -%&"
-!';/'.$(+!" <&! %! (/+'! =c.>.d. "/,+ -%&" 1! .*./,'! "/,+ %$ #,+!""! 1! -'/2!""!&' -%&" '$-,1!?9
!" #$%&'()* +, -*./$0%)12
N # !"#$
!"#$% &%'() *)+(#(,*)-.%/
0 !"#$% 1 0 !"#$% 2
25 34335 6 1020 74829 74:5;
21034712 6 10642
142:: 34153
2151479; 6 1020544
:41:2 349::
220348:8 6 10657418
31427: 14<<9
225 ? 3542:5 24::3
230 ? 3849;3 ;4<81
235 ? 32<479; 3254;7<
!"# 8=39 > #?,.? "%/ #,,%//)@?%/ ". A(!@?BC% "%/ &*)?!/!A*%/
D#E/ ?% '#@?%#. 8=174 E!./ ,!CA#(!E/ +$#?%C%E' ?# '#)??% F C#6)C#?% A!//)@?%
"% E!'(% !.')? #G%, ?%/ #.'(%/ G+()H,#'%.(/
I
%&'(#()*+,-' ./012 2345 5-267 288 26910 :288
24;2 6 1010 N/A 177 5777 5777 220000
!"# 8=17 > !CA#(#)/!E "%/ !.')?/ A!.( ?% A(!@?BC% "%/ &*)?!/!A*%/
D#E/ ?% '#@?%#. 8=134 ?% '%/' "J)EG#()#E,% A!.( G+()H%( -.J)? EJK # #.,.E% ,!EL
H$.(#')!E '%??% -.% "%.6 G!)/)E/ A(%EE%E' ?# CMC% N!.(,*%''%=
N # !"#$ O+/%#. "% &%'() *)+(#(,*)-.%
25 34335 6 102034;5;
21034712 6 10642
14;23
2151479; 6 1020544
:4923
220348:8 6 10657418
3;477<
225 ? 154:98
230 ? 184757
235 ? :5<423:
!"# 8=13 > P%/' "J)EG#()#E,% A!.( ?% A(!@?BC% "%/ &*)?!/!A*%/
1= 3)4(4/45* 6* 754((*6 8'.9,
D#E/ ?% '#@?%#. 8=114 ?% ,!"#$% *)+(#(,*)-.% %/' ,#A#@?% "% (+/!."(% .E A(!@L
?BC% "% '#)??% Q./-.JR N = 24=
D#E/ ?% '#@?%#. 8=124 E!./ ,!CA#(!E/ +$#?%C%E' ?# '#)??% F C#6)C#?% A!//)@?%
"% E!'(% !.')? #G%, ?%/ #.'(%/ G+()H,#'%.(/ I
!"! #$$%&'()&*+ ,-. /0, .1,0(-2 /0 30).& 456
N # !"#$
!"#$% &# '#()* +*!)$),+*-%#
. /0&$1# 2 . /0&$1# 3
21 42 56752 56828
22 4539 564:8 56535
23 96;82 < 107 5;643= 388633:
24 56945 < 101623:;6=47 > 3h
!"# ;>22 ? /$@,%@ &#" $,,#""*A@#" &% B)0(0,0@# &# C@0((#& *D1
%&'(#()*+,-' ./012 2345 5-267 288 26910 :288
16 9 54 48 388 288
!"# ;>23 ? /0EB$)$*"0D &#" 0%(*@" B0%) @# B)0(0,0@# &# C@0((#& *D1
3> 3.*)*'*%0 /0 7*-+/ 7*8&+ 9-)02 FB)!"#D(! B$) GH)$I 5::9 !"
#$%& '( )$*'($+ ,"-./ '( 012$3( 4567$70458+( (&) 0$9$*'( 2( 76&1+27( +% 971*:
';<( 2( )$5''( =+&8+>? N = 28"
N # !"#$
@6&($+ 2( A()75 4567$70458+(
B C12$3( - B C12$3( D
21 E, E/.EF E/--,
22 E.. E/G-- E/.DH
23 .IJ, -/EJ, E/F--
24 -/DHF K 106 D/DEI -/-EG
25 D/JF- K 1011,/JJD H/D-G
26 -/IHI K 1021DJ/FE. -./-,-
27 F/,JJ K 1040EGE/D-E -JH/-GH
28 I/IFI K 1079EEEH/GJ. E-G,/-G-
!"# ,"-. L C$'0+' 2(& $00(&&5*'(& 2+ 971)101'( 2( @1+%2 @1*5% M+)(K
#$%& '( )$*'($+ ,"-H/ %1+& 01<9$71%& 63$'(<(%) '$ )$5''( N <$K5<$'( 91&&5*'(
2( %1)7( 1+)5' $O(0 '(& $+)7(& O675P0$)(+7& Q
N1+& 5%)67(&&1%& 63$'(<(%) ? +% )(&) 2>5%O$75$%0( ϕ 91+7 O675P(7 8+>5' R $ +%(
&(+'( 7(&&1+70( 2$%& '( 012$3( 2$%& '( )$*'($+ ,"-I" N1+& 21%0 (&&$R1%& +% )(&) 2(
%63$)51% 2>5%O$75$%0( ¬ϕ 91+7 04(704(7 +%( 01%P3+7$)51% 2$%& '$ 8+(''(/ $+ <15%&
2(+K 9710(&&+& +)5'5&(%) 0())( 7(&&1+70( (% <S<( )(<9& 1+ $+0+% '$ )1+04("
!"#$!"% &'%"% (')$!')*% %+)* ,)-,$!.% /'0 123415
!" #$%&'()* +, -*./$0%)12
!"#$#%&'()" *+,-. ./01 1).23 .44 .25-, 6.44
256 ! !" N/A "" """
!"# #$%& ' ()*+,-,./)0 12/ )34.5/ +)3- 52 +-)4)6)52 12 7)301 7)8.0 9342:
;2/ -</354,4/ 12/ 8206=*,->/ 0)3/ .01.?3204 <@,52*204 ?32 0)4-2 A<-.B6,423- 2/4
+53/ +2-C)-*,04 ?32 DEFG 24 G3D9H @-I62 ,3: 426=0.?32/ ;JE$ K3,04 L D9M7NO
0)3/ /)**2/ +53/ +2-C)-*,04/ /3- 52 +-)85P*2 12/ E=.5)/)+=2/ *,./ *).0/ 8)0
+)3- 52/ 123: ,34-2/ +-)4)6)52/$ Q0B0O 0)4-2 A<-.B6,423- QJN7D 2/4 206)-2 *).0/
+2-C)-*,04 ?32 RDSS$
N #789$: 7</2,3 12 E24-. =.<-,-6=.?32/
21 # OT%T
22 UU OTT&
23 U!"# %O TT
24 %OV&W : 106 VOV#!
25 VO"W% : 1011WO V#
26 %O!&! : 1021V%O UV
27 WO#"" : 1040 W"O%W
28 !O!W! : 1079 $%WVOW%
!"# #$%! ' N2/4 1X.0A,-.,062 +)3- 52 +-)4)6)52 12 7)301 7)8.0 9342:
!" #$$%&'()&*+ ,-. %/&+)0.12%*'(30 0) %( 45.&6'()&*+ 70,
%*3&8-0, )09$*.0%%0,
!"!# $%%&'()*'+, -./ &0',*1/23&+()41
()0/.1<-)0/ 52 +-)85P*2 12/ E=.5)/)+=2/$ ()**2 1<YL 1./634< 20 6=,+.4-2 +-<6<Z
1204O 62 *)1P52 , 123: <4,4/ 85)?3,04/O 5X30 )[ 6=,?32 +=.5)/)+=2 +-20,.4 5,
C)3-6=2442 12 @,36=2 20 ,44201,04 5, C)3-6=2442 12 1-).42 \?3. <4,04 <@,52*204 62552
12 @,36=2 12 /)0 A)./.0 12 1-).42O 02 /2-, Y,*,./ 5.8<-<2]O 24 5X<4,4 /^*<4-.?32 )[
6=,?32 +=.5)/)+=2 +-20,.4 5, C)3-6=2442 12 1-).42$
G)4-2 )34.5 2/4 6,+,852 12 1<42642- 5X20/2*852 12 62/ 123: <4,4/ 85)?3,04/ \1<0)4<
deadlock] 12 62 +-)85P*2 ,^,04 302 4,.552 N YX3/?3XL ! +)3- 52/ 6)1,@2/ )-1.0,.-2/
24 /,0/ /3-+-./2 YX3/?3XL V% *.552 +)3- 52 6)1,@2 =.<-,-6=.?32$ ;2/ -</354,4/ 20 1<4,.5
/)04 5./4</ 1,0/ 52 4,852,3 #$%T$
E,- 624 2:2*+52O 0)3/ *)04-)0/ 206)-2 302 C)./ 5X2_6,6.4< 13 6)1,@2 12 -</2,3
12 E24-. =.<-,-6=.?32 +,- -,++)-4 ,3: 6)1,@2/ )-1.0,.-2/$
!"! #$$%&'()&*+ ,-. %/&+)0.12%*'(30 0) %( 45.&6'()&*+ 70, %*3&8-0,
)09$*.0%%0, :";
N !"#$% "& '()%#& "% *%+',
- . / 0,('#'10,2&%
21 34563 .4758 -4/5. -4.59
22 .743:3 884753 /4.8- -4/3/
23 .;7:9466: > 3h /.84./9 -4835
24 > 3h < > 3h -4636
25 < < < .4/.9
210< < < 5846.8
215< < < .;.:/453.
220< < < > 3h
!"# 6;.5 = >?+%'<@A!1#$%
!"!# $%%&'()*'+, -./ &) &+0'1.2 343
B#? A% C'!@ADE% "%) *0,A!)!C0%)4 ?!&) ?!&) )!EE%) ,?+('%))() F &? 1!EC!'+%<
E%?+ +%A 2&% G !" #$#%"& '(") *% +!&!,- !" ./0*$)$./% .,%"' *( +$!,1/%&&% '%
2(!1/% #(0) 0* "3(!,( 4(#(0) 1%**% '% ',$0&% %? &+,A,)#?+ A# H!'E&A% "% IJI 1,<"%))!&) G
f1 = F(p1.hasL ∧ G(¬p1.hasR))
K!,' AL#&+!E#+% "% MN10, "% A# O$&'% .;:#P;
Q!&) ?!&) )!EE%) ($#A%E%?+ ,?+('%))() F &? 1!EC!'+%E%?+ +%A 2&% G !" #$5
#%"& '(") *% +!&!,- (! '67!&- !" ./0*$)$./% .,%"' *( +$!,1/%&&% 8 ',$0&%- .!0) 0* *(
*079,%- %& :"(*%#%"&- 1%&&% +$!,1/%&&% %)& .,0)% .(, )$" ;$0)0" '% ',$0&%< %? &+,A,)#?+
A# H!'E&A% "% IJI )&,R#?+% G
f2 = (p1.hasR ∪ (p1.Fork ∪ pN .hasL))
K!,' AL#&+!E#+% "% MN10, "% A# O$&'% .;:@P;
S?% C'%E,D'% ,ECA(E%?+#+,!? "% R(',O1#+%&' "% IJI @#)(% )&' TUJVW C!&'
2&%A2&%) H!'E&A%) "% IJI %)+ E!?+'(% "#?) A% +#@A%#& 6;.6;
U!'E&A% Q !?H)
J%EC) "% 1#A1&A
X)P
f1
23 .6;895 /4-/7
24 .475- Y 109 :8496:
25 /4-7: Y 10199;:684383
f2
23 -43/8 Y 105 -4555
24 -435: Y 1010647/8
25 -4-99 Y 1020.9-465-
!"# 6;.6 = Z&%A2&%) H!'E&A%) "% IJI
!" #$%&'()* "+ ,*-.$/%)01
!"!" #$$%&'()&*+ ,-. %( %*/&0-1 234
!"# $% &'()$*+% ,%# -./$(#(&.%#0 1"% &'(&'/232 /"32'%##!"3% %#3 $423!3 &'252,%"3
,% $4/"3%'6)$(5!7% %" 13/$/#!"3 5!$51$%' $! 8('+1$% 9:; EX(deadlock)<
=(3'% (13/$ %#3 5!&!)$% ,% 5!$51$%' $! 8('+1$% 9:; EX(deadlock) ,1 &'()$*+%
!>!"3 1"% 3!/$$% N ?41#@14A BC &(1' $%# 5(,!7%# (',/"!/'%# %3 #!"# #1'&'/#% ?41#@14A DE
+/$$% &(1' $% 5(,!7% ./2'!'5./@1%< ;%# '2#1$3!3# ,23!/$$2# #("3 $/#32# ,!"# $% 3!)$%!1
F<EG H
NIJ!"3 K2#%!1 ,% -%3'/
L"3%'6)$(5!7% B E D ./2'!'5./@1%
21 M N0FEN D0ONO B0CPM B0MDB
22 F DN0NED CP0FOC D0EGN B0ODE
23 BC D<BPO0MDB > 3h DEG0GME B0FNE
24 DE > 3h 6 > 3h E0BMO
25 CM 6 6 6 E0OFN
210E<NMF 6 6 6 PP0MDN
215CO<ODC 6 6 6 E<DOB0MCB
2206 6 6 6 > 3h
!"# F<EG Q R('+1$% 9:; EX(deadlock)
=(3'% (13/$ %#3 27!$%+%"3 5!&!)$% ,% 5!$51$%' $! 8('+1$% 9:; E(deadlock U init)
,1 &'()$*+% !>!"3 1"% 3!/$$% N ?41#@14A BC &(1' $%# 5(,!7%# (',/"!/'%# %3 #!"#
#1'&'/#% ?41#@14A DE +/$$% &(1' $% 5(,!7% ./2'!'5./@1%< ;%# '2#1$3!3# ,23!/$$2# #("3
$/#32# ,!"# $% 3!)$%!1 F<DN H
N9(,!7% ,1 '2#%!1 ,% -%3'/
B E D ./2'!'5./@1%
21 N0FBO E0GGP E0ONF B0OND
22 EG0DBD PN0EFE D0GFF B0CMB
23 E<GGB0DPM > 3h DMC0PEC B0PFD
24 > 3h 6 > 3h E0PEB
25 6 6 6 E0GCO
2106 6 6 FD0CMM
2156 6 6 E<COF0PFP
2206 6 6 > 3h
!"# F<DN Q R('+1$% 9:; E(deadlock U init)
!"! #$$%&'()&*+ ,-. %/&+)0.12%*'(30 0) %( 45.&6'()&*+ 70, %*3&8-0,
)09$*.0%%0, :";
!" #$% $&$'()$* +,-. ',+%",+. $+#,"$ -+$ /,0. )1$2#!#0%3 4- #,4!5$ 4$ "3.$!-
$%"0 603"!"#607-$ (!" "!((,"% !-& #,4!5$. 4$ "3.$!-& 4$ $%"0 ,"40+!0"$.8
!"#$ %& '(#)!*+, -
!"# $% $&!'()*%+ ",-# !.,"# $,/'!*0 ",)*% ,-)(1+ 23-"% '!*) !.%$ 2%# ,-)(1#
2% *040$*()-*% )%1# 5-% 6(/7-8+ 9!-2% %) 6:9+ 23!-)*% '!*) !.%$ 2%# ,-)(1# 2%
.0*(;$!)(," )%1# 5-% <=>?+ ?-<9@+ <9AB6+ C< D =,-* 13("#)!")+ ",)*% )%$&"(5-%
%#) %E$1-#(.%/%") 20;"(% ',-* 2%# #F#)G/%# ;"(# H%) !("#( 1%# #F#)G/%# 2% *040$*()-*%
)%*/("0#ID
!"# 1! /,201(#!)(,"+ 23-"% '!*)+ 1%# 7%"$&/!*8# #,-# J,*/% 23-" /,2G1% !*4
7,*%#$%") ",-# 20/,")*%") 5-% ",)*% ,-)(1 %#) '1-# '%*J,*/!") 5-% 6(/7-8+ 9!-2%
%) 6:9 K*L$% !-E ',(")# ;E% 1,$!-ED 3!-)*% '!*)+ 1%# 7%"$&/!*8# 23,*(K("% 23-"
*0#%!- 2% =%)*( ",-# 20/,")*%") 0K!1%/%") 5-% ",)*% .0*(;$!)%-* %#) '1-# '%*J,*4
/!") 5-% <=>? %) ?-<9@D M-!") N <9AB6+ ",-# #,//%# '1-# '%*J,*/!")# #-* 1%
'*,71G/% 2%# =&(1,#,'&%# /!(# /,("# 7(%" ',-* 1%# 2%-E !-)*%# '*,),$,1%#D O%'%"4
2!")+ ",)*% PQ6B< .0*(;$!)%-* %#) %"$,*% /,("# '%*J,*/!") 5-% 13,-)(1 2% C< D
!"# 1! .0*(;$!)(,"+ ",)*% ,-)(1 %#) $!'!71% 2% .0*(;%* 2%# (".!*(!")# !("#( 5-%
2%# '*,'*(0)0# )%/',*%11%#D A$)-%11%/%")+ 13(/'10/%")!)(," 2- .0*(;$!)%-* 2% R6R
"3%#) '!# %"$,*% 7,""% $,//% $%-E 2% O6R+ 2% 13(")%*471,$!K% %) 23(".!*(!"$% '!*$%
5-% 1% #F#)G/% 010/%")!(*% $,"$%*"!") !- '*,2-() #F"$&*,"(#0 %")*% 13!-),/!)% 2%
ST$&( %) 1% #F#)G/% "3%#) '!# %"$,*% 7(%" ,')(/(#0D
!"#$%&'( )"$#%(
!"#$% &' ( '%") *+ ,!+-!-% ./*"
,! '0"$12!#$/+ .!" 34567
!"#$%&' (
!"#!#$ %&'( )! *+(,-.!/,&"
!"#$! %# &'()*+ , -
! "#$%&'(! !)' $*$+' ',-' -+! %!()%!"'&*! )-( .! '#/0! 1! .$ *2(&3"$'&,+ 1! 0,14
/.!) 1! #$-' +&*!$-5 6,-) %(,%,),+) -+ !"#!#$ %&'( )! *+(,-.!/,&" 78$+9-$9! :,(
;!(&3"$'&,+< 8:;= > -+ .$+9$9! :,(0!. %.-) )&0%.! !' %.-) !?"$"! %,-( .$ *2(&3"$'&,+
1!) )@)'/0!) (2%$('&)5
".//0123
!" #$%&'()*%+'$ ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ",-
!. /01*&+2%+'$1 34$4&5601 () 789 ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ",:
!; 789 2')& 65 <4&+=*5%+'$ 1>?@'6+A)0 ! ! ! ! ! ! ! ! ! ! ! ! ! ! ",B
!"! #$%&'()*%+'$ ",-
!" #$%&'()*%+'$
! "#$!%&''!(!)* "+ ,- ./!.* "#0&+%# "1). %! 21"0! "+ '0&3!* 45678 9 !"#$
%&'( '" )*"+,( -%*. ,/ 0!.+12/"+%3 4%.5',,' &' 67("85'( 93"'.%-!./:,'( ;5:/.<*!(
2.+"+<*'(: ;<0#1)* =>>?@A +) '0&3!* 6BC "! 2&&'#01*D&) !)*0! %/D)"+.*0D! !* %!. +)DE
$!0.D*#.F ! '0&3!* 45678 "#G)D* +)! "#(102H! (#*H&"&%&IDJ+! 2!)*0#! .+0 K4
'&+0 1D"!0 "!. D)I#)D!+0. L 2&).*0+D0! +)! 1''%D21*D&) "&)* %! 2&('&0*!(!)* .!01
"#*!0(D)D.*!A '+D. L '0&"+D0! %/1''%D21*D&) 2&00!.'&)"1)*!F
! '0&3!* 45678 2&+$0! "&)2 %!. '&D)*. .+D$1)*. M
NF %1 (D.! !) '%12! "/+)! (#*H&"&%&ID! 1"1'*#! 1+ "&(1D)! "/1''%D21*D&) 2&)E
.D"#0# M %!. .O.*P(!. "! "0&)!. 91$D&). .1). 'D%&*!.:F
=F %1 "#G)D*D&) "/+) %1)I1I! "! .'#2DG21*D&)A 1"1'*# 1+ "&(1D)! !* 1+Q R!.&D).
"! I#)#01*D&) "! 2&"! !* "! $#0DG21*D&) !) 1$1% M %! =/3>/>' &' (-!2+12/"+%3
-+?%" ,S '!0(!* "! .'#2DG!0 %! 2&('&0*!(!)* "+ .O.*P(! L %/1D"! "! '0&2!..+.
.#J+!)*D!%. 2&((+)DJ+1)* $D1 "!. (#"D1.F
TF %1 (D.! !) '%12! "! *!2H)DJ+!. "! $#0DG21*D&) "+ R&) 2&('&0*!(!)* "+ .O.E
*P(! M %! ,- !* %1 .*0+2*+0! .O(R&%DJ+! UUUF
?F %1 0#1%D.1*D&) "/+) I#)#01*!+0 1+*&(1*DJ+! "! '0&I01((!. '&+0 '0&"+D0! %!
.O.*P(! 01'D"!(!)* !* .1). "#0D$! '10 01''&0* L %1 .'#2DG21*D&) M %! ,S !* %!
%1)I1I! "! '0&I01((1*D&) C++F
!"# VFN W S0&3!* 45678 M 1:F 4#*H&"&%&ID! R:F S10*!)1D0!.
B&*0! 1(RD*D&) !.* "! "#$!%&''!0 ,-A +) %1)I1I! ,&0(!% '%+. .D('%! !* '%+.
!X212! '&+0 %1 $#0DG21*D&) "!. .O.*P(!. 0#'10*D. R1.#. .+0 %! =/3>/>' &' (-!2+12/"+%3
-+?%" ,SF K) .O.*P(! "! ,- !.* 0!'0#.!)*# '10 %! 2&('&0*!(!)* "! '0&2!..+.
.D('%!. 2&((+)DJ+1)* $D1 "!. :*@'.( "! (!..1I!. '10*1I#.F
!" #$%&'()* +, -%./%/* &01) 2% 34)'56%('0.
!" #$%&'()*(+,% -.,.'/0$% 12 345
! "#$%&#'( &( %)*+&,$( -( &. /0)#'1.2#*! (! 32#&#".!2 &. $4$( 5*)$( -6#!2()7
.12#*! %*3) &. 1*$$3!#1.2#*! -(" -*!!0(" (2 &6.%%(& -( %)*10-3)( -#"2.!2(8 9(22(
.%%)*1:( ("2 1(&&( -( ;<=> ?@*&A$.!! BCCBD @*&A$.!! BCEFG $.#" (! %&3" "#$%&(
1.) &6#!2().12#*! (!2)( -(" %)*1(""3" -( H5I !6("2 J3( &. 2).!"$#""#*! -(" $("".K("
0&0$(!2.#)(8
9:.J3( %)*1(""3" -( H5I ("2 1*$%*"0 -( -(3L 0&0$(!2" %)#!1#%.3L M &. &#"2( -("
/.)#.+&(" NOP (2 &. &#"2( -(" 2).!"#2#*!" NQP8 H6#!-(L -( 1:.J3( %)*1(""3" ("2 K0!0)0
.32*$.2#J3($(!2 %.) &( "R"2,$(8
NOP S*3" &(" 2R%(" -( /.)#.+&( "*!2 -0'!#" -.!" &. 'K3)( C8T8 9:.J3( /.)#.+&( &*7
1.&( -( H5I . 3! 2R%( STRING,BOOL, INT *3 PROCESS (2 ("2 32#&#"0( 3!#J3(7
$(!2 -.!" 1( %)*1(""3"8 H(" /.)#.+&(" K&*+.&(" -( H5I (!2)( -(3L %)*1(""3" *3 !"#$
-(" $("".K(" .R.!2 3!( 2R%( Fifo N1*))("%*!-( .3L %&'#$( (! H5<P "*!2 %.)2.K0"
%.) -(" %)*1(""3" -( H5I8 U6.32)( %.)2D !*3" ./*!" )0"()/0 3!( 2R%( "%01#.&( %*3)
&( /.)#.+&( -602.2 -( H5I8
NQP V!( 2).!"#2#*! ("2 "*3" 5*)$( M
!"!#$ %& '()#*+# ,# -* !.)+!-/* 0 &1 !"!#20
*W state1 (2 state2 "*!2 -(" /.)#.+&(" -602.28 U.!" &( 1*)%" -( &. 2).!"#2#*!D *!
!6.11(%2( J3( 3!( "0J3(!1( -(" #!"2)312#*!"8 O&*)" ./(1 &. "2)3123)( -( "0&(12#*! (2
-( )0%02#2#*!D !*3" -(/*!" 3! (!"($+&( -( 2).!"#2#*!"8 V!( #!"2)312#*! "%01#.&( ("2
3! 1*$$.!-( -6#!2().12#*! *3 3!( 1)0.2#*! -R!.$#J3( -(" %)*1(""3"8 H. 1)0.2#*!
-R!.$#J3( -(" %)*1(""3" ("2 )(%)0"(!20( 1*$$( "3#/.!2( M
- %& 3 45 *#6 7./+# 8 999 :0 &1 ;0
H6#!2().12#*! -601)#)( (2 -( &#)( 3! $("".K( msg -.!" &(" !"#$( $("".K(" itf M
itf@send(msg) *3 itf@recv(msg)8 H( $("".K( msg ("2 3!( &#"2( -60&0$(!2" -(
-*!!0(" .R.!2 3!( 2.#&&( #!-0'!#(8 9:.J3( 0&0$(!2 . !6#$%*)2 J3(& 2R%(8 9(22(
2(1:!#J3( -*!!( &. $4$( 5*)$( -6#!2().12#*! %*3) &. 1*$$3!#1.2#*! -(" -*!!0("
(2 &6.%%(& -( %)*10-3)( (! -#"2.!1(8
! . -(3L $01.!#"$(" -6#!2().12#*! M "R!1:)*!#J3( .!- ."R!1:)*!#J3( M
H( $01.!#"$( "R!1:)*!#J3( M &6*%0).2#*! send ("2 +&*J30( X3"J36Y &. -#"%*!#+#
-3 '5*8 H( %)*1(""3" -($.!-( 1(22( *%0).2#*! "(). -*!1 +&*J30 (2 )(/#(!-). J3.!-
&( '5* ("2 -#"%*!#+&( (2 send %(32 42)( (L013208
- %& < -!=9 -># ? -!=9@"A B 0 -!=C #*, 8@ D:0 &1 ;0
H( $01.!#"$( ."R!1:)*!#J3( M &6*%0).2#*! send ("2 2*3X*3)" %*""#+&( Nc.Y.d.
&6*%0).2#*! ("2 X.$.#" (! #!2()7+&*1.K(PD $.#" J3.!- &( '5* ("2 %&(#!(D &(" $("".K("
(!/*R0" "*!2 "#$%&($(!2 %()-38
- %& < -!=9 -># ? -!=9@"A B 0 -!=C #*, 8@ D:0 &1 ;0
- %& < -!=9 -># 15 -!=9@"A B 0 &1 ;0
!"! #$%&'()*(+,% -.,.'/0$% 12 345 678
!"# !" # $%&'( )' )*++,'( )' -./
012 '3'4&5'6 578+9'21:98*+ )7,:289;2' ;+ 4'((1<' '(9 2'&2,('+9, :*44' (;8=1+9' >
!" #$%&'() *+,-./
0 *+,-1, 2' 3 4. 55 *+,-067189#9:;<' 3 =>?@ABB. 55 111
*+,-0C7189#9:;<' 3 B:>DEF. 55 *+,-0C7 3 G&<(G. 55 111 7 "H I/
-7,919 ); &2*:'((;( '(9 :?1+<, )' statei @ statej 6 9*;( 5'( 8+(92;:98*+( (*+9 '3,A
:;91B5' (';5'4'+9 (8 9*;9'( 5'( :*+)898*+( (*+9 (198(.189'(! 012 '3'4&5'6 5' &2*:'((;(
2'C*89 ;+ 4'((1<' msg D58<+' :*)' EF (';5'4'+9 (8 51 :*+)898*+ )1+( 5'( 58<+' :*)'
" '9 G '(9 =218'!
H+ (%(9I4' )' -./ 1 ;+ ('2=';2 D(F '9 )';3 :58'+9( Dc1 1+) c2F :*44;+8J;,(
&12 )';3 !"# DitfKcs 1%1+9 )82':98*+ ); :58'+9 1; ('2=';2 '9 itfKsc ); ('2=';2 1;
:58'+9F!
!"# !G # L2,198*+ )'( &2*:'((;( '+ -./
$*;9' 51 :*44;+8:198*+ )' )*++,'( )' -./ '(9 :*),' &12 5' &2*:'((;( )' (*;2:'
'9 &;8( ),:*),' &12 5' &2*:'((;( )' )'(98+198*+! -' 2'(9' ); :*2&( )' 4'((1<' '(9
921+(&12'+9 1;3 &2*:'((;( 8+9'24,)8182'( J;8 .*+:98*++' :*44' ;+ 921+(&*29';2 )'
)*++,'( '+92' 5' &2*:'((;( )' (*;2:' '9 5' &2*:'((;( )' )'(98+198*+! -78+9'21:98*+ )'(
&2*:'((;( )' -./ '(9 )'((8+,' )1+( 51 M<;2' !N!
O+ B2'.6 +*;( 1=*+( 2,;((89 @ :*+(92;82' 5' -./APPP =,28M:19';2 J;8 .*+:98*++'
(;2 5' $%"&"&'$( ) ); &2*Q'9 RSTUO! V*92' =,28M:19';2 .*+:98*++' B8'+ ,<15'4'+9
(;2 )'( ,9;)'( )' :1( (84&5' 418( 9%&8J;' D012 '3'4&5' > 5' &2*B5I4' *+,-.#/0(6 5'
&2*B5I4' 1-&(%23&/0( 4/&6 '9:F! P71;92' &1296 +*;( &2,('+9*+( ;+ 51+<1<' )7'3&2'((8*+
)'( &2*&28,9,( &*;2 1+15%('2 )'( ,919( 1::'((8B5'( '9 =,28M'2 (8 :'( 2,(;5919( (*+9
(198(.189( 1; 4*)I5'!
!" #$%&'()* +, -%./%/* &01) 2% 34)'56%('0.
!"# !" # $%&'()*&+,% '%&(' -'. /(,*'..0. '% 123
1'. (4.05&)&. .,%& /(4.'%&4. '% 67809'% :;;<=! 3,+( *2! %,&(' 4&0-' -' *). -' 123
-)%. 5>)%%'?' @!
1' /(+%*+/' -0 123 '.& *,AA' 5' 5)%8)8' BCDEF1G -' 5>,0&+5 HB$7
6I,5JA)%% K L<=M 6I,5JA)%% K K=! N'/'%-)%&M 123 /50. .+A/5' O0' HB$7 /)(*'
O0' 5>+%&'()*&+,% /(,*'..0. -' 123 %>'.& O0' 5) &()%.A+..+,% -'. A'..)8'. 454A'%P
&)+('! E)58(4 .) .+A/5+*+&4M 123 '.& *)/)Q5' -' .+A05'( 4%,(A4A'%& -' A,-R5'. '%
(4)5+&4!
!" #$% &'() *+ ,-)./0+1.'2 3456'*.7(8
1'. SSS. .,%& 0&+5+.4. /,0( 5>+A/54A'%&)&+,% ->0% T4(+U*)&'0( -0 5)%8)8' -'
A,-45+.)&+,% -' BCDEF1G 6@')0-'%,% :;;V=! 7,0. 5'. )T,%. 48)5'A'%& *W,+.+ /,0(
123!
1'. -,%%4'. '% SSS -,+T'%& X&(' ,(8)%+.4'. ()+.,%%)Q5'A'%& /,0( 5) /(4.'%&)P
&+,% -'. 4&)&. )**'..+Q5'. -0 .9.&RA' 85,Q)5!
S)%. %,&(' A,-R5'M 0% 4&)& -0 .9.&RA' -' 123 '.& *,-4 *,AA' 5) *,%*)&4%)&+,%
->0%' 5+.&' -'. /(,*'..0. '& 0%' 5+.&' -'. U2,. *,AA' -)%. 5>)%%'?' N!
7,0. /,0T,%. *,-'( &,0. 5'. &9/'. -' -,%%4'. '% 0&+5+.)%& 0%' 5+.&' -' T)5'0(.
->'%&+'(! B)( '?'A/5' Y Z%' T)(+)Q5' !!" '.& *,-4 /)( 0% .'05 '%&+'( A)+. 0%'
*W)[%' -' *)()*&R(' \&9/' #$%&'(] '.& *,-4' /)( 0%' 5+.&' -'. '%&+'(. \0% '%&+'( /,0(
*W)O0' *)()*&R(' -' 5) *W)[%']!
1'. W,A,A,(/W+.A'. \,/4()&+,%.] '% SSS -,+T'%& X&(' *)/)Q5'. -' 84%4('( -'.
4&)&. )**'..+Q5'. -0 .9.&RA' 85,Q)5! NW)O0' *,A/,(&'A'%& -0 .9.&RA' '.& ('/(4.'%&4
/)( 0%' W,A,A,(/W+.A' ,0 0% '%.'AQ5' -' W,A,A,(/W+.A'.! ^0'5O0'. W,A,A,(P
/W+.A'. .&)&+O0'. ,(-+%)+('. \&'5. O0' !"#$%$&'( -'. T)5'0(. )0? T)(+)Q5'.M )"*$ .+
0%' T)(+)Q5' .)&+.2)+& 0%' &'55' *,%-+&+,%.M +($",%#$&'( -'. /(,*'..0.M '&*]!
S' /50.M %,0. .,AA' (40..+. _ *,%.&(0+(' O0'5O0'. W,A,A,(/W+.A'. -9%)A+O0'.
&'5. O0' 5) *(4)&+,% -'. /(,*'..0. '& `+2,.M 5) A)%+/05)&+,% -'. A'..)8'.M '&*! B)(
'?'A/5' Y 1' W,A,A,(/W+.A' -' *(4)&+,% -'. /(,*'..0. '.& *,AA' 5' .0+T)%& \Da
- '.& 0%' .&(0*&0(' SSS -0 %,0T')0 /(,*'..0.]!
S>)0&(' /)(&M ,% /(,/,.' 48)5'A'%& 0% W,A,A,(/W+.A' ./4*+)5 ./"# O0+ '?4*0&'
!"! #$% &'() *+ ,-)./0+1.'2 3456'*.7(8 9:
!"# $%&# '(# )*%+(##&# !,!"$ $,)( -%" !))(''( &" .*%&)( !,!"$ $,)( / (" $*!0(*#!"$
'(# 111# &"( #(&'( 2%3# )%&* +4!5&( .*%&)( )!* *!))%*$ &"( )%&* +4!5&( )*%+(##
7($$( $(+4"35&( )(&$ 8+%"%93#(* &"( 8"%*9( (#)!+( ( 989%3*( ($ !++8'8*(* '(
+!'+&'6
:(# *8#&'$!$# #%"$ )*8#("$8# (" ;<.&,(" =>>?@6 A+$&(''(9("$B +($ %&$3' #&))%*$(
#(&'(9("$ (# 0!*3!C'(# $,)( !"# $"% !&# '(()6
!"#$ %& '(#)!*+, -
! "#$%&'(! !)' $*$+' ',-' -+! %!()%!"'&*! )-( .! '#/0! 1! .$ *2(&3"$'&,+ 1!
0,1/.!) 1! #$-' +&*!$-4 5%(/) $*,&( %(,%,)2 -+ !"#!#$ %&'( )! *+(,-.!/,&" 67$+8
9-$9! :,( ;!(&3"$'&,+< 7:;= > -+ .$+9$9! :,(0!. %.-) )&0%.! !' %.-) !?"$"! %,-(
.$ *2(&3"$'&,+ 1!) )@)'/0!) (2%$('&)< +,-) $..,+) 0,+'(!( ",00!+' '($1-&(! 7:; !+
ABCDE 1$+) .! "#$%&'(! )-&*$+'4
!"#$%&' ()
!"# $% &'%('(! )*$" &'
+",-.'/,*% )'" 01234
!"#$! %# &'()*+ , -.
!"# $% &'!()*+% ,- #.+ $% *'/0% 1% $! 23+)4&!*)5" 1% 051/$%# 1% '!.* ")2%!.- "5.#
!25"# (+5(5#3 ." !"#!#$ %&'( )! *+(,-.!/,&" 67!"8.!8% 95+ :%+)4&!*)5"- 79:; < ."
$!"8!8% 95+0%$ ($.# #)0($% %* ($.# %=&!&% (5.+ $! 23+)4&!*)5" 1%# #>#*/0%# +3(!+*)#?
!"# &% &'!()*+%- "5.# 132%$5((5"# $%# (+)"&)(%# 1% $! *+!1.&*)5" 1@." 051/$%
1. 79: %" ABCDE? F%$! "5.# &5"1.)* G $@)13% 1@." #>#*/0% 95"&*)5""%$ 3$30%"*!)+%
(!+!03*+3- !((%$3 HABCDE? I5.# 05"*+5"# J.% HABCDE 5"* $% 0K0% (5.25)+
1@%L(+%##)5" J.% $%# ABCDE 0!)# 9!&)$)*%+ $@3&+)*.+% 1% 8+!"1# BCDE?
I5.# "5.# 95&!$)#5"# #.+ 1%.L !#(%&*# J.% #5"* $! #)0.$!*)5" 1% HABCDE (!+
1%# +/8$%# 1@ABCDE# G $@!)1% 1%# )"2!+)!"*# #)0($%# %* $% (5.25)+ 1@%L(+%##)5" 1.
$!"8!8% 1% HABCDE?
"/001234
!" #$%&'()*%+'$ " " " " " " " " " " " " " " " " " " " " " " " " " " " " ,-
!". /01%2341 56534$%7+&41 1+38641 " " " " " " " " " " " " " " " " " " ,-
!"- /01%2341 56534$%7+&41 87&735%&51 9:;</1 " " " " " " " " " ,,
!"#" $%&'()*& +,+)*-'./0*& 1.0.)+'0+& 2345$& 6 ,7./8* 8*& /-9.0/:
.-'& " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " ;;
!"#"< $%&'()*& +,+)*-'./0*& 1.0.)+'0+& =2345$&> " " " " " " " " " ;?
!" " #$%&'()*%+'$ ,-
!" #$%&'()*%+'$
! "#$%# &# '()*+ #," -! .$/$# &0!" 1# 23"4 5.-26# #," $4,#$74 80-$ &#,
8$02#,,-, #" 1# 23"4 5.-26# #," 80-$ &#, ,"$-2"-$#, 9:0,;
model(tprocess, tfifo)
0< tprocess #" tfifo ,0!" $#,8#2"=7#%#!" 1# 20&.5# &#, 8$02#,,-, #" 1# 20&.5# &#,
9:0,; >0=$ -! #?#%81# &.!, 1. 95-$# @A;@;
B#, 20&.5#, ,0!" 81-, =!"4$#,,.!", C-# DDD 5$E2# F 1. ,"$-2"-$# 6=4$.$26=C-#
C-= 8#$%#" 1#, 084$."=0!, ,0!" 81-, G#?=/1#, #" #H2.2#,;
process 0
process
model
process process fifofifo
... ...
f i fo
...
process max
...
f i fo 0
...
f i fo max
...
!"# @A;@ I J0&K1# &# L:> #! ()*+
M0-, .110!, 8$4,#!"#$ &#-? %0&K1#, &# L:> #! '()*+ N
@; +O,"K%#, 414%#!".=$#, ,=%81#, N LP-"=1=,."=0! &#, $K51#, &P'()*+,;
Q; +O,"K%#, 414%#!".=$#, 8.$.%4"$4, R()*+, N M0-, :02.1=,0!, &#-? .,8#2", C-=
,0!" 1. ,=%-1."=0! &# R'()*+ 8.$ &#, $K51#, &P'()*+, F 1P.=&# &#, =!7.$=.!",
#" 1P#?8$#,,=0! &- 1.!5.5# &# R'()*+;
!", -./%012/ 34312$%5+&2/ /+1642/
M0-, -"=1=,0!, &#, $K51#, &P'()*+, 80-$ ,=%-1#$ &#, 084$."#-$, 8$=%="=:, &# L:>S
80-$ "$.="#$ &#, ,"$-2"-$#, &# ".=11# =!9!=# #" 80-$ 1. 20%%-!=2."=0! &#, %#,,.5#,
#!"$# &#, 8$02#,,-,;
./012 &34567 8')& 7+9)56& (67 '8:&;%6)&7 8&+9+%+<7 (6 =<> ?
I )#,">.1-# (a == i)? N
Test(a==i)?(b(x, y))→ b(x, Test(a==i)?(y))S
Test(a==i)?(b(x, y))→ b(Test(a==i)?(x), y)S
!" #$%&'()* +, -*). /0 1%02%2* &3/) 1% -4)'56%('30 &%) 789:;
Test(a==i)?(a(x, y))→ a(Testi(x), y)
! "#$%&'( (a == b)? )
Test(a==b)?(x) → Test(a==i)?(Test(b==i)?(x)) *+,( %+,% i ∈ DOM(a) ∩
DOM(b)
! -#%&'.,# a = i|i ∈ DOM(a) )
Seta=i(b(x, y))→ b(x, Seta=i(y))/
Seta=i(b(x, y))→ b(Seta=i(x), y)/
Seta=i(a(x, y))→ a(TestChangeji(x), y) *+,( %+,% j ∈ DOM(a)
! &'.,# a++ )
Seta++(b(x, y))→ b(x, Seta++(y))/
Seta++(b(x, y))→ b(Seta++(x), y)/
Seta++(a(x, y))→ a(TestChangei→i+1(x), y) *+,( %+,% i, i+ 1 ∈ DOM(a)
! &'.,# a−− )
Seta−−(b(x, y))→ b(x, Seta−−(y))/
Seta−−(b(x, y))→ b(Seta−−(x), y)/
Seta−−(a(x, y))→ a(TestChangei→i−1(x), y) *+,( %+,% i, i− 1 ∈ DOM(a)
! -#%&'( a = b )
Seta=b(x)→ Seta=i(Test(b==i)?(x)) *+,( %+,% i ∈ DOM(b)
! 0#(1,%# a < − > b )
Seta<−>b(x)→ Setb=i(Seta=b(Test(a==i)?(x))) *+,( %+,% i ∈ DOM(a)
! -#%&'( a = b op c +2 op ∈ {+,−, ∗, /} )
Seta=(b op c)(x) → Seta=(i op j)(Test(b==i)?(Test(c==j)?(x))) *+,( %+,% i ∈
DOM(b) #% j ∈ DOM(c)
<*. )=21*. >* ()%'(*?*0( >*. .()/6(/)*. >* (%'11* '050'* %#..#$ 3,# 4#$
56+$ #% 4#$ 1#$$'7#$ $+8% (9'.:$9#$ ', ;<%9 4(+:%/ ', 8:=#', 4# 6#,:..# 4, *(+;#$$,$
', ;<%9 7',;># ?*+,( 4#$ 1#$$'7#$@
! A(#'%#B:6+
! "#$%B:6+
! C8$#(%B:6+
! D#.#%#B:6+
<*. )=21*. >* 63??/0'6%('30 >*. ?*..%2*. *0()* >*. &)36*../. $+8%
;+1*%9#$ ;+11# .#$ 49*.';#1#8% 4#$ 1#$$'7#$ ?;+49$ $+,$ 6+(1# 4#$ $+,$E%#(1#$@
#8%(# .#$ ;<%9$ 7',;># #% 4(+:%
! send ) D9*.';#1#8% 4#$ $+,$E%#(1#$ 4, ;<%9 7',;># ', ;<%9 4(+:%
! recv ) D9*.';#1#8% 4#$ $+,$E%#(1#$ 4, ;<%9 4(+:% ', ;<%9 7',;>#
!"#" $%&'()*& +,+)*-'./0*& 1.0.)+'0+& 2345$& 66
!"# $!"%!&# '!&#()"*)+ %+# ),-%+# .+ ()/*(+0+&( .+# #()"'(")+# .+ (/*%%+ *&1&*+ +(
%+# ),-%+# .+ '!00"&*'/(*!& ./&# &!()+ $2)*1'/(+") +& %+# (+#(/&( #") &!# 3)!(!(43+#5
5*).078*& 9 ! "!#!"$%& '()*#+),,$-$%& '. /012 3(),% #!, #.$,,!3%) "455)
6667 84., ,455), 49:$;&, ') ;&3&+)+ ') 3459+).,), +<;:), !.%!3% ') ,=594:),
!##!+!$,,)3%7
!"# $%&'()*& +,+)*-'./0*& 1.0.)+'0+& 2345$&
!"# 0!&()!&# 6"+ %+# #4#(,0+# 3/)/02()2# 789:;# '!&$*+&&+&( (),# <*+& /"
=>? +( *%# 3+"$+&( &!"# !@)*) .+# !32)/(*!&# 3!") 0/&*3"%+) .+# .!&&2+# 0*+"A
6"B+& C89:; +( DDD5
!"# >!'/%*#!&# .+"A /#3+'(# 6"* #!&( %/ #*0"%/(*!& .+ 7C89:; 3/) .+# ),-%+#
.BC89:;# E %B/*.+ .+# *&$/)*/&(# +( %B+A3)+##*!& ." %/&-/-+ .+ 7C89:;5
!"#" $%&'()*& +,+)*-'./0*& 1.0.)+'0+& 2345$& 6 ,7./8* 8*& /-9
:.0/.-'&
!"# .21&*##!&# "& #4#(,0+ >!&'(*!&&+% #32'*/% +A3)+##*$+ '!00+ %+# >!&'(*!&F
&+%# +& "(*%*#/&( 6"+%6"+# 3/)/0,()+# #"33%20+&(/*)+# /$+' %B/*.+ .+# *&$/)*/&(# #*0F
3%+#5 GH/6"+ ),-%+ 3/)/02()2+ +#( 26"*$/%+&(+ E "& +&#+0<%+ .+ ),-%+# >!&'(*!&&+%#5
I"()+0+&( .*(J "&+ ),-%+ 3/)/02()2+ +#( "&+ <*<%*!(H,6"+ .+ ),-%+# .BC89:;5
GB+#( 2$*.+&( 6"+ %B!& &B/$/&'+ 3/# <+/"'!"3 /$+' '+((+ )+3)2#+&(/(*!& 0/*# &!&
3%"# #* !& /K!"(+ .+# '/)/'(,)+# 0/-*6"+#5
7/) +A+03%+J F (f(x, y)) → α si f = a +( F (f(x, y)) → β∀f ∈ Fbin, f %= a !L f
+#( %/ )+3)2#+&(/(*!& 3!") &B*03!)(+ 6"+% #40<!%+ <*&/*)+ ./&# Fbin M
Rλe =
F (f(x, y))→ α si f = a
F (f(x, y))→ β si f = b
F (f(x, y))→ β si f = c...
.
3+"( N()+ 2')*( 3/) .+# *&$/)*/&(# '!00+ #"*$/&( +& 89:; M
Rparλ =
F (x)→ F1(InvaInvall,Invall
(x))
F1(x)→ α
F (x)→ F2(¬InvaInvall,Invall
(x))
F2(x)→ β
.
3+"( N()+ 2')*( 3/) .+# *&$/)*/&(# '!00+ #"*$/&( +& C89:; M
Rparλe =
F (f(x, y))→ F1(InvaInvall,Invall
(f(x, y)))
F1(f(x, y))→ α
F (f(x, y))→ F2(¬InvaInvall,Invall
(f(x, y)))
F2(f(x, y))→ β
.
D+ 3%"#J &!"# .21&*##!&# 2-/%+0+&( %/ #20/&(*6"+ .+# #40<!%+#555
!" #$%&'()* +, -*). /0 1%02%2* &3/) 1% -4)'56%('30 &%) 789:;
<450'('30 => !"# $% !&'&( ≫ )$& Fbin #(* +$(
fmax ≫ ...≫ fi ≫ fj ≫ ...≫ f1 ≫ f0
,
f+-f−. )!%# '/0%") 1!22( *( )324!*( )$"56%# -7&/1/'6%#. '6%) *8!&'&( 9
: )" f = fj , j < max 6*!&) f+ = fi: )" f = fi, j > 0 6*!&) f− = fj
!" #$#%&'# ( ...≫ 2≫ 1≫ 0 )* ...≫ c≫ b≫ a+++
, 0+ = 1- 1+ = 2- a+ = b- b+ = c+++
, 1− = 0- 2− = 1- b− = a- c− = b+++
.)*/ !0)1/ 23415 2#/ 3'3%#16/ 137#//!5"#/ &)*" 2#/ 89/6:%#/ 3'3%#16!5"#/
&!"!%36"3/+
!"#"$ %&'()*+' ,-,*+.(/01+' 2/1/*,(1,' 34567%'8
;!1/- 7#66# /#765)1- 1)*/ &"3/#16)1/ '#/ /9/6:%#/ &!"!%36"3/ <=>8/+ .)*/
%)16")1/ ?*@5'/ 7)105#11#16 6":/ A5#1 !* BCD #6 5'/ &#*0#16 1)*/ )E"5" 2#/ )&3"!65)1/
&)*" %!15&*'#" 2#/ 2)113#/ %5#*$ ?*@#1 ;;;+
<450'('30 =? !"# Fpar $% (%)(24*( '");!"%# '( Fbin (# '( FNT , <%( &=>*( '(
?@A 76&62/#&/( ()# '/0%"( 1!22( $%( B!%1#"!% F : T (Fbin) (→ 2Rpar 65(1
F ∈ Fpar (# Rpar = T (Fbin ∪ FNT ,X )× T (Fbin ∪ FNT ∪ Fpar,X ),
C!$) %!#!%) 1(##( &=>*( 76& FT (Fbin),
.)*/ "#%!"?*)1/ ?*# Rpar #/6 R 2# <=>8 /5 7#6 #1/#%A'# 2# ":F'#/ 1# 7)165#16
!*7*1 /9%A)'# 2# Fpar+
:@21*. A* B89:; &3/) A*. 3&4)%(*/). &)'C'('D. A* ED- F
, TESTa(⊥,⊥) G)* InvaInvall,InvallH #/6 *1 #1/#%A'# 2# ":F'#/ 7)%%# /*50!16/ (
Rparλe =
{
Test(a(x, y))→ a(x, y)}
.
, SETa(⊥,⊥) #/6 *1 #1/#%A'# 2# ":F'#/ 7)%%# /*56 (
Rparλe =
{
Set(f(x, y))→ a(x, y)}
.
, TESTV ALUEa(⊥,i(⊥,⊥)) #/6 *1 #1/#%A'# 2# ":F'#/ 7)%%# /*56 (
Rparλe =
TestV alue(f(x, y))→ A(TESTa(⊥,⊥)(f(x, y)))
A(f(x, y))→ a(TESTi(⊥,⊥)(x), y)
TestV alue(f(x, y))→ B(¬TESTa(⊥,⊥)(f(x, y)))
B(f(x, y))→ f(x, TestV alue(y))
B(f(x, y))→ f(TestV alue(x), y)
.
!"#" $%&'()*& +,+)*-'./0*& 1.0.)+'0+& 2345$& 67
! "#$% &! #''() *+&! #&%,( -.-%/0( *+12345 '#,#06%,6 TESTi(⊥,⊥)
7 TESTV ARa(⊥,b(⊥,⊥)) (-% &! (!-(08)( *( ,/9)(- :;00( -&$% <
Rparλe =
TestV ar(f(x, y))→ A(TESTa(⊥,⊥)(f(x, y)))
A(f(x, y))→ a(x,TESTVALUEb(⊥,Top(x))(y))
TestV ar(f(x, y))→ B(TESTb(⊥,⊥)(f(x, y)))
B(f(x, y))→ b(x,TESTVALUEa(⊥,Top(x))(y))
TestV ar(f(x, y))→ C(TEST¬(b∨a)(f(x, y)))
C(f(x, y))→ f(x, TestV ar(y)),
C(f(x, y))→ f(TestV ar(x), y)
.
! "#$% &! #''() *+&! #&%,( 12345 '#,#06%,6 TESTVALUEa(⊥,i(⊥,⊥))
#=(: )# ,/9)( Top(i(x, y)) → i(⊥,⊥) !( 9#,*( >&( )# )(%%,( )# ')&- ?#&%( *&
%(,0(@
7 SETV ALUEa(⊥,i(⊥,⊥)) (-% &! (!-(08)( *( ,/9)(- :;00( -&$% <
Rparλe =
SetV alue(f(x, y))→ A(TESTa(⊥,⊥)(f(x, y)))
A(a(x, y))→ a(SETi(⊥,⊥)(x), y)
SetV alue(f(x, y))→ B(¬TESTa(⊥,⊥)(f(x, y)))
B(f(x, y))→ f(x, SetV alue(y)),
B(f(x, y))→ f(SetV alue(x), y)
.
7 PLUSPLUSa(⊥,⊥) (-% &! (!-(08)( *( ,/9)(- :;00( -&$% <
Rparλe =
Plusplus(f(x, y))→ A(TESTa(⊥,⊥)(f(x, y)))
A(f(x, y))→ f(Plus(x), y)
Plus(f(x, y))→ f+(x, y)
Plusplus(f(x, y))→ B(¬TESTa(⊥,⊥)(f(x, y)))
B(f(x, y))→ f(x, P lusplus(y)),
B(f(x, y))→ f(Plusplus(x), y)
.
7 MINUSMINUSa(⊥,⊥) (-% &! (!-(08)( *( ,/9)(- :;00( -&$% <
Rparλe =
Minusminus(f(x, y))→ A(TESTa(⊥,⊥)(f(x, y)))
A(f(x, y))→ f(Minus(x), y)
Minus(f(x, y))→ f−(x, y)
Minusminus(f(x, y))→ B(¬TESTa(⊥,⊥)(f(x, y)))
B(f(x, y))→ f(x,Minusminus(y)),
B(f(x, y))→ f(Minusminus(x), y)
.
!" #$%&'()* +, -*). /0 1%02%2* &3/) 1% -4)'56%('30 &%) 789:;
:*<%)=/*. > !" #$%&!" '(#()*+#*!" ',-# ./ ",0+ 102,)'&$+!" )(1" &! #!"+! '!-+
3+#! 2,0"+#-1+ 4! &( )3)! )(01$#! 5
6 7!+/(# a = b
6 8!#)-+! a < − > b
6 7!+/(# a = b op c ,9 op ∈ {+,−, ∗, /}:::
:*<%)=/*. " ;,-" ),0+#,0" <-! &!" "="+$)!" '(#()*+#*" 8>?@7" 2,0A1!00!0+
+#$" B1!0 (- ./ !+ 1&" '!-A!0+ 0,-" ,C#1# 4!" ,'*#(+1,0" ',-# )(01'-&!# 4!" 4,00*!"
)1!-D <-E!0 F>?@7 !+ GGG:
!"#$ %& '(#)!*+, -.
!"# $%!&# '($)*+*&, +!&,-' .!++*&, ,-$/"0-* "& +!/1)* /" 234 *& 56789:
;*)$ &!"# .!&/"0, < )=0/'* /="& #>#,1+* 3!&.,0!&&*) ')'+*&,$0-* ?$-$+',-'@ $??*)'
A56789@ ?)"# *B?-*##03 C"* )*# ')'+*&,$0-*#: ;D$C"* -1()* ?$-$+',-'* *#, 'C"0%$E
)*&,* < "& *&#*+F)* /* -1()*# ')'+*&,$0-*#:
G$&# .* .D$?0,-*@ &!"# $%!&# /'%*)!??' )*# ?-0&.0?*# /* )$ ,-$/".,0!& /="& +!/1)*
/" 234 *& 56789: ;*)$ &!"# .!&/"0, < )=0/'* /="& A56789: !"# $%!&# +!&,-' C"*
A56789 !&, )* +H+* ?!"%!0- /=*B?-*##0!& C"* )*# 56789 +$0# 3$.0)0,*- )='.-0,"-*
/* (-$&/# 6789: !"# &!"# #!++*# 3!.$)0#'# #"- /*"B $#?*.,# C"* #!&, )$ #0+")$,0!&
/* A56789 ?$- /*# -1()*# /=56789# < )=$0/* /*# 0&%$-0$&,# #0+?)*# *, )* ?!"%!0-
/=*B?-*##0!& /" )$&($(* /* A56789:
!# $+F0,0!&# *#, /* /'%*)!??*- "&* *B,*�!& /* A56789 $%*. /*# -1()*#
/=56789# < )=$0/* /*# 0&%$-0$&,# .!+?)*B*# $0� C"* )$ &'($,0!& /*# 0&%$-0$&,#
.!+?)*B*#:
!"#$%&' ((
!"#$%&'!" ()")*+$,
!"# $! %!&'() '*+,&(-.) !"##" #$%&" '()')&" *"& ! "#$% &% '()(*'+",'% -./*)
"+.//%0 +,-./0 12 2)13"41 #5'" 6" &5&#%7"& 6" (898!(:#1(" ')1( *4 38(:;!4#:)2
&57<)*:=1" 6" 7)6%*"& 68!:64<*"&> ?)1& 7)2#()2& =1" 2)#(" 7)6%*" 4 *4 '1:&&42!"
6@"A'("&&:)2 6"& &5&#%7"& 6" (898!(:#1(" "# =1@:* "&# <:"2 464'#8 B *@8#16" 6" '()9
'(:8#8& 6" &C("#8 "# 6" '()'(:8#8& 6" *)D:=1" #"7')("**" 6" 7)6%*"&>
?)1& 43)2& 7:& "2 83:6"2!" 12" &)1& !*4&&" 6" &5&#%7"& E)2!#:)22"*& *"& (0()
$%/"1+'% +F,-./0 "# *"& (0($%/"1+'% 2 &'.+"% +.F,-./0 '(8&"(342# *4 '1:&&42!"
6@"A'("&&:)2 6"& &5&#%7"& E)2!#:)22"*&> G" '*1& 2)1& 43)2& 8#4<*: 6"& #"!$2:=1"&
6@4!!8*8(4#:)2 6"& !4*!1*& 4<)1#:&&42# B 12 )1#:* 6" 38(:;!4#:)2 &57<)*:=1" "H!4!">
I"& #"!$2:=1"& 6@4!!8*8(4#:)2 &)2# "J"!#:3"7"2# 4''*:!4<*"& 2)2 &"1*"7"2# ')1(
2)#(" 7)6%*" 74:& 8D4*"7"2# ')1( *"& 41#("& E)(74*:&7"& <4&8& &1( *"& &5&#%7"& 6"
(898!(:#1(" "# *"& 41#)74#"& 6@4(<("&>
F2 !4*!1*42# *4 !*K#1(" (8L"A:3" "# #(42&:#:3" 6"& #"(7"& 4!!"&&:<*"& &1(#)1# ')1(
6"& &5&#%7"& 6" (898!(:#1(" 6" D(426"& #4:**"& *" #"7'& 6" !4*!1* 6"3:"2# #(%& *)2D>
?)1& 6"3)2& ;A"( 12 68*4: +*"& (8&1*#4#& 6" #4:**" "A')2"2#:"**" 642& *" !$4':#(" M &)2#
(84*:&8& 43"! 12 68*4: 6" 3 $"1("&0> G42& !" !4&9*B 2)1& 2" &43)2& '4& "A4!#"7"2#
6@)N 3:"2# *" '()<*%7" O P"& ("&&)1(!"& +*4 787):(" "# *" '()!"&&"1(0 2" &)2# '4&
&1H&42#"& ')1( *1##"( !)2#(" *@"A'*)&:)2 !)7<:24#):("> Q1 6"*B 6"& '1:&&42!" 6"
!4*!1* 12 41#(" '()<*%7" '"1# '()3"2:( 6@12" &'8!:;!4#:)2 <)D1"8 (8&1*#42# 4:2&: &1(
6"& !4*!1*& :2;2:&> R)1( '4**:"( !"##" 6:H!1*#8 6"& (8&1*#4#& D828(41A 6" #"(7:24:&)2
"2 (898!(:#1(" &)2# 4''*:!4<*"& &1( 2)& 7)6%*"&> G" <)2& 8#4#& 6" *@4(# 6"& ("!$"(!$"&
'"13"2# S#(" #()138"& "2 TU42#"74 VWWX Y$*"<1&!$ VWWVZ>
!"# $! %!&'() )/%+&(0)"'!$) 2)1& 43)2& :7'*42#8 12 "2&"7<*" 6@)1#:*& #"*&
=1@12 38(:;!4#"1( 6@F,-./ 12 )1#:* 6" #(42&E)(74#:)2 41#)74#:=1" 6"& 7)6%*"& 6"
(8&"41A 6" R"#(: )(6:24:("& "# $:8(4(!$:=1"& 3"(& 6"& F,-./& "# 12 )1#:* '"(7"##42#
6" 68!(:(" "# 38(:;"( 6"& '()'(:8#8& 6" *)D:=1"& #"7')("**"& "# 6"& '()'(:8#8& 6"
&C("#8> ?)1& 43)2& !)7'4(8 2)#(" 38(:;!4#"1( 6@12" '4(# 43"! 6"& )1#:*& 6" (89
8!(:#1(" #"*& =1" -:7<1[ \416" "# -Y\ &1( 6"& 7)6%*"& 4(<)("&!"2#& 6@41#("
'4(# 43"! 6"& )1#:*& 6" 38(:;!4#:)2 #"*& =1" /R]? ?1/\^ /\Q.- )1 _/GG &1(
6"& 7)6%*"& 6" (8&"41A 6" R"#(:> P"& "A'8(:7"2#4#:)2& ("26"2# B 687)2#("( =1"
2)#(" )1#:* "&# #(%& !)7'8#:#:E>
R)1( *"& $.�% 1'3.'% *%/" #"*& =1" *"& '()#)!)*"& -QR RR "# PFR 2)#("
38(:;!4#"1( ')&&%6" 12" '"(E)(742!" &:D2:;!4#:3"> R)1( *"& '( %1,4 &% 5%"'+
61'1$("'( 61' ,/ /.$3'% &% ., '( %1,4 +&%/"+7,% #"*& =1" *" '()<*%7" 6"&
R$:*)&)'$"& *" '()#)!)*" 6" /*)##"69.:2D "# *" '()#)!)*" 6" .)1269.)<:2 \1#"A
74*D(8 *4 D828(4#:)2 41#)74#:=1" 6"& (%D*"& 6" (898!(:#1(" 2)1& 43)2& 6"& '("7:"(&
!" #$%&'()* + #,-./01',- 23-3)%/*
!""# $ %&'( )&!* +*,& &-)'./!0 /!& 12*,& 324#'& 5.0,(,"5./!& )2!, '& )'.(6& ,0"!,7
.8 4& 2! ,0 &(!- .4&1*./!& "219.&1* *,# :.&1 (9&" '& *&"51./!& 4;(""0'0,(*.21
<"$ $4$ =>? @$ %&)&14(1*A )2!, '& !"#$%&' ($ )$*!+ ,%!%-"*!"# ,%! &. ./-0!$ ($
1$*/.# *&' /!& '& B *#3& 4& C(1:(1 &* '& B *#3& 4& >DEA 2F '& )'.(6& ,0"!, .8 4&
12*,& 324#'& 5.0,(,"5./!& 1;& * )( ())'."(:'&A '& *&"51./!& 4& =>? 1& 21* 421"
)( &G"("& $ H2! 233& I '( ,&"5&,"5& 4;!1& 2'!*.21 &G"("& *&' /!& ';!*.'. (*.21
.1*&''.6&1* 4! "24(6& 4& *&,3& I ';(.4& 4& B *#3& )(,(30*,0 <?J>KLE@$
M"*!&''&3&1*A 4(1 ';0*!4& 4& ),2),.0*0 4& '26./!& *&3)2,&''& A 12*,& 90,.N7
"(*&!, ,& *& &1"2,& 4& '.3.*&A &1 )(,*."!'.&, )2!, '& ),2),.0*0 4& =K= <O2., "8$ 4&
,0 !'*(* 4& "5().*,& P@$ %&'( )&!* +*,& &-)'./!0 /!& '& "('"!' 4! ),24!.* B1"5,21. 0
&1*,& ';(!*23(*& 4& QR"5. &* '& B *#3& <),0 &1*0 4(1 '( &"*.21 S$T$U@ 1;& * )(
:.&1 2)*.3. 0$ H2! (921 421" :& 2.1 4;0*!4.&, ),2821403&1* '& B *#3& 0'03&17
*(.,& "21"&,1(1* I "& ),24!.* )2!, /!& "& B *#3& 0'03&1*(.,& (4()*& 3.&!- (!-
*&"51./!& 4;(""0'0,(*.21$
!"#!$%&'()!*
!"#$%%& '((') *+ !+ !"#$%%&, -+ ./011/0, *+ 2&3&4& 54 .+ #6781/+ !"#$%& '&!!
()*!$ +,!-./0"+ 90 :!;6(', </%$=5 '>(>?'((' /@ 1!-2#&! 3)2!4 /0 +)56#2!&
7-/!0-!, A&B51 >C'D>EE+ FA8G0B58H;58%&B, '(('+ '>
!"#$%%& '((E) *+ !+ !"#$%%&, !+ I5B&J, .+ #6781/ 54 !+ K5LG05+ '&!! !"#$%& ()*!$
+,!-./0" 8 9 7/5#$%2/)0:;%4!* 966&)%-,< M35 ./$80&% /@ I/BGN &0# !%B5"8&GN
*8/B8&==G0B, '((E+ G<, OP, '>, 'Q, PC
!80/%# OQPP) !0#8R !80/%# 54 *&$% :8$"G%%5+ 9 $/0!%& %$")&/2,5 2) 4)$=! >?!*:6)/02
!@#%2/)04 )0 2&%04/2/)0 4A42!54+ 90@+ *8/N511+ I544+, </%+ 'Q, 0/+ ', A&B51 CSD
EE, OQPP+ >, O'
-&: OQQP) T+ -&: 54 M+ UGAV/W+ M58= 85W8G4G0B &0# &%% 43&4+ :&="8G#B5
X0G<581G4J *8511, OQQP+ G<, OP
-&3&8 OQQS) K+ 98G1 -&3&8, Y8GN& !+ T8/3=, :3&8%51 2+ Z&/0&, Z&8J [+ \&N345%,
Y08GN/ 2&NGG, !"5%&8#/ *&8#/ 54 T&"G/ F/=50LG+ 9$"!;&%/- B!-/4/)0 B/%:
"&%54 %0* ',!/& 966$/-%2/)04+ 90 9::![6Q], </%$=5 O( /@ C)&5%$ (!2,)*4
/0 7A42!5 B!4/"0, A&B51 OSOD'(E, OQQS+ GGG, O>
-&%%&0# '((S) Y=G%G5 -&%%&0#, *&$% -8&$058, K&#$ ^/A54L, *G5885HY4G5005 2/85&$
54 !04/G05 K5G%%51+ ')5 8 D/""A;%-./0" &!E&/2/0" )0 F%=%+ 90 :/0@5850N5
/0 K5W8G4G0B M5N30G_$51 &0# !AA%GN&4G/01 H KM!6(S *8/N55#G0B1 /@ 435
OP43 :/0@5850N5 /0 K5W8G4G0B M5N30G_$51 &0# !AA%GN&4G/01, </%$=5 >C]]
/@ 13+7, A&B51 ]ED>S, *&8G1?T8&0N5 T8&0N5, (E '((S+ FA8G0B58H;58%&B+ ](
-5&$#50/0 '((C) ;+ -5&$#50/0+ B/%"&%55!4 *! B!-/4/)0 *! B)00G!4 6)#& $%
HG&/>-%2/)0 *! 7A42I5! (%2G&/!$4+ M3`15 #5 #/N4/8&4 50 90@/8=&4G_$5, I9*E,
X0G<+ *&8G1 ;9, '((C+ O>P
-/GN3$4 '((S) a+ -/GN3$4, M+ Z5054, M+ .50150 54 I+ I5 K/$b+ !E&/2/0" 966&)?:
/5%2/)04 J)& C%42 D&)2)2A6/0" )J 72%2/- 90%$AK!&4< 90 *8/N55#G0B1 /@ 435
OP43 :/0@5850N5 /0 K5W8G4G0B M5N30G_$51 &0# !AA%GN&4G/01, </%$=5 >C]] /@
1!-2#&! 3)2!4 /0 +)56#2!& 7-/!0-!, A&B51 >PDE', '((S+ ](
-/GN3$4 '((P) a/3&0 -/GN3$4, *G5885H:J8G%%5 \5&= 54 7%B& ^/$N30&850V/+ 96:
6&)?/5%2/)0 ;%4!* 2&!! &!"#$%& 5)*!$ -,!-./0"+ U/8#GN ./$80&% /@ :/=A$4G0B,
</%+ O>, A&B51 'OED'>O, '((P+ G<
-/GN3$4 '(O() a+ -/GN3$4, .+H2+ :/$<85$8 54 [HM UB$J50+ C#0-2/)0%$ '!&5
!E&/2/0" 7A42!54+ K515&8N3 85A/84, I9T7, '(O(+ G<, O'C, O]O
!!" #$$%&!'()(*+)(,-'.%&,'/%)+%!$*0%1*.$%&!'($+
-/$&cc&0G '((() !3=5# -/$&cc&0G, -50B4 ./011/0, 2&8N$1 UG%11/0 54 M&J11G8
M/$G%G+ !"#$%& ()*!$ +,!-./0"+ 90 :!;6((, </%$=5 OPCC /@ 1!-2#&! 3)2!4
/0 +)56#2!& 7-/!0-!+ FA8G0B58H;58%&B, '(((+ G<, '>
-/$&cc&0G '((') !+ -/$&cc&0G 54 M+ M/$G%G+ L?2&%6)$%2/0" '&!! '&%04J)&5%2/)04+
90 :!;6(', </%$=5 '>(>?'((' /@ 1!-2#&! 3)2!4 /0 +)56#2!& 7-/!0-!, A&B51
]>QD]E]+ FA8G0B58H;58%&B, '(('+ OP, '>
!" #$%&$'()*+,$-
!"#$%& '(()* +, !"#$%&- .,/0, 1234"53"- +, 678875"9- +, :2";2%- <, 023%75"-
=, >$472&/?;5&- @, >27&"5%$3;- @, A5B5C 5& 6, D3&"5, !"#$%&' (&" )#*%+,%&'
-#.()%!*($ (/0#12/, <% +2"E$8 E5&F2;9 G2" 5EH5;;5; ;79&"7H3&5; 9I9&5E9 J
F2K &2 E$9&5" &F5 L2EC85M7&I- C$B59 NONP'NN, :83K5" ?L$;5E7L >3H879F5"9-
'((), 74- NQ- N)Q
!"I$%& NRST* A, =, !"I$%&, !"#$%&"'() *+,-!./$0' 1-! &--+("2 3425/.-2 6"2.#%
4+"/.-2 !""" #$%&'%()*+&' +& (+,-.)/$'0 1+2 34560 &+ 70 -%8/' 9::;9<=0
>+?) =<79 ***0 @0 =A0 =50 9<
BC$D%&) =<<AE F " C$D%&) 7809-+.5 &--+("2 0"2.#4+"/.-2 :./$ -!)(!() 9.2"!8%
)(5.'.-2 ).",!"0' >3G 3+,-.) H.$1 0 1+2 A@0 &+ 50 -%8/' A<5;5=70 =<<A
***0 @0 =A0 =50 9<
BC.$(I =<<AE J F C.$(I0 " G 32%$K/ /) L M G(G*22%& 7809-+.5 6-)(+ ;$(5<.2, =
10207/"/(' "2) &(8-2) !&N+$,%)*+& %&O 3+,-.)%)*+& PH-/(*%2 *''./ N+$ Q/')
-%-/$' N$+, M!3H<RS0 1+2 <70 &+ A0 -%8/' =65;=7=0 =<<A ***0 =50 =6
B3*%$O+ ARRRE T 3*%$O+0 T MU))8/& /) F H*,*&*(/%&. >?5.(2/ 7809-+.5 7/"/(%
7#"5( ;-2'/!45/.-2 1-! *'825$!-2-4' 78'/(0' !& !3>#VWXARRR0 1+2.,/
=7A6 +N @(5/4!( A-/(' .2 ;-0#4/(! 75.(25(0 -%8/' =R5;=AA H-$*&8/$ Y/$2%80
ARRR ***0 =@
B3*%$O+ ARRA%E T 3*%$O+0 F M J+&/' !!!0 G%$,+$')/*& F G 0 > H G*&/$ /) F H*4
,*&*(/%&. 76*BC = 7/-5$"'/.5 6-)(+%5$(5<.2, *2"+8D(! 1-! B(+."9.+./8 "2)
C.0.2, !& ZHW0 -%8/ 6@60 ARRA =55
!!" #$$%%%&'(&)'*&+,)$ '-.*,/$01234$
B3*%$O+ ARRAQE T 3*%$O+ /) F H*,*&*(/%&. E'.2, >),(%F"+4() G(5.'.-2 G.",!"0'
1-! 7809-+.5 (2(!"/.-2 -1 7$-!/('/ H"/$' !& [G3>Z XRA \ V$+(//O*&8' +N
)I/ @)I !&)/$&%)*+&%2 3+&N/$/&(/ +& [+$,%2 G/)I+O' *& 3+,-.)/$4>*O/O
Z/'*8&0 -%8/' A69;A:50 M+&O+&0 ]L0 ARRA H-$*&8/$4Y/$2%8 =55
B3*%$O+ ARR5E T 3*%$O+0 F G G%$,+$')/*& /) F H*,*&*(/%&. 7"/4!"/.-2 E2%
9-42) !& #>3>H0 -%8/' 5:<;5<50 ARR5 =@0 :R0 :=0 =A6
B3*%$O+ ARR@E T 3*%$O+ B("5$"9.+./8 7(/ (2(!"/.-2 1-! H(/!. A(/' = ;"2 &!4/(
3-!5( &( 70"!/ I !& !3>#VWXR@0 1+2.,/ 5R<< +N @(5/4!( A-/(' .2 ;-0#4/(!
75.(25( H-$*&8/$4Y/$2%80 ARR@ =55
B3*%$O+ ARR6E T 3*%$O+ /) > J ^. 7"/4!"/.-2%&"'() 7809-+.5 B("5$"9.+./8 *2"+%
8'.' E'.2, ;-2J425/.K( "2) G.'J425/.K( H"!/./.-2.2, !& 3_>FG"0 -%8/'
=@9;=9=0 ARR6 =55
B3*%$O+ ARR9E T 3*%$O+0 F G%$,+$')/*& /) F H*,*&*(/%&. C$( '"/4!"/.-2 "+%
,-!./$0 1-! '809-+.5 '/"/(%'#"5( (L#+-!"/.-2 !&) J H+N)` #++2' #/(I&+2
#$%&'N 0 1+2 70 &+ =0 -%8/' @;A60 ARR9 =A6
B3*%$O+ ARR:E T 3*%$O+0 T MU))8/& /) > H G*&/$ >L#+-./.2, .2/(!+("K.2, '(0"2%
/.5' .2 '809-+.5 '/"/(%'#"5( ,(2(!"/.-2 [+$,%2 G/)I+O' *& HD')/, Z/'*8&0
1+2 5=0 &+ =0 -%8/' 95;=RR0 ARR: =55
!"#!$%&'()!* +,-
!"#$%%" &''&( )* !"#$%%"+ ,* !-$./0+ ,* 1"2345"6-"$+ 7* 1"2345"6-"$+ 8* 9":%;.0+
8* <;=0."+ <* >0?$:%"$3" 0% )* @$4450--$* !"#$ $%&'()* + , -* ./%*")!&0%
1))2 3)& "456)2(0 #)7%2 89%0:(*;* A3 9.;4* A3%0.3$%";3$- !;3B0.0340 ;3
!;#C2%0.D)"E0E F0."G4$%";3 H!)F &''&I+ =;-2#0 &J'J ;B <%0=!&% )=%' (*
8)5/!=%& "0(%*0%+ !;C035$603+ K03#$./+ L2-M &''&* >C."360.DF0.-$6* NO+
NP&
!!" #$$%&'()*+,'!*+!-*+!$*
!-$./0 NQRS( ,E#23E 8* !-$./0+ ,* )--03 ,#0.:;3 0% )* 9.$:$E >":%-$* -!=)5>=(0
$%&(?0>=()* )3 @(*(=%A"=>=% 8)*0!&&%*= "4'=%5' B'(*; 1%5/)&>2 <);(0 "/%0A
(?0>=()*'* )!8 @.$3:* 9.;6.$#* T$36* >M:%*+ =;-* R+ 3;* &+ C$60: &JJU&SP+
NQRS* J+ NN+ N&
!-$./0 NQRQ( ,* !-$./0+ K* T;36 0% V* 848"--$3* 8)5/)'(=()*>2 5)7%2 09%0:(*;* A3
9.;400E"36: ;B %50 7;2.%5 )332$- >M#C;:"2# ;3 T;6"4 "3 4;#C2%0. :4"0340+
C$60: POPUPS&+ 9":4$%$W$M+ XL+ Y>)+ NQRQ* A,,, 9.0::* NR+ &O
!-$./0 &'''( ,* 8* !-$./0+ Z* 1.2#?0.6 0% K* )* 90-0E* 8;E0- 4504/"36* 8A@
9.0::+ &'''* """+ J
!-$=0- &''N( 8* !-$=0-+ 7* K2.[3+ >* ,/0.+ 9* T"34;-3+ X* 8$.%\DZ-"0%+ L* 80:0620.
0% L* 7* ]20:$E$* #>!7% , "/%0(?0>=()* >*7 C&);&>55(*; (* D%E&(=(*; <);(0*
@50;.0%"4$- !;#C2%0. >4"0340+ &''N* P'
!!" #$$(.&/0*-'*&+&-*0/&*
!;#;3 &''&( ^* !;#;3+ 8* K$2450%+ <* 1"--0.;3+ 7* L$4_20#$.E+ K* T26"0`+
>* @":;3 0% 8* @;##$:"* @.00 $2%;#$%$ %0453"_20: $3E $CC-"4$%";3:*
!!" #$$111*2,."".*&%+)34+4405*6,$!.!.$+ &''&* NR
!;2=.02. NQQQ( L*D8* !;2=.02.* .*A=9%AF4 $%&(?0>=()* )3 <(*%>& 1%5/)&>2 <);(0*
A3 >C."360. F0.-$6+ 0E"%02.+ 9.;4* ;B 78aQQ+ C$60: &OPU&bN+ NQQQ* T04%2.0
X;%0: "3 !;#C2%0. >4"0340* Q+ NNP
!;2=.02. &''&( L*D8* !;2=.02.+ ,* ,34.03$`+ ,* 9$=";%)E0%+ K* 9;"%.03$2E 0%
9* c$4.03"0.* G>=> G%0('()* G(>;&>5 3)& C%=&( %= -*>24'('* A3 A!)@9X+
=;-2#0 &PS'+ C$60: NUN'N* >C."360. F0.-$6+ &''&* """+ NJ+ NO+ bN+ N&O
!;2=.02. &''J( L*D8* !;2=.02.* 8)*=&(6!=()* H 2I>2;)&(=95% 7% 2> JK&(?0>=()** 8dD
#;".0 Ea5$?"-"%$%";3 e E"."60. E0: .0450.450:+ Y3"=0.:"%d f;.E0$2g N+ T$f<A+
&''J* Q
!;2=.02. &''O( L*D8* !;2=.02. 0% h* @5"0..MD8"06* L(%&>&09(0>2 G%0('()* G(>A
;&>5' =) MN/2)(= #)7%2 "=&!0=!&%* A3 7Z<@,+ C$60: JJPUJOb+ &''O* NJ+ NO+
bN+ N&O+ NPP
!;2=.02. &''R( L*D8* !;2=.02. 0% K*D@* X62M03* 1&%% G>=> G%0('()* G(>;&>5'*
A3 F0!Z>* T00E:+ YV+ &''R* "=
K03/0. NQQR( 1* K03/0.+ L* 80:0620. 0% !* @$-4;%%* C&)=)0)2 "/%0(?0>=()* >*7
-*>24'(' (* #>!7%* A3 9.;4* &3E c<T) c;./:5;C+ 9;3% e 8;2::;3
H7.$340I+ NQQR* P'
!! "#$%#&'()*+#,
!"#$%&'()* +,,-. /0 !"#$%&'()* ") 10230 1&45665470 85679&&: &; )%"&#")(<5= <&>2
?4)"# $<("6<"@ A&=4>" B@ <%5?()#" C D E"'#()" FG$)">$@ ?5H"$ IJJKLI-0 M=2
$"A("# F<("6<" 349=($%"#$ B0 N0 O/&#)%28&==567P@ +,,-0 Q=$& 5$ D E"$"5#<%
#"?&#) JRS@ TEU0 (A@ +S
!(== +,S,. !0 T0 !(==0 V#5<" )%"&#G ;&# 54)&>5)(< %("#5#<%(<5= A"#(W<5)(&6 &; $?""72
(67"?"67"6) <(#<4()$0 XUV 3#"$$@ Y5>9#(7H"@ XQ@ ZFQ@ +,S,0 +S@ I[
M>"#$&6 +,S-. M0 M>"#$&6 ") M7>467 Y=5#:"0 !"#"$%&#'(')* $+##&$%)&,, -#+-&#.
%'&, +/ -"#"00&0 -#+*#"1, 2,')* 34-+')%,0 Q4)&>5)5@ T56H45H"$ 567 3#&H#5>2
>(6H@ ?5H"$ +C,K+S+@ +,S-0 J@ +I
\"4(==57" I--J. ]0 \"4(==57"@ V0 ]"6") ") N0 N(") V#("> V&6H0 5&"$!"6'0'%7 8)"07,',
+9&# : 5&;#'%')* <7,%&1,0 1QE@ A&=0 LL OL2JP@ ?5H"$ LJ+KLSL@ I--J0 (A@
I[@ L-
]5$)(6 I--+. 30 ]5$)(6 ") !0 ^77&4_0 =",% >:> %+ ?@$!' 82%+1"%" :#"),0"%'+)0
U6 ]0 B"##G@ 80 Y&>&6 ") Q0 \(6:"=@ "7()"4#$@ 3#&<""7(6H$ &; )%" +L)% Y&62
;"#"6<" &6 Y&>?4)"# Q(7"7 N"#(W<5)(&6 OYQN`-+P@ 64>a#& I+-I 7" T"<)4#"
/&)"$ (6 Y&>?4)"# F<("6<"@ ?5H"$ [LKC[0 F?#(6H"# N"#=5H@ I--+0 ,
]"6") I---. V0 ]"6") ") \0 b=5G0 5&;#'%')* /+# #7-%+*#"-!'$ A#+%+$+0 B&#'3$"%'+)0
U6 3#&<0 +R)% YQ!M Y&6;0@ 3())$94#H% O3"60@ ZFQP@ A&=4>" +SL+ &; >C8D0
F?#(6H"#2N"#=5H@ I---0 L-
]"6") I--+. V0 ]"6") ") N0 N(") V#("> V&6H0 5&"$!"6'0'%7 8)"07,', +/ :
5&;#'%')* <7,%&1, ;'%! !"#$%0 U6 3#&<0 S)% T3QE Y&6;0@ 85A565 OY495P@
A&=4>" II[- &; >C8D@ ?5H"$ C,+KR-I0 F?#(6H"#2N"#=5H@ I--+0 (A@ L-
!!" #$$%%%&'(')*&+($,-.!'/0-$1-2-!$!'3405$0
]"6") I--L. V0 ]"6")@ c02X0 V56H2V5=?(6 ") N0 N(") V#("> V&6H0 B&#'3$"%'+)
+/ +-7 A#+%&$%'+) #7-%+*#"-!'$ A#+%+$+0 2,')* 8--#+4'1"%'+), +/ :
5&;#'%')* <7,%&1,0 U6 U6 3#&<""7(6H$ &; d&#:$%&? &6 U$$4"$ (6 )%" V%"&#G
&; F"<4#()G@ I--L0 L-
]"#)% +,,[. E0 ]"#)%@ !0 3"="7@ X0 c0 N5#7( ") 30 d&=?"#0 <'1-0& +).%!&.E7 "2.
%+1"%'$ 9&#'3$"%'+) +/ 0')&"# %&1-+#"0 0+*'$0 U6 3FVN@ ?5H"$ LK+S@ +,,[0 J@
++
](=="#&6 +,,[. E0 ](=="#&6 ") F0 V($&60 5&*20"# :#&& >")*2"*&, ")F 5&;#'%& <7,.
%&1,0 \4675>"6)5 U6;&#>5)(<5"@ A&=0 IJ@ ?5H"$ +[RK+R[@ +,,[0 +S
](==("#$ I--J. \0 ](==("#$@ 10230 N"=4 ") \0 b&60 G&)&#"%'+) +/ H',%#'62%&F A#+.
*#"1, ') :!&'# :"#*&% I4&$2%'+) I)9'#+)1&)%0 U6 UMMM U6)"#65)(&65= d&#:2
$%&? &6 E5?(7 FG$)"> 3#&)&)G?(6H@ ?5H"$ +IRK+LJ@ I--J0 (A@ +[
]#5; +,,C. F0 ]#5;@ B0 F)"e"6 ") ]0 Tf))H"60 +1-+,'%'+)"0 J')'1',"%'+) +/ =')'%&
<%"%& <7,%&1, K,')* D)%&#/"$& <-&$'3$"%'+),0 \&#>5= Q$?0 Y&>?4)0@ A&=0 S@
6&0 [@ ?5H"$ C-RKC+C@ +,,C0 S@ +L[
]4?)5 +,,L. Q0 ]4?)5 ") Q0 T0 \($%"#0 5&-#&,&)%"%'+) ")F ,716+0'$ 1")'-20"%'+)
+/ 0')&"#07 ')F2$%'9& ?++0&") /2)$%'+),0 U6 UYYQ!`,L@ ?5H"$ +++K++C@ +,,L0
(((@ +J
!"#!$%&'()!* +,-
!"#$% &''() *+ !"#$%+ !"#$%&'( )**+(,! -#!$%&*! .,!&/#+,%&*!+ ,-. $-/0102
3%45/61/ 7/8895 :51;/401$<2 &''(+ 1112 &(
=98>?%55 &'@A) !+B+ =98>?%55+ 01( .*"(+ 21($3(4 56 7+ CDDD E4%5F%0$1950 95
G9H$I%4/ D5615//41562 ;98+ JK2 59+ L2 ?%< &'@A+ (2 &(2 &KJ2 &(A2 &(@
!!" #$$%"&'())!*+),$%"&'$- .!&%"&'* !,/+
=98>?%55 &''&) !+B+ =98>?%55+ ./0165 %5M ;%81M%$195 9H F9?#"$/4 #49$9F980+
,4/5$1F/ =%882 &''&+ (2 &(2 &(A2 &(@
="86%%4M &''') =+ ="86%%4M2 ,+ N+ O1881%?0 /$ =+ P+ *5M/40/5+ 89#&',+(!$( 21($3:
&!; *< 2*=>&!,%&*!,+ 2&4$#&%? #?&!; )**+(,! 8@/4(??&*! A&,;4,=?+ CDDD
E4%50%F$1950 9H 39?#"$/4Q*1M/M ./01652 ;98+ &@2 59+ R2 &'''+ 1112 &K
S/0$/5 &''R) T+ S/0$/52 U+ 7%8/42 7+ 7%4F"02 *+ ,5"/81 /$ D+ G-%-%4+ 5B=:
>*+&$ .*"(+ 21($3&!; C&%1 D&$1 E??(4%&*!,+ F,!;#,;(?+ C5 E=DUPDEC3*V
3U7,:EDP G3CDW3D2 #%6/0 (J(X(KL+ G#4156/4QY/48%62 &''R+ 1;2 J(2 @K
S98Z0 &''K) E+ S98Z02 [+ V15 /$ =+ ./ 7%5+ 5&G&!; ,!" H(4&I$,%&*! *< 2*==#:
!&$,%&*! )#J(4? <*4 2*==#!&$,%&!; 64*$(??(?+ C5 C33*.\'K2 ;98"?/ &@JL2
#%6/0 AA]XAA(2 &''K+ &(
7%"^9465/ &''') V+ 7%"^9465/+ )&!,4B A($&?&*! K4,/1?+ C5 G*G\''2 ;98"?/ &A'(
9H F($%#4( 7*%(? &! 2*=/#%(4 5$&(!$(2 #%6/0 &]&X&&A+ G#4156/4QY/48%62 &'''+
1112 &(
7%"^9465/ J]]]) V+ 7%"^9465/+ E! !$4(=(!%,+ L!&9#( D(/4(?(!%,%&*! <*4 D(;:
#+,4 04((?+ W94M1F B9"45%8 9H 39?#"$1562 ;98+ R2 59+ (2 #%6/0 J']XK&&2 J]]]+
1112 &(
7/0/6"/4 J]]K) B+ 7/0/6"/42 7+ ,%89?159 /$ W+ 7%4$_QU81/$+ 89#,%&*!,+ E>?%4,$:
%&*!?+ C5 ,49F+ &'$- 3*.D 395H+2 71%?1 [/%F- `N8+2 :G*a2 ;98"?/ JR(& 9H
F($%#4( 7*%(? &! 2*=/#%(4 5$&(!$(2 #%6/0 JX&A+ G#4156/4QY/48%62 J]]K+ K]
715%$9 &'']) G+ 715%$92 W+ C0-1"4% /$ G+ T%b1?%+ 51,4(" )&!,4B A($&?&*! A&,:
;4,=? C&%1 E%%4&>#%(" 8";(? <*4 8M$&(!% )**+(,! -#!$%&*! .,!&/#+,%&*!+
C5 .*3\']2 #%6/0 LJXLR+ *37cCDDD2 CDDD 39?#"$/4 G9F1/$< ,4/002 &'']+
1112 &K
715/4 &''') *+ G+ 715/4 /$ !+ 31%4M9+ 8M$&(!% D(,$1,>&+&%B 5(% K(!(4,%&*! ,!"
5%*4,;( L?&!; A($&?&*! A&,;4,=?+ C5 ,49F//M1560 9H $-/ J]$- C5$/45%$195%8
395H/4/5F/ 95 *##81F%$195 %5M E-/94< 9H ,/$41 W/$02 #%6/0 AXJL2 V95M952
:S2 &'''+ G#4156/4QY/48%6+ 1112 A2 &(2 &L2 'A
W6"</5 J]]A) .+QE+ W6"</5+ F<HN F,!;#,;( <*4 H(4&I$,%&*!+ C5 G$"M/5$ 0/001952
R$- GF-998 95 7UM/88156 %5M YD41H<156 9H #%4%88/8 ,49F/00/02 #%6/0 KKAXK(&+
[94M/%"d 39?#"$/4 GF1/5F/ V%^94%$94< `V%[PCa %5M 3WPG2 J]]A+ 1;2 &L2
&(@
W6"</5 J]]R) .+QE+ W6"</5+ F<H:AAA 21($3(4+ C5 .9F$94%8 G<?#901"?2 L$-
CDDD C5$/45%$195%8 395H/4/5F/ 95 P/0/%4F-2 C559;%$195 %5M Y10195 H94 $-/
N"$"4/2 #%6/0 &ALX&AA+ =%5912 Y1/$5%?2 J]]R+ 1;2 &L2 &('
!" #$%&$'()*+,$-
!"#$%&'(" )**)+ ,- !"#$%&'("- ./012($/ 3456(' 62 3$78 7$976362:- ;5762:$7<
=$7#1:> ?42/42> @A> )**)- 60> BCB
D2&$#6 BEFF+ .- D2&$#6- !" "#$%&'( )%*+, %- .&%*&'#/- G2 H!I;> 51:$' JCKLF>
BEFF- J> M
N$O$# BEEE+ H- N$O$#- 01123%4"/ 5'6 0" 7%&" 89$&"//+:"- G2 .;G.PQEE> 04#&8$
BFJ) 4R )",;<&" 3%;"/ +6 5%#$<;"& =,+"6,"> 51:$' )EJKS*F- ;5762:$7 =$7#1:>
BEEE- 666> BJ
;6'3#1 BEM)+ .- D- ;6'3#1 $3 ,- T- I#17U$- !" ,%#$("9+;> %- $&%$%/+;+%6'( (+6"'&
;"#$%&'( (%*+,/- G2 ;V!I QM) W D74($$/62:' 4R 3"$ R4&73$$23" 122&1# .IT
'X854'6&8 42 V"$47X 4R (485&362:> 51:$' BLEKBCM> P$9 Y47U> PY> @;.>
BEM)- .IT- J> B)
V"6$77X<T6$: )**J+ Y- V"6$77X<T6$:- ",!6+?<"/ $%<& (" 7%4"(25!",@+6* 4" /$A,+2
B,';+%6/ 4" C'<; 3+:"'<- V"Z'$ /$ /4(34713 $2 G2R478136[&$> ?GDC> @260-
D176' =G> )**J- BJ> BL> FB> B)L
V"6$77X<T6$: )**M+ Y- V"6$77X<T6$:> .- \18$] $3 H- A47/42- 0<+(4+6* "D,+"6;
#%4"( ,!",@"&/ </+6* C+"&'&,!+,'( ="; 4",+4+%6 4+'*&'#/ '64 '<;%#';+, /';2
<&';+%6- H&2/18$231 G2R47136(1$ D$376 P$3'> 04#- B<)L> )**M- BJ> BL> FB> B)L>
BSS
V"6$77X<T6$: )**E+ Y- V"6$77X<T6$:> ^- D4637$21&/> .- \18$] $3 H- A47/42- C+"&2
'&,!+,'( ="; 1",+/+%6 1+'*&'#/ '64 E"*<('& 7%4"(/- G2 V.I.;> 51:$' BKBL>
)**E- BSS
!!" #$$%&'()*+",)-.$ /&-!01.($222$
!!" #$$/&3.4(-&.5()6(!$".&7(4!/$8399:$
!!" #$$'*/+)4&*&.19&)(93$ -18+&$;<22$ -
=17/6 BEMC+ T- Y- =17/6 $3 D- _4#5$7- F<;%#';'2 !"%&";+, ",!6+?<"/ -%& 7%4'(
)%*+,/ %- .&%*&'#/- `- I485&3- ;X'3- ;(6-> 04#- S)> 24- )> 51:$' BMSK))B>
BEMC- J> E
_4#5$7 BEEM+ D6$77$ _4#5$7 $3 a$7217/ a46:$#43- G"&+->+6* />/;"#/ H+;! +6B6+;"
I<; &"*<('& /;';" /$',"/- G2 I.=QEM> =!?@T, BJ)F !H ?$(3&7$ P43$' 62
I485&3$7 ;(6$2($> 51:$' MMEKF- ;5762:$7<=$7#1:> BEEM- 60> )J
_4#5$7 )**B+ D- _4#5$7- 5%6/;&<,;+6* F<;%#';' -&%# "#$%&'( )%*+, J%&#<('/ K
F <;%&+'(- G2 ?$(3&7$' 42 H4781# T$3"4/' 62 D$7R47812($ .21#X'6' bH67'3
,,Hc,&74 ;&88$7 ;("44# 42 V7$2/' 62 I485&3$7 ;(6$2($d> 04#&8$ )*E*
4R )",;<&" 3%;"/ +6 5%#$<;"& =,+"6,"> 51:$' )CBK)FF- ;5762:$7<=$7#1:> `&#X
)**B- B*> BBS
e123$81 )**S+ \- e123$81- V$78 7$976362: 'X'3$8> ("153$7 3$786213642- I18<
%76/:$ @260$7'63X D7$''> @A> )**S- 60> BCB
!!"#"
!"#$"%
!" #$%&'()* +! ,-.(/0*. 1* 23435)'(6)* 7895('899*:.
!"!" #$%&'() *!"! +,-#.
!"#$" % !"#"$%&%"' (
) α→{l→r} β ⇒ Fl→r(α)→⋆Rλ
β * +,-"' -. /01'%&%"'2 %- ,3%$&, 4', #"$%&%"' p /,
α ,& 4', $45$&%&4&%"' σ : X → T (Fbin) &,--,$ 64, α|p = lσ ,& β = α[rσ]p7 +"%&
α′ 4' &,!8, 9"'$&!4%& 9"88, -, $4%:.'& * α′ = α[Fl→r(α|p)]p7 ;-.%!,8,'&2
,' 4&%-%$.'& $,4-,8,'& -,$ !<=-,$ /, 9%!94-.&%"'2 c.>.d. -,$ !<=-,$ /, -. ?"!8,
Fl→r(a(x, y))→ a(Fl→r(x), y) ,& Fl→r(a(x, y))→ a(x, Fl→r(y))2 '"4$ .:"'$
Fl→r(α)→⋆Rλ
α′. !"#$
+,-"' -. @A#"&@<$,2 %- ,3%$&, 4', $45$&%&4&%"' σ : X → T (Fbin) &,--, 64, α|p =
lσ7 .! 9"'$064,'&2 '"4$ .:"'$ Fl→r(α|p) = Fl→r(lσ)7 B%'$%2 Fl→r(α|p) =
Fl→r(l)σ7 +,-"' -. 9"'$&!49&%"' /, α′2 %- ,3%$&, 4', !<=-, =0'0!0, /, Rλ2 c.>.d
Fl→r(l) → r2 4', $45$&%&4&%"' σ : X → T (Fbin) ,& 4', #"$%&%"' p &,--,$ 64,
α′|p = Fl→r(l)σ7 C' 9"'$064,'9,2 '"4$ #"4:"'$ 9"'$&!4%!, -, &,!8, α′[rσ] 64%
,$& 0=.- > β $,-"' -. 9"'$&!49&%"' /, α′7 D"4$ .:"'$ /"'9 *
α′ →Rλ β. !"%$
B%'$%2 ,' 4&%-%$.'& EB7(F ,& EB7GF2 '"4$ #"4:"'$ /0/4%!, 64, Fl→r(α)→⋆Rλ
β7
.! 9"'$064,'&2 α→{l→r} β ⇒ Fl→r(α)→⋆Rλ
β7
) α →{l→r} β ⇐ Fl→r(α) →⋆Rλ
β * +"%& R′λ -, HIJ+ &,- 64, R′λ = Rλ \
{Fl→r(l) → r |l → r ∈ R}7 CK,9&%:,8,'&2 '"4$ #"4:"'$ 8"'&!,! 64,
R′⋆λ ({Fl→r(α)}) = ∅7
C' 9"'$064,'9,2 9,-. %8#-%64, 64,2 #"4! "5&,'%! -, &,!8, β ∈ R⋆λ({Fl→r(α)})2
-. !<=-, Fl→r(l) → r /"%& L&!, .##-%640, .4 8"%'$ 4', ?"%$7 B9&4,--,8,'&2
'"4$ /,8.'/"'$ 64, 9,&&, !<=-, ,$& .##-%640, ,3.9&,8,'& 4', ?"%$7 CK,9&%:,M
8,'&2 #"4!$4%:"'$ ,' %'/49&%"' $4! -, 9@,8%' /, !0M09!%&4!, 64% '"4$ 8<',
> #.!&%! /, Fl→r(α) > β c.>.d %- ,3%$&, t0, . . . , tn ∈ T (Fbin ∪ FNT ) &,- 64,
t0 = Fl→r(α) →Rλ t1 →Rλ . . . tn−1 →Rλ tn = β7 D"4$ 8"'&!"'$ /N.5"!/ 64,
[email protected], ti . 4', #"$%&%"' p .4 #-4$ &,--, 64, ti(p) = Fl→r7 +"%& Pn -. #!"#"$%&%"'
$4%:.'&, * #"4! &"4& p, p′ ∈ Pos(tn)2 $% tn(p) = tn(p′) = Fl→r .-"!$ p = p′7
) P0 * 4%$64, t0 = Fl→r(α) ,& α ∈ T (Fbin)2 ε ,$& -N4'%64, #"$%&%"' /, t0 "O
Fl→r .##.!.P&7 P0 /"'9 ,$& :!.%7
!" #$$%&% #' ()%*+%,
Pn ⇒ Pn+1 ! "#$$%&%'& (#) Pn &%*+ ,-.*/ 0.- 1%'&2(#)'+3 tn &.+*&4.*+ 5
6. $-%$-*2+2 ! p, p′ ∈ Pos(tn)3 &* tn(p) = tn(p′) = Fl→r/ 7) $6#&3 $#*&(#)
tn →Rλ tn+13 *6 )8*&+) '21)&&.*-)9)'+ #') #'*(#) $%&*+*%' p :) tn +)66) (#)
tn(p) = Fl→r/ 0%#-&#*,%'& )' -2;21-*,.'+ 6<.'.6=&) :) 1.& !
Fl→r(a(x, y)) → a(Fl→r(x), y) )&+ .$$6*(#2) 5 6. $%&*+*%' p ! >6 :%'1 )8;
*&+) #') &#?&+*+#+*%' σ : X → T (Fbin) +)66) (#) tn|p = Fl→r(a(x, y))σ =
Fl→r(a(σ(x), σ(y)))/ 0.- 1%'&2(#)'+3 tn+1 = tn[a(Fl→r(x), y)σ] =
tn[a(Fl→r(σ(x)), σ(y))]/ 0#*&(#) σ(x), σ(y) ∈ T (Fbin) )+ ∀p′ ∈ Pos(tn)
+)66)& (#) p &= p′3 tn(p′) /∈ FNT 3 tn+1 &.+*&4.*+ [email protected])9)'+ 5 ∀p, p′ ∈
Pos(tn+1)3 &* tn+1(p) = tn+1(p′) = Fl→r/ 06#& $-21*&29)'+3 6<#'*(#) $%;
&*+*%' p′ :) tn+1 +)66) (#) tn+1(p) = Fl→r )&+ p′ = p.0/
Fl→r(a(x, y)) → a(x), Fl→r(y) )&+ .$$6*(#2) 5 6. $%&*+*%' p ! 6) 1.& :)
$-)#,) )&+ &*9*6.*-) .# $-212:)'+/
Fl→r(l)→ r )&+ .$$6*(#2) 5 6. $%&*+*%' p ! >6 :%'1 )8*&+) #') &#?&+*+#+*%'
σ : X → T (Fbin) +)66) (#) tn|p = Fl→r(l)σ = Fl→r(lσ)/ A' 1%'&2(#)'1)3
tn+1 = tn[rσ]/ 0#*&(#) r ∈ T (Fbin,X ) )+ σ : X → T (Fbin)3 rσ )&+
.*'&* #' +)-9) :) T (Fbin) )+ tn+1 ∈ T (Fbin)/ B*'&*3 tn+1 &.+*&4.*+ 5 6.
$-%$-*2+2 ! ∀p, p′ ∈ Pos(tn+1)3 &* tn+1(p) = tn+1(p′) = Fl→r/
C%#& ,)'%'& D#&+)9)'+ :) 9%'+-)- (#) 1E.(#) +)-9) :.'& 6) 1E)9*' :) -2;
21-*+#-) 1%'+*)'+ )8.1+)9)'+ #' &=9?%6) Fl→r )+ :F& (#) 6. -F@6) Fl→r(l)→ r
)&+ .$$6*(#2)3 #' +)-9) :) T (Fbin) )&+ %?+)'#/ G) HIJ" Rλ ') $)#+ :%'1 $6#&
K+-) .$$6*(#2)/ L)6. 9%'+-) 6. -),)':*1.+*%'/
B*'&*3 6) 1E)9*' :) -2;21-*+#-) '%#& 9)'.'+ 5 $.-+*- :) Fl→r(α) 5 β )&+
:) 6. 4%-9) &#*,.'+) ! t0 = Fl→r(α) →Rλ\{Fl→r(l)→r} t1 →Rλ\{Fl→r(l)→r}. . . tn−1 →{F (l)→r} tn = β/ B,)1 #') *':#1+*%' +-F& $-%1E) .# $-212:)'+3
'%#& $%#,%'& 9%'+-)- (#<*6 )8*&+) #') $%&*+*%' p :) α )+ #') &#?&+*+#+*%'
σ : X → T (Fbin) +)66)& (#) tn−1 = α[F (l)σ]p/ C%#& $%#,%'& [email protected])9)'+ 9%';
+-)- (#) tn−1|p.0 = α|p/ A' 1%'&2(#)'1)3
lσ = α|p. !"#$
7) $6#&3 tn = β = tn−1[rσ]p/ B*'&*3
β = α[rσ]p. !"%$
")6%' 6. 1%'&+-#1+*%' :) Rλ3 l → r ∈ R/ B*'&*3 :<.$-F& MB/NO )+ MB/PO3 *6
)8*&+) #') -F@6) l → r ∈ R3 #') $%&*+*%' p ∈ Pos(α) )+ #') &#?&+*+#+*%' σ :
X → T (Fbin) +)66)& (#) lσ = α|p )+ β = α[rσ]p/ 0.- 1%'&2(#)'+3 α→{l→r} β/
A' -2Λ '%#& .,%'& 9%'+-2 (#) α→{l→r} β ⇐ Fl→r(α)→⋆Rλ
β3 1%'16#.'+
6. $-)#,)/
!"! #$%&'()* +! ,-.(/0*. 1* 23435)'(6)* 7895('899*:. ";"
<)*6=* > !"#$%&' (
) R⋆(E) ⊆ R⋆λ(E
′) * +#,- α ∈ E '- β ∈ R⋆({α}). +'/#0 /1 2"30,-,#0 2'
R⋆4 ,/ '5,6-' l1 → r1, . . . ln → rn ∈ R '- t0, . . . , tn -'//'6 78' α →{l0→r0}
t0 →{l1→r1} . . . →{ln−1→rn−1} tn−1 →{ln→rn} tn = β. 9' : ;+ Rl0→r0 $'&<
=/,- /'6 >#02,-,#06 1--'028'6 6=">,3"'6 ? /1 =$#=#6,-,#0 (. @1$ >#06"78'0-4
6'/#0 /1 =$#=#6,-,#0 (4 t0 →l0→r0 t1 ,&=/,78' 78' Fl0→r0(t0) →⋆Rl0→r0
t1.
A1/!'8$'86'&'0-4 Fl0→r0(t0) &∈ E′. B'='0210-4 t0 = α '- α ∈ E. C,06,4
G(t0) ∈ E′. DE'>-,F'&'0-4 G(t0) →G(x)→G(Fl0→r0 (x)) G(Fl0→r0(α)). D0 >#0<
6"78'0>'4 G(t0) →⋆Rλ
G(t1) →G(x)→x t1 ,&=/,78' 78' t1 ∈ R⋆λ(E
′). D0
G"0"$1/,610- >' =$#>'66864 0#86 #H-'0#06 /' >!'&,0 2' $"<">$,-8$' >,<2'66#86 *
G(t0) →⋆Rλ
G(t1) . . . →⋆Rλ
G(tn) →G(x)→x tn = β. @1$ >#06"78'0-4 β ∈
R⋆λ(E
′).
C,06,4 0#86 =#8F#06 2"28,$' 78'
R⋆({α}) ⊆ R⋆λ({G(α)}). !"#$
) R⋆λ(E
′) ⊆ R⋆(E) * I-82,#06 R⋆λ({G(α)})
! "#$$%&%'& (#)#'* +,-.* /* .0 1%+2* G(x) → G(Fl→r(x)) *&3 0$4
$.5(#6* 75'&58 G(α) →G(x)→G(Fl→r(x)) G(Fl→r(α))) 9%3*: (#* G(x) →
G(Fl→r(x)) '* $*#3 $0& ;3+* 0$$.5(#6* /,& (#* .* &<2=%.* Fl→r 0$$0+0>3
"5 R⋆l→r(Fl→r(α)) = ∅ 0.%+& 0#?#' 3*+2* /* T (Fbin) $*#3 ;3+* 033*5'3 /*
Fl→r(α) 75'&58 $#5&(#* Rl→r $*+2*3 .0 /6@'535%' /* Rλ /0'& .* $+%$%&535%'
A8 '%#& $%#B%'& /6/#5+* (#* .0 +,-.* l → r ∈ R '* $*#3 $0& ;3+* 0$$.5(#6*
&#+ α
"5 R⋆l→r(Fl→r(α)) #= ∅ 0.%+& 5. *C5&3* #' 3*+2* /* t1 ∈ T (Fbin) (#5 $*#3
;3+* 033*5'3 /* Fl→r(α) D#5&(#* Fl→r(α)→⋆Rl→r
t18 '%#& 0B%'& G(α)→+Rλ
G(t1) E* $.#&8 /)0$+,& .0 $+%$%&535%' A8 α→{l→r} t1 9%3*: (#* t1 ∈ R⋆λ(E
′)
$#5&(#* .0 +,-.* G(x)→ x $*#3 ;3+* 0$$.5(#6* &#+ G(t1) 75'&58 ?* $+%?*&&#&
$*#3 ;3+* 536+6 *3 '%#& $%#B%'& ?%'&3+#5+* #'* &6(#*'?* 3*..* (#* G(α)→+Rλ
G(t1) →+Rλ
G(t2) . . . →+Rλ
G(tn) 0B*? ti ∈ R⋆λ(E
′) &5 .0 +,-.* G(x) → x *&3
0$$.5(#6* &#+ ?F0(#* 3*+2* G(ti) G?)*&3 $%&&5=.* $#5&(#* ti ∈ T (Fbin)H E*
$.#&8 &*.%' .0 $+%$%&535%' A8 '%#& 0B%'& ti →{l→r} ti+1 $%#+ i = 1, . . . , n−1
9%#& $%#B%'& /%'? /6/#5+* (#* tn ∈ R⋆({α}) I5'0.*2*'38 '%#& $%#B%'&
/6/#5+* (#*
R⋆({α}) ⊇ R⋆λ({G(α)}). !"#$
! "#$$%&%'& (#)0#?#'* +,-.* /* .0 1%+2* G(x) → G(Fl→r(x)) *&3 0$$.5(#6*
D0+ ?%'&6(#*'38 0#?#'* +,-.* /*
⋃
l→r∈RRl→r $*#3 ;3+* 0$$.5(#6*
J)#'5(#* 3*+2* 033*5'3 /* T (Fbin) *&3 /%'? %=3*'# *' 0$$.5(#0'3 .0 +,-.*
G(x)→ x 75'&58 α ∈ R⋆λ({G(α)}) E)0$+,& .0 /6@'535%'8 α ∈ R⋆({α})
D%#+ ?%'?.#&5%'8 $%#+ 3%#& α ∈ E8 /* G7 KH *3 G7 LH8 '%#& %=3*'%'& (#*
R⋆(E) = R⋆λ(E
′)
!" #$$%&% #' ()%*+%,
()%*+% - !""! #
$%&&'(')( %) *!+ ,-$ .')(*/%0* .'""! 12)( +! +!""! #3 $!+') +! *45'/6"! #7 )'%(
'8*!)')( R⋆λ(E
′) = R⋆(E)3
!"!# $%&'()* +!#! ,-./$0
()%*+% . /()%*+% ,*001$02% 3% 45 6)767,1217$ "8
! &'0)* .+5 !(* 1! .')(*/%0/! %) 9:,-$ Rλe ;%0 0"&+5"!)*! +! &/'.!((%( 1!
/5<5./0*%/!3
! 9:,-$ Rλe !(* .')(*/%0* .'""! +! (%0=2)* >
Rλe = Rl→rvisit ∪R
l→rTV ∪Rl→r
check ∪Rl→rσ ∪Rl→r
GS ∪Rl→rσ−apply
? α →{l→r} β ⇒ Fl→r(α) →∗Rλe
β > 9@!.*0=!"!)*7 (0 α →{l→r} β 2+'/( 0+
!A0(*! %)! &'(0*0') p 1! α !* %)! (%8(*0*%*0') σ : X → T (Fbin) *!++!( ;%!
α|p = lσ 2)1 β = α[rσ]p3 B &2/*0/ 1! +2 15C)0*0') 1! Rλe 7 )'%( &'%=')(
151%0/! ;%! Fl→r(α) →∗Rλe
α[Fl→r(α|p)]p3 B0)(07 !) 2&&+0;%2)* 1! +2 /6D+!
Fl→r(x)→ F l→rrewrite(F
l→rcheck(F
l→rcopy(F
εl→r(x)))) (%/ α[Fl→r(α|p)]p 7 )'%( '8*!)')(
α[F l→rrewrite(F
l→rcheck( F
l→rcopy(F
εl→r(α|p))))]p3 E%0(;%! σ !(* %)! (%8(*0*%*0') 1! X
*' T (Fbin)7 !* 1F2&/6( +F!)(!"8+! 1!( /6D+!( Rl→rTV ∪ R
l→rcheck ∪ R
l→rσ ∪ Rl→r
GS 7
α[F l→rrewrite(F
l→rcheck(F
l→rcopy(F
εl→r(α|p))))]p &!%* G*/! /5<5./0*( !) α[F
l→rrewrite(t)]p 'H
t !(* 15C)0 .'""! +! (%0=2)* >
? t(p) = σ(xi)7 t(p′) = ⊤X ∈ Fbin 2=!. p′ = (2.)i p = p′.1 !* xi !(* +2 ith
=2/028+! 1! l +% 1% .I*5 D2%.4! 1! l
? t(prmp) =⊥X∈ Fbin 2=!. prmp = (2.)m2 !* m !(* +! )'"8/! 1! =2/028+!(
2&&2/20((2)* 12)( l
B0)(07 +! *!/"! t /!&/5(!)*! .'""! %) *!/"! +2 (%8(*0*%*0') σ3 :0)2+!"!)*7
+! 9:,-$ Rl→rσ−apply )'%( &!/"!* 1! /!.')(*/%0/! r !* +2 =2+!%/ 1!( =2/0<
28+!( !(* .'&05! (J"8'+! &2/ (J"8'+!3 !( %)0;%!( *!/"!( 2..!((08+!( 1').
(')* rσ3 B0)(07 !) /5(%"57 F l→rrewrite(t) →
∗Rl→rσ−apply
rσ ∈ T (Fbin)3 9) .')<
(5;%!).!7 α[F l→rrewrite(t)]p →
∗Rl→rσ−apply
α[rσ]p ∈ T (Fbin) !* &%0(;%! α[rσ]p = β7
β ∈ R∗λe({Fl→r(α)}).
? α →l→r β ⇐ Fl→r(α) →∗Rλe
β > 2 &/!%=! &'%/ .! .2( !(* &/'.4!
2% .2( (0"0+20/! */20*5 12)( +2 &/!%=! 1! +2 &/'&'(0*0') # 12)( +! (!)(
;%! K!( &2( 1! /5<5./0/! (')* '/1'))5( !* 15&!)12)*( 1% *!/"! 0)0*02+
c.L.d Fl→r(α)3 :0)2+!"!)*7 &%0(;%! +2 4J&'*46(! Fl→r(α) →∗Rλe
β7 1M
2%A 15C)0*0')( 1! Rl→rvisit !* Rl→r
TV 7 0+ !A0(*! %)! &'(0*0') p ∈ Pos(α)
*!++! ;%! Fl→r(α) →∗Rl→rvisit∪R
l→rTV
α[F l→rrewrite(F
l→rcheck(F
l→rcopy(F
εl→r(α|p))))]p !*
F l→rrewrite(F
l→rcheck(F
l→rcopy(F
εl→r(α|p)))) →
∗Rλe
β3 KM 2% &0+! 1!( (J"8'+!( 1! FNT
(%/ α|p7 )'%( &'%=')( "')*/!/ ;%! Fl→rrewrite(F
l→rcheck(F
l→rcopy(F
εl→r(α|p)))) →
∗Rλe
β ⇔ (Rl→rσ−apply)
∗((Rl→rGS )
∗((Rl→rσ )∗((Rl→r
check)∗(F l→r
rewrite(
F l→rcheck(F
l→rcopy(F
εl→r(α|p)))))))) = {β}. E2/ .')(5;%!)*7 (!+') 15C)0*0')(
1! Rl→rσ 7 Rl→r
GS !* Rl→rcheck7 &'%/ %) *!/"! t ∈ T (Fbin)7 )'%( &'%=')( "')*/!/
!"! #$%&'()* +! ,-.(/0*. 1* 23435)'(6)* 7895('899*:. ";<
!" #$ (Rl→rcheck)
∗({F l→rcheck(F
l→rcopy(F
εl→r(t)))}) #= ∅ %&'(# $& ")$#*" !+" #!,#*$*!*$'+
σ : X → T (Fbin) *"&&" !" lσ = t- .+ %//&$ !%+* 0'11" !+ (2#!&*%* 3" +'*("
2*!3" 3" 0%#4 $& ")$#*" σ : X → T (Fbin) *"& !" lσ = α|p- 5%# #"!&"1"+*
(Rl→rcheck)
∗({F l→rcheck(F
l→rcopy(F
εl→r(t)))}) #= ∅4 1%$# $& ")$#*" 26%&"1"+* !+ *"(1"
t′ 3" T (Fbin) *"& !" (Rl→rcheck)
∗({F l→rcheck(F
l→rcopy(F
εl→r(t)))}) = {t′}- 7" *"(1"
t′ "#* !+" ("/(2#"+*%*$'+ 3" σ4 "* &8")20!*$'+ 3" &8.9:;< Rl→rσ−apply #!( t′
+'!# /"(1"* 3" ("0'+#*(!$(" r "+ #!,#*$*!%+* 0=% !" >%($%,&" xi 3" r /%(
σ(xi)- 78!+$ !" *"(1" %00"##$,&" 3" t′ 3'+0 "#* rσ- ?$+#$4 "+ (2#!124 $&
")$#*" !+" /'#$*$'+ 3" α "* !+" #!,#*$*!*$'+ σ : X → T (Fbin) *"&&"# !"
α|p = lσ4 β = α[rσ]p "* Fl→r(α) →∗Rλe
β- @'!# /'!>'+# 323!$(" !"
α→l→r β ⇒ Fl→r(α)→∗Rλe
β !$ 0'+0&!* &% /("!>"-
=)*6>* ? :=2'(A1" B
C*%+* 3'++2 &8.9:;< Rλe =⋃
l→r∈R(Rl→rvisit ∪ R
l→rTV ∪ R
l→rcheck ∪ R
l→rσ ∪ Rl→r
GS ∪
Rl→rσ−apply) ∪ RFP %>"0 RFP = {G(a(x, y)) → G(Fl→r(a(x, y))) | l → r ∈ R ∧ a ∈
Fbin} ∪ {G(a(x, y)) → a(x, y) | a ∈ Fbin}4 &% /("!>" "#* *(A# /('0=" D 0"&&" 3!
*=2'(A1" E-
=)*6>* @ A=)*6>* .655'95(* 1* :% &)8&8.'('89 <B
C*%+* 3'++2 &8.9:;< Rλe 4 &% /("!>" "#* *(A# /('0=" D 0"&&" 3" &% /('/'#$*$'+ B-
7" /'$+* 0&2 "#* 3" 0'+#*(!$(" !+ .9:;< Rλe !$ $1/&21"+*" &" /('0"##!# 3"
(2F20($*!(" "+ 0'+#$32(%+* H 0'11" !+ #G1,'&" #/20$%& "+ Fbin "* "+ $6+'(%+* &"#
(A6&"# 3" 0$(0!&%*$'+-
7" .9:;< Rλe "#* 0'+#*(!$* 0'11" &" #!$>%+* H
Rλe = RH(t)→α
TV ∪RH(t)→α
check ∪RH(t)→ασ ∪R
H(t)→α
GS ∪RH(t)→α
σ−apply
!"!# $%&'()* +!#! ,-./,$0
=)*6>* ; A=)*6>* .655'95(* 1* :% &)8&8.'('89 +B
C*%+* 3'++2 &"# ;.9:;<# Rλe→I 4 Rλe→II 4 Rλe→III 4 Rλe→A "* Rλe→B4 &% /("!>"
"#* *(A# /('0=" D 0"&&" 3" &% /('/'#$*$'+ B-
7" /'$+* 0&2 "#* 3" 0'+#*(!$(" !+ ;.9:;< !$ $1/&21"+*" &" /('0"##!# 3" (2F
20($*!("-
7" ;.9:;< "#* 0'+#*(!$* 0'11" &" #!$>%+* H
Rλe→∪ = RH (a(x,y))→α
TV ∪RH (a(x,y))→α
check ∪ RH (a(x,y))→ασ ∪R
H (a(x,y))→α
σ−apply
'I
J RH (a(x,y))→α
TV = {FH (a(x,y))→α(x)→ FH (a(x,y))→αrewrite (
FH (a(x,y))→α
check (FH (a(x,y))→αcopy (F ε
H (a(x,y))→α(x))))} >% K*(" *(%3!$* "+
2C7D2,. 0'11" #!$* H
!" #$$%&% #' ()%*+%,
RH (a(x,y))→α
TV =
FH (a(x,y))→α(a(x, y))→ F 1TV (F
εH (a(x,y))→α
(a(x, y)))
F 1TV (a(x, y))→ F 2
TV (FH (a(x,y))→αcopy (a(x, y)))
F 2TV (a(x, y))→ F 3
TV (FH (a(x,y))→α
check (a(x, y)))
F 3TV (a(x, y))→ F 4
TV (FH (a(x,y))→αrewrite (a(x, y)))
F 4TV (a(x, y))→ a(x, y)
.
!"# ∀a ∈ Fbin$ R
H (a(x,y))→α
check %&' "( -./0-1 (!"& %#)%''*(' +% ,-#./%# &. "( '%#)% +%
T (Fbin) 0!(0!#+% *,%0 H (a(x, y))1
23* 4.0*'.!( +% 0%' -./0-1 %&' #%(,!5- *# 4* #-&%(0% +" &5)6!4%
(!(7'%#).(*4 F εH (a(x,y))→α
1 8'*(' +!((- "( '%#)% t ∈ T (Fbin) 0!(7
0!#+*(' *,%0 H (a(x, y))9 (RH (a(x,y))→α
check )∗({F εH (a(x,y))→α
(t)}) %&' {t′} !:
PosX (H(a(x, y))) = {11, 12} %' ⊕x,11,⊕y,12 ∈ Fbin ;
R=check
F εH (a(x,y))→α
(H(x))→ H(F 1H (a(x,y))→α
(x))
F 1H (a(x,y))→α
(a(x, y))→ a(F 11H (a(x,y))→α
(x), F 12H (a(x,y))→α
(y))
F 11H (a(x,y))→α
(x)→ ⊕x(x)
F 12H (a(x,y))→α
(y)→ ⊕y(y)
.
$ RH (a(x,y))→ασ %&' "( -./0-1 (!"& %#)%''*(' +% 0!(&'#".#% "(% 4.&'%
!#+!((-% +% '%#)%& .(+%<-& *# +%& ,*#.*64%& +% H(a(x, y))1 =%''% 4.&'%
%&' #% #-&%('-% *# "( '%#)% %' &-)*('.>"%)%(' *" 0!"#*('9 (!"& !",!(&
0!(&.+-#%# 0%''% 4.&'% 0!))% 4* &"6&'.'"'.!( #-&"4'*(' +" *& 0!(0!#7
+*(0%1 ?!.' t′ 4% '%#)% #-&"4'*(' +% (RH (a(x,y))→α
check )∗({F εH (a(x,y))→α
(t)})1
(RH (a(x,y))→ασ )∗(F
H (a(x,y))→αcopy (t′)) )@(% A "( "(.>"% '%#)% t′′ #% #-&%('*('
4* &"6&'.'"'.!( #-&"4'*(' +% 4* 0!(0!#+*(0% %('#% t %' H(a(x, y))1
$ RH (a(x,y))→α
σ−apply %&' "( -./0-1 & -0./*(' 43* 4.0*'.!( +% 4* &"6&'.'"7
'.!( 6.%(7B!#)-% #-&"4'*(' +% t %' H(a(x, y)) &"# 4% '%#)% α1 C.(&.9
(RH (a(x,y))→α
σ−apply )∗(FH (a(x,y))→αrewrite (t′′)) = {ασ}1
()%*+% 2 DE-!#@)% F
?%4!( 4%& #! !&.'.!(& G %' 4% 'E-!#@)% H1
!" #$%&'()* +! ,-%./%('01 2*3 343(56*3 7.76*1(%')*3
()%*+% 3 4()%*+% 5% 67 8)989,:;:9$ <=
(Rλe ∪ (R′λe→I ∪R
′λe→II))
⋆(E′) = R⋆λe(E
′).
*,%0 E ⊆ T (Fbin) %' E′ = {H(t)|t ∈ E}1
!"! #$%&'()* +! ,-./0'1%('-2 &%) 34567 89:
R⋆λe(E
′) ⊆ (Rλe ∪ (R′λe→I
∪R′λe→II
))⋆(E′)! "#$%& '()*$+& %$,-+ Rλe ⊆ (Rλe ∪
(R′λe→I
∪R′λe→II
))
(Rλe ∪ (R′λe→I
∪R′λe→II
))⋆(E′) ⊆ R⋆λe(E
′). .-/0%/)(-+% $+ &01)&1+& ,#1+1,2%$
*$ "1% 3
(Rλe ∪R′λe→I
)⋆(E′) ⊆ R⋆λe(E
′) 3
R′λe⋆H⋆ (a(x,y))→a(x,y)(E
′) ⊆ Rλe⋆H (a(x,y))→a(x,y)(E
′)
R′λe⋆H⋆ (a(x,y))→a(G⋆ (x),y)(E
′) ⊆ Rλe⋆H (a(x,y))→a(G (x),y)(E
′) "10
Hn (a(x, y)) →⋆H (a(x,y))→a(G (x),y) a(Gn (x), y) 1($" n %/4%155$+&
601+*7
8)5),1)0$5$+& 9-/0 ,$ 0$%&$7
(Rλe ∪R′λe→II
)⋆(E′) ⊆ R⋆λe(E
′) 3
8)5),1)0$5$+& 9-/0 ,$% 1/&0$% "1%7
;)*<=* 8> ?;)*<=* 1<@@'2@(* .* 0% &)-&-1'('-2 +A
(Rλe ∪ (R′λe→I ∪R
′λe→II) \ (Rλe→I ∪Rλe→II))
⋆(E′) = R⋆λe(E
′).
1($" E ⊆ T (Fbin) $& E′ = {H(t)|t ∈ E}7
:#190;% ,1 90-9-%)&)-+ < +-/% -=&$+-+%
(Rλe ∪ (R′λe→I ∪R
′λe→II))
⋆(E′) = R⋆λe(E
′).
1($" E ⊆ T (Fbin) $& E′ = {H(t)|t ∈ E}7
:#1/&0$ 910&! 1/ 5-5$+& *-++'! +-/% 9-/(-+% 90-/($0 >/#/+$ 0;6,$ $+ R′λe→I
$%& 199,)"1=,$ %$/,$5$+& %) ,1 0;6,$ "-00$%9-+*1+&$ $+ Rλe→I $%& 199,)"1=,$7 :$ 9,/%
,$ 0'%/,&1& *#/+ 91% *$ 0'?'"0)0$ *$ ,1 0;6,$ $+ R′λe→I
$%& )*$+&)>/$ @ "$,/) *$ ,1 0;6,$
"-00$%9-+*1+&$ $+ Rλe→I 7 8)5),1)0$ 9-/0 R′λe→II
1($" Rλe→II 7 A$,1 +-/% 9$05$&
*#-=&$+)0 ,$ 0'%/,&1& *#/+ 91% *$ 0'?'"0)0$ %1+% /&),)%$0 ,1 0;6,$ $+ (Rλe→I ∪Rλe→II)7
A$,1 5-+&0$ ,1 0$($+*)"1&)-+7
!" #$%&'()* +! ,-./0'1%('-2 &%) 34567
;)*<=* 88 ?;)*<=* 1<@@'2@(* .* 0% &)-&-1'('-2 9A
B/ 5-5$+& *-++'! +-/% 9-/(-+% 90-/($0 >/#/+$ 0;6,$ $+ Rλe $%& 199,)"1=,$ %/0 tm
%$/,$5$+& %) ,1 &01+%)&)-+ "-00$%9-+*1+&$ $+ T $%& C01+"D)%%1=,$ %/0 m7 EF$"&)($?
5$+&! +-/% 9-/(-+% 90-/($0 >/#), $G)%&$ /+$ C-+"&)-+ f *$ ,1 ",H&/0$ &01+%)&)($ R⋆λe
%/0 tm0 @ ,#$+%$5=,$ *$% 510>/16$% 1""$%%)=,$% *$ "$ 0'%$1/ .$&0)7
!8 #$%&'()* 9! :/)';<%('-2 &%) 34567
;)*<=* 8B ?;)*<=* 1<@@'2@(* .* 0% &)-&-1'('-2 C A
deadlock = R⋆λe(init)\R
−1λe (R
⋆λe(init))
deadlock ⊆ R⋆λe(init)\R
−1λe (R
⋆λe(init)) 3 I+&/)&)($5$+& %$,-+ deadlock ⊆
R⋆λe(init) $& deadlock ∩R
−1λe (R
⋆λe(init)) = ∅
!" #$$%&% #' ()%*+%,
R⋆λe(init)\R
−1λe (R
⋆λe(init)) ⊆ deadlock ! "#$%&'($)$*& +,-%$ ./$ 01'2 $3'0&$ t
,%%$00'42$ $& '2 *1$0& +,0 5,*0 R−1λe (R
⋆λe(init))6 ,27-0 '2 57'& 8&-$ /* 9&,& 427:
./,*&;
()%*+% - .()%*+% ,*//0$/1% 2% 34 5)656,0106$ 7 8
<$27* 2, %7*0&-/%&'7* 51/* &$2 0=0&>)$ 5$ -9:9%-'&/-$ RSyncλe 6 '2 (, 0/'(-$ 2$ %7)+7-&$:
)$*& 5/ 0=0&>)$ $* 0=*%?-7*'0,*& ,($% 2$ %?$)'* @A@ 5$ 21,/&7),&$ 5$ BC%?';
D'*0'6
(RSyncλe ∪ Rω
λe ∪ RMλe)
⋆({Sync(init)}) #= ∅
c.E.d. M $0& (9-'F9 +,- 2, G7-)/2$ f ;
()%*+% 9 .()%*+% ,*//0$/1% 2% 34 5)656,0106$ : 8
M, s ² EXφ1 ⇐⇒ ∃π, s0 = s⇒M, s1 ² φ1; "#$%&'($)$*&6 *7/0 74&$*7*0
EXRλe (u) = R−1λe (u)
M, s ² AXφ1 ⇐⇒ ∀π, s0 = s⇒M, s1 ² φ1; H7/0 +7/(7*0 %,2%/2$- ,/&-$)$*&
$* /&'2'0,*& ! %7))$ 0/'(,*& !
M, s ² AXφ1
⇐⇒ s #= s′ avec M, s ² EX¬φ1
⇐⇒ s #∈ S = {s′|∃π, s0 = s′ ⇒M, s1 ² ¬φ1}
"#$%&'($)$*&6 *7/0 74&$*7*0
AXRλe(u) = R⋆
λe(u)\(Rλe−1(R⋆
λe(u)\u))
!!"#" $
!"#$ #$ %&' #" ()*
!" #$%&'(()( *$+,&+*-.
! "!#$%& '()*!#!$%+& &,$ '!-, *& "#.)&,,/, !"# !/ '&,,./, .0 .- )#(& '&,
"#.)&,,/, /$%*%,(, '!-, *& ,1,$23& 4567 .- /$%*%,& '.-) ,&/*&3&-$ *&, ".%-$&/#, '&,
"#.)&,,/, '!-, *& ).'& '& *8!/$#& "#.)&,,/,5
!"#$%% &'()*+ ,
-'!%.
/(0" (1023$!-$!24 #25#6 (1023$!-$!24 #2#57
/(0" (1024 #289($)125#:6 (1024 #289($)12#5:7;;;
3$!-$! %:6 %<7
4 # ! #:6 ! #<6 ! #=6 ! #>7
89($)1 #:6 #<6 #=6 #>7
1!')%.
: ?@ (1023$!-$!24 #25# .A )$B /C/D*E+7 FF/(0" 0!"5 5$G(' 1" #9'%%
(1023$!-$!24 #2#5 .A )$B /C/D*E+7 FF/(0" 0!"5 #9'%% 1" 5$G('
@H <7
< ?@ (1024 #289($)125#: .A )$B /C/D*I+7
(1024 #289($)12#5: .A )$B /C/D*I+7
@H =7
;;;
J ?@ %: .A )$B 3$!-$!* (102#5 K@ (1023$!-$!24 #2#56
(1025# K@ (1023$!-$!24 #25# +7
@H E7
E ?@ %< .A )$B 3$!-$!* (102#5 K@ (1023$!-$!24 #2#56
(1025# K@ (1023$!-$!24 #25# +7
@H L7
L ?@ #: .A )$B 89($)1* (102#5 K@ (1024 #289($)12#5:6
(1025# K@ (1024 #289($)125#: +7
@H I7
;;;
<< ?@ ! #: .A )$B 4 #* () M12#5 K@ (1024 #289($)12#5:6
() M125# K@ (1024 #289($)125#:6
1'!N$12#5 K@ (1023$!-$!24 #2#56
1'!N$125# K@ (1023$!-$!24 #25# +7
@H <=7;;;
O7FF !"#$%% &'()
!" #$$%&% '( )*+,% ,% -./ ,+ 012
!" #$%&'(()( *'$+')$
! "#!$$% 3%45%4 &'! ()')&% $%)#% *+,-./% !"#$%&'%()%*+ !0!&, )& 1!2!*3,2%
4.2*%# "),5 % $%26%)2 !,,%&/ /%$ !11%#$7 %8+"),% /%$ *+,-./%$7 2%&/ /%$ 2+$)#,!,$
.9,%&)$ !)8 "#:%&,$7 %, 2%6:%&, ; #'+,!, :&:,:!#5 % "./% /) 12."%$$)$ <%26%)2 %$, !)=
/%$$.)$5 >)!&/ #! *+,-./% %$, !11%#+%7 #% 1!2!*3,2% !",)%# %$, :&"2+*%&,+ ? :@&%
"./% ABC5
!"#$%% &$!'$!() *
'+!%,
-./" .0/1#23 .0/12#4
5!"#$%% #6.$704
.70 7824
9+0+ 2%:;<4
0!+7%,
= >? .0/12#@!$#' (2%:)4
; (2%:A%.B$ C D) EE (2%:;=<A9+0+FG $ C 5HIJK&&) EE
(2%:;L<A9+0+FG $ C 5HIJK&&) EE (2%:;M<A9+0+FG $ C &FHNOP) EE
(2%:;M< C Q! #Q) EE (2%:;R<A9+0+FG $ C NOF) < 4
SS#"79.0."7 /"! !$#$.'.7: + 2$%%+:$
?T L4
L >? #6.$70 ,C 2%:;=<4 782 ,C 2%:;R<4
?T M4
M >? 782 ,C 782 U L4
?T R4
R >? 2%:A%.B$ ,C D4
2%:;=<A9+0+FG $ ,C 5HIJK&&4 2%:;=< ,C OVWW4
2%:;L<A9+0+FG $ ,C 5HIJK&&4 2%:;L< ,C #6.$704
2%:;M<A9+0+FG $ ,C &FHNOP4 2%:;M< ,C Q! #Q4
2%:;R<A9+0+FG $ ,C NOF4 2%:;R< ,C 7824
?T D4
D >? ; .0/1#2A%.B$ X .0/1#2A2+Y < 4 SS%G7#Z!"7"8% #"79.0."7 /"! %$79.7: + 2$%%+:$
.0/1#2@%$79 (2%:)4
?T =4
[4SS !"#$%% &$!'$!
!"! #$%&'(()( *+,'-. /01
!" #$%&'(()( *+,'-.
! "#!$$% !"#$% &!'( )*% +,-"./),% !"#$%&'%()%*+ !) $%,0%), 1 '2*% "-/% 345
$' #% +!,!67(,% ,-)"+ %$( '*&.,'%), 8 9: % +,-"%$$)$ ;#'%*( %$( "-/. "-66% $)'0!*( <
!"#$%% &'($)*+,-
./!%0 1(2" (*23#45 (*234#6
()* #"7)*6
8/*/ 4%9:;6
*!/)%0
< => #"7)* 0? <6
>@ A6
A => : #"7)* @ B ; 6
>@ C6
A => : #"7)* D? B ; 6
>@ E6
E => 4%9F%(G$ 0? H6
4%9:<;F8/*/IJ $ 0? KLM&NOO6 4%9:<; 0? *P(%6
4%9:A;F8/*/IJ $ 0? KLM&NOO6 4%9:A; 0? QRSS6
4%9:E;F8/*/IJ $ 0? OILTQU6 4%9:E; 0? V! #V6
4%9:W;F8/*/IJ $ 0? TQI6 4%9:W; 0? #"7)*6
>@ W6
W => : (*23#4F%(G$ D (*23#4F4/X ;6
(*23#4Y%$)8 +4%9,6
>@ H6
H => (*234#Y!$#. +4%9,6
: +4%9F%(G$ ? H, ZZ +4%9:<;F8/*/IJ $ ? KLM&NOO, ZZ
+4%9:A;F8/*/IJ $ ? KLM&NOO, ZZ +4%9:A; ? *P(%, ZZ
+4%9:E;F8/*/IJ $ ? OILTQU, ZZ +4%9:E; ? V! #V, ZZ
+4%9:W;F8/*/IJ $ ? TQI, ; 6
>@ B6
B => #"7)* 0? 4%9:W;6
>@ <6
[6\\ !"#$%% &'($)*
!" #$$%&% '( )*+,% ,% -./ ,+ 012
!" #$%&'(()( *+,-.
!"# $%& '&()% )% $!#* +% ,')-!34- %#& .%#/0"#!1+% /0(. +! $0,,("-$!&-0" %"&.%
+%# $+!##%# 2+-%"& %& 3%.4%.5 !"# (" /.%,-%. &%,/#* -+ +-& +% ,%##!6% )7!$&-4!&-0" )!"#
+% !"#$% )% $+-%"& /(-# +7%"40-% !( #%.4%(.5 !"# (" #%$0")% &%,/# -+ +-& +% ,%##!6%
.%&0(."' $0..%#/0")!"& 8/!.,- +7!(&.%# ,%##!6%#9 %& +7%"40-% !( $+-%"&5 :% ,')-!
;/$ %#& %" ,<,% =0.,% /0(. +! $0,,("-$!&-0" )%# )0""'%# (&-+-#!"& +% /.0&0$0+%
;/$5 :% /.0$%##(# ;/$ %#& $0)' $0,,% #(-4!"& >
!"#$%% & #'()
*+!%,
-./" .0 123#45 .0 1234#5 2+!6$23#45 2+!6$234#7
8!"#$%% .0 128!"#$%%5 2+!6$28!"#$%%7
9+2+ 4%6:;7
2!+0%,
< => .0 123#4?!$#* '4%6(7
: '4%6:<;@9+2+AB $ C 8&DEFGG( HH '4%6:I;@9+2+AB $ C 8&DEFGG( HH
'4%6:J;@9+2+AB $ C GA&KLM( HH '4%6:J; C N! #N( ; 7
>O I7
I => .0 128!"#$%% ,C 4%6:<;7 2+!6$28!"#$%% ,C 4%6:I;7
>O J7
J => : 2+!6$234#@%.P$ Q 2+!6$234#@4+R ;7
2+!6$234#?%$09 '4%6(7
>O S7
S => 2+!6$23#4?!$#* '4%6(7
: '4%6:<;@9+2+AB $ C 8&DEFGG( HH '4%6:<; C 2+!6$28!"#$%%( HH
'4%6:I;@9+2+AB $ C 8&DEFGG( HH '4%6:I; C .0 128!"#$%%( HH
'4%6:J;@9+2+AB $ C GA&KLM( HH '4%6:J; C N! #N( ; 7
>O T7
T => : .0 1234#@%.P$ Q .0 1234#@4+R ;7
.0 1234#?%$09 '4%6(7
>O <7
U7VV !"#$%% & #
!!"#" $
!"#$ %&'
!" #$%&' $( )#$*+' ', ---
!"# $%!$!#!&# "& '!()*+ (,"& -.). (" #/#.01+ (+ 234 '!11+ 5) '!&').-&).6!&
(,"&+ 56#.+ (+# $%!'+##"# +. "&+ 56#.+ (+# 73!# '!11+ #"68)&. 9
proc0→ proc1→ ...→ procN → procMAX →
fifoM → ...→ fifo1→ fifo0→ fifoMAX
!: proc ::= PROC(type); IDENT ; var0; ...varL; +. fifo ::=
FIFO(ident);FIRST ; ...
2+# $%!'+##"# +. 5+# 73!# $)%.6'"560%+# #!&. '!11+ 5+# #"68)&.# 9
proc0 ::= procMAIN ;
procMAX ::= PROC(MAX); IDENT (MAX);PNUM ;
fifoMAX ::= FIFO(MAX);FIFONUM ;
!1# %-#+%8-# $!"% (+# 8)%6);5+# <<< 9 =>?@A BCB?A C<D EA DFEA GEHEDA
=>?@ IJA BCB? IJK 2+# 8)%6);5+# (" 1+##)*+ !&. "& ./$+ &!&L(-7&6+ +. #!&
&!1 +#. (6M-%+&. '!11+ "& 8)%6);5+ 5!')5 9
var ::= nom;NEXT (val1);NEXT (val2); ...NEXT (valP );
!. /012%34#,5 $( )#$*+' ', ---
H$%0# N"+5N"+# O!1!1!%$O6#1+# #.).6N"+# !%(6&)6%+# .+5# N"+ 5,)M+'.).6!& (+#
8)5+"%# )"P 8)%6);5+#A 5+# E+#.# +. 5,6&.+%)'.6!& (+# $%!'+##"#A &!"# #!11+# %-"##6
Q '!&#.%"6%+ N"+5N"+# O!1!1!%$O6#1+# (/&)16N"+# .+5# N"+ 5) '%-).6!& (+# $%!'+#L
#"# +. B63!#A 5) 1)&6$"5).6!& (+# 1+##)*+#A +.'K =)% +P+1$5+ 9 2+ O!1!1!%$O6#1+
(+ '%-).6!& (+# $%!'+##"# +#. '!11+ 5+ #"68)&. R?: +#. "&+ #.%"'."%+ <<< ("
&!"8+)" $%!'+##"#S 9
ProcCreate(d)(e, x) =
d ˆ ex−→ Id #6 Re T =>?@S
U Rx T JHFS
ex−→ ProcCreate(d) #6&!&
2) #.%"'."%+ (+ <<< (+ 5,+P+1$5+ +#. $)%"+ ()&# 5+# '!()*+# )" (+##!"# 9 2,-.).
6&6.6)5 (" #/#.01+ 234 9
!" #$$%&% '( ))) *+,- ./0
!"#$%& '()*+$%& ,+-+)$%& ../012 3456788
!"#$9-:& '()*+$9-:& !"#*;9$<& ..=7>0?@A
B'B"$9-:& B'B"*;9$C<%%& ..=7>0?@A
! "#$# %&'($!# $)*+% ,$ -*"$#'.! /0!$1'2&3 /3% )*.-3%%&% 3# 45.% 6
!"#$%& '()*+$%& ,+-+)$<& ../012 3456788
!"#$<& '()*+$<& ,+-+)$;*()B& :$;*()B& 9,D$;*()B& ..874E74 8
!"#$F& '()*+$F& ,+-+)$;*()B& G$;*()B& 9,D$;*()B& ..6@172A 6<
!"#$F& '()*+$H& ,+-+)$;*()B& G$;*()B& 9,D$;*()B& ..6@172A 6F
!"#$9-:& '()*+$9-:& !"#*;9$I& ..=7>0?@A
B'B"$C<%<& ..>1>5 1A>J68
B'B"$C<%%& ..>1>5 1A>J86
B'B"$9-:& B'B"*;9$C<%F& ..=7>0?@A
78$&#*3 )$*#9 .! )*.).%3 ":$,313!# &! ;.1.1.*);'%13 %)"-'$, !"# 2&' 3<"-
/$!% #.&% ,3% )*.-3%%&% $0$!# #0)3 $ =.! $))3,,3 &! :*.&)3 $0$!# #0)3 $> 3! #*$(3*%$!#
,3% 777% &!3 %3&,3 5.'% ).&* -;$2&3 :*.&)3 )$* *$)).*# &!3 ).&* -;$2&3 )*.-3%%&%?
@3##3 #3-;!'2&3 )3&# "-.!.1'%3* &!3 "!.*13 3%)$-3 /3 1"1.'*3 3# $--","*3* ,3
-$,-&,?
Exec(t, gh)(e, x) =
ex−→ Exec(t, gh)&gh %' =e A BCD@>
E =x A t>
ex−→ Exec(t, gh) %'!.!
B&'% .! ,$!-3 ,3 ;.1.1.*);'%13 ).&* ,3 %0%#+13 :,.F$, 3! &#','%$!# ,$ -.1).G
%'#'.! /3% ;.1.1.*);'%13% !"# /3 #.&% ,3% :*.&)3%?
!"! #$%&'()*+, -. /*-012 2+ 333 456
!"#$%&' ()*+,(
!"#$%&'#() *+,-(.#/01 21 ,(23.1* 4 .5 21
*+*'3,1* 21 "!6!%"#'0"1* 2!2#!*
!"#$! %
&'((' ()*"' +,-+-"' #. .-#/'0# (1+' 2' "1"(*$'" 2' ,!3!4,5(#,'6 0++'7! 7'" ! "#$% &% '()
(*'+",'% -./*"+.//%0 8 9-#" $-.(,-." :#' .-(,' $-2*7' 0 70 +#5""0.4' 2;'<+,'""5-. 2'" "1"(*$'"
2' ,!3!4,5(#,' '( :#;57 '"( =5'. 020+(! > 7;!(#2' 2' +,-+,5!(!" 2' "?,'(! '( 2' +,-+,5!(!" 2' 7-@5:#'
('$+-,'77' 2' $-2*7'"8
9-#" 0/-." $5" '. !/52'.4' #.' "-#" 470""' 2' "1"(*$'" A-.4(5-..'7"6 7'" (0($%/"1+'% '( 7'" (0()
$%/"1+'% 2 &'.+"%6 +,!"',/0.( 70 +#5""0.4' 2;'<+,'""5-. 2'" "1"(*$'" A-.4(5-..'7" '( 2'" ('4).5:#'"
2;044!7!,0(5-. 2'" 4074#7" 0=-#(5""0.( > #. -#(57 2' /!,5B40(5-. "1$=-75:#' 'C404'8
D0." 70 +0,(5' '<+!,5$'.(07'6 .-#" 0/-." 4-$+0,! .-(,' -#(576 2;#.' +0,( 0/'4 2'" -#(57" 2' ,!3!4,5(#,'
('7" :#' E5$=#F6 G0#2' '( EHG6 2;0#(,' +0,( 0/'4 2'" -#(57" 2' /!,5B40(5-. ('7" :#' IJK96 9#IGL6
IGM E6 NIDD8 9-" ='.4)$0,F" 2!$-.(,'.( 7;'C4045(! 2'" "1"(*$'" A-.4(5-..'7" !7!$'.(05,'"
+-#, 70 /!,5B40(5-. 2' $-2*7'"8
G-(" 47!" % /!,5B40(5-. "1$=-75:#'6 "1"(*$'" 2' ,!3!4,5(#,' 2'" (',$'"6 250@,0$$'" 2' 2!453
"5-." =5.05,'"6 07@-,5()$' 2' "0(#,0(5-.6 7-@5:#' ('$+-,'77' 75.!05,'8
!"#$%&' "$()%*'+)',&-. #/0)( $- 1)21&3&-. 0!03)"0
M="(,04( %
E)5" J)D ()'"5" +,-+-"'" ()' ()'-,'(5407 A-#.20(5-." -A 0 .'O A-,$07 (--7 A-, "1$=-754 /',5B3
40(5-. -A B.5(' $-2'7"8 I-$' 0++,-04)'" ,'2#4' ()' +,-=7'$ -A "1"('$ /',5B40(5-. (- ()' ,'04)0=575(1
+,-=7'$ 5. (',$ ,'O,5(5.@ "1"('$" PE I"Q8
K. -#, 0++,-04)6 "(0('" 0,' '.4-2'2 =1 (',$" 5. 0 RDD375F' $0..', 0.2 ()' (,0."5(5-. ,'70(5-. 5"
,'+,'"'.('2 =1 0 .'O ,'O,5(5.@ ,'70(5-. "- 4077'2 3,/*"+./10 4%'$ 5%6'+"+/7 8! "%$ PSE I"Q8
S5,"(6 O' ")-O ()0( SE I" 0,' 0" '<+,'""5/' 0" E I"8 E)'.6 O' A-4#" -. 0 "#=470"" -A SE I"6 "-
4077'2 90%$%/"1'! 3,/*"+./10 4%'$ 5%6'+"+/7 8! "%$ PTSE I"Q6 0.2 O' ")-O ()0( TSE I"
+,'"',/' ()' SE I" '<+,'""5/'.'""8 E)' $05. 02/0.(0@' -A TSE I" 5" ()0( ()'1 0,' O'77 020+('2 A-,
044'7',0(5-. ('4).5:#'" #"#0771 #"'2 5. "0(#,0(5-. 07@-,5()$" -. RDD375F' 20(0 "(,#4(#,'"8
H#, '<+',5$'.(" ")-O ()0( A-, O'773F.-O. +,-(-4-7" P'8@8 E,'' M,=5(',6 J',4-70('6 -#.2 -=5.
G#('< +,-(-4-7"6888Q -#, (--7 5" .-( -.71 ='((', ()0. -()', ,'O,5(5.@ (--7" "#4) 0" E5$=#F6 G0#2' -,
EHG6 =#( 07"- 4-$+'(5(5/' O5() -()', $-2'734)'4F'," "#4) 0" IJK96 9#IGL -, IGM E8 G-,'-/',6
5( 40. 07"- =' 0++75'2 (- $-2'734)'4F5.@ 5./0,50.( +,-+',(5'" O)54) 0,' 0 +0,(54#70, "#=470"" -A 75.'0,
('$+-,07 7-@54 A-,$#70 PUEUQ8
V'1O-,2" % "1$=-754 /',5B40(5-.6 (',$ ,'O,5(5.@ "1"('$6 =5.0,1 2'45"5-. 250@,0$6 "0(#,0(5-.
07@-,5()$6 75.'0, ('$+-,07 7-@54
!"#$!%#&$' ()*+,#$-!%&./' 0#+(!-'+%!1'
()2$13!+4
5!%6 7*89 :+&;'$4&%3 ()2$13!+4
</' 3#+!$( (' =&+>&9 56?6 @ABC
0DEBF@A 2< G8HI J'('K L