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6GhG6Q` 6GhG- i?2 M2ti QT2`�iQ` Q7 GhG Bb bTHBi BMiQ � r2�F U��V �M/ bi`QM; U�+V p2`bBQMX h?2bi`QM; M2ti `2[mB`2b i?2 2tBbi2M+2 Q7 � M2ti bi�i2 iQ #2 i`m2- r?BH2 i?2 r2�F M2ti QT2`�iQ` Bbp�+mQmbHv i`m2 B7 i?2`2 Bb MQ M2ti bi�i2X h?mb- BM i?2 }M�H bi�i2- i?2 r2�F QT2`�iQ` 2p�Hm�i2biQ i`m2- �M/ i?2 bi`QM; M2ti iQ 7�Hb2X h?2 bi`QM; �M/ r2�F M2ti QT2`�iQ`b �`2 /m�H Q7 2�+?Qi?2`- rBi? ¬�+ ' ⌘ ��¬'X
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.2}MBiBQM kXkXj U6GhG avMi�tVX � 6GhG 7Q`KmH� ' Bb BM/m+iBp2Hv /2}M2/ �b 7QHHQrb,
' ::= a | ? | ¬' | ' _ ' | ' ^ ' | �+ ' | �� ' | 'l'
�+ Bb T`QMQmM+2/ �b ǵbi`QM; M2tiǶ �M/ �� �b ǵr2�F M2tiǶX
6GhG mb2b i?2 b�K2 �##`2pB�iBQMb �b GhG ULQi�iBQM kXRXkVX
.2}MBiBQM kXkX9 U6GhG b2K�MiB+bVX .2}M2 i?2 b�iBb7�+iBQM `2H�iBQM 7Q` 6GhG ✏6GhG�b 7QHHQrbX *QTv i?2 bi�M/�`/ GhG b2K�MiB+b BM .2}MBiBQM kXRXe- �M/ `2TH�+2 i?2 `mH27Q` i?2 M2ti QT2`�iQ` rBi?,
� ✏ �+' Bz �1 ✏ ' �M/ �1 6= ✏
� ✏ ��' Bz �1 ✏ ' Q` �1 = ✏
� 7mHH /2}MBiBQM Bb ;Bp2M BM i?2 �TT2M/Bt �b .2}MBiBQM dXRXkX
G2KK� kXkX8 U1tT�MbBQM G�rb 7Q` 6GhGVX AM 6GhG- i?2 7QHHQrBM; 2tT�MbBQM H�rb�`2 2[mBp�H2M+2b,
⇤' ⌘ ' ^��⇤' U9V⌃' ⌘ ' _�+⌃' U8V
l' ⌘ ' _ ( ^�+( l')) UeV
1t�KTH2 kXkXe U6GhG 2t�KTH2bVX G2i r, y, g 2 P �M/ ' = ⇤(g ! (¬rly)) #2 �M 6GhG7Q`KmH�- i?2 b�K2 i`�{+ HB;?i 7Q`KmH� 7Q`KmH� 7`QK 1t�KTH2 kXRXNXh?2 i`�+2 ({g}, {r}) pBQH�i2b 'X h?2 i`�+2 ({g}) b�iBb}2b Bi- �b /Q2b ({g}, {y}, {r})XG2i = ⇤⌃g- i?2 b2+QM/ T`QT2`iv BM 1t�KTH2 kXkXkXh?2 i`�+2 ({g}, {r}) pBQH�i2b - �b /Q2b ({g}, {y}, {r})X h?2 i`�+2 ({g}) b�iBb}2b X� T`QQ7 Q7 G2KK� kXkX8 Bb �p�BH�#H2 BM ("GaRy- TX 8Xj)X
GhG3
AM GhG3- � M2r i`mi? p�Hm2 Bb BMi`Q/m+2/ iQ `2T`2b2Mi mM+2`i�BMivX h?2 mM+2`i�BMiv Bb i?�ii?2`2 �`2 bQK2 bm{t2b r?B+? b�iBb7v i?2 i`�+2 �M/ bQK2 r?B+? /Q MQiX GhG3 mb2b i?2 b�K2bvMi�t U.2}MBiBQM kXRXRV �M/ �##`2pB�iBQMb ULQi�iBQM kXRXkV �b GhGX
.2}MBiBQM kXkXd UGhG3 b2K�MiB+bVX h?2 b2K�MiB+b 7Q` GhG3 Bb /2}M2/ #v +�b2b `2HvBM;QM bi�M/�`/ GhG b2K�MiB+b U.2}MBiBQM kXRXeVX h?2 MQi�iBQM �� `272`b iQ i?2 +QM+�i2@M�iBQM Q7 i?2 BM}MBi2 i`�+2 � iQ i?2 }MBi2 i`�+2 �X .2}M2 i?2 b�iBb7�+iBQM `2H�iBQM 7Q`GhG3 ✏GhG3 �b 7QHHQrbX
� ✏GhG3 ' =
8><
>:
> B7 �� ✏ GhG' 7Q` �HH � 2 P+
? B7 �� ✏ GhG' 7Q` �HH � 2 P+
? Qi?2`rBb2
q?2`2 ? Bb � i`mi? p�Hm2 +�HH2/ ǵBM+QM+HmbBp2ǶX h?2 p�Hm2 ? Bb MQi T�`i Q7 i?2 bvMi�t Q7GhG3- `�i?2` Bi Bb T�`i Q7 i?2 b2K�MiB+ K2i�@H�M;m�;2 ("GaRy- TX e8N)X
Ai K�v MQi Q#pBQmb QM }`bi `2�/BM; i?�i `mMiBK2 p2`B}+�iBQM mbBM; GhG3 Bb /2+B/�#H2X � `mM@
e
iBK2 p2`B}+�iBQM T`Q+2/m`2 7Q` GhG3 Bb ;Bp2M BM �`�7�i 2i �HX (�`�Yy8- TX e)X h?2 T`Q+2/m`2Bb /Qm#Hv 2tTQM2MiB�H rBi? `2bT2+i iQ i?2 bBx2 Q7 i?2 BMTmi 7Q`KmH�X1t�KTH2 kXkX3 UGhG3 2t�KTH2bVX G2i r, y, g 2 P �M/ ' = ⇤(g ! (¬rly)) #2 �M GhG3
7Q`KmH�- i?2 b�K2 i`�{+ HB;?i 7Q`KmH� 7Q`KmH� 7`QK 1t�KTH2 kXRXNXh?2 i`�+2 ({g}, {r}) pBQH�i2b 'X h?2 i`�+2 ({g}) Bb BM+QM+HmbBp2- �b Bb ({g}, {y}, {r})XG2i = ⇤⌃g- i?2 b2+QM/ T`QT2`iv BM 1t�KTH2 kXkXkXh?2 i`�+2 ({g}, {r}) Bb BM+QM+HmbBp2- �b Bb ({g})- �M/ �HbQ ({g}, {y}, {r})X
_o@GhG_o@GhG Bb #�b2/ QM 6GhG �M/ GhG3X _o@GhG ;�BMb i?2 +�T�#BHBiv iQ `2T`2b2Mi mM+2`i�BMiv7`QK GhG3- r?BH2 ;BpBM; KQ`2 mb27mH BM7Q`K�iBQM i?�M bBKTHv ǵi?2 p2`B}+�iBQM r�b BM+QM+Hm@bBp2Ƕ- #v /Bz2`2MiB�iBM; #2ir22M BM+QM+HmbBp2 +�b2b mbBM; i?2 r2�F �M/ bi`QM; M2ti QT2`�iQ`b7`QK 6GhGX_o@GhG mb2b i?2 b�K2 bvMi�t �b 6GhG U.2}MBiBQM kXkXjV �M/ i?2 �##`2pB�iBQMb Q7 GhGULQi�iBQM kXRXkVX
.2}MBiBQM kXkXN U_o@GhG b2K�MiB+bVX h?2 b2K�MiB+b 7Q` _o@GhG �`2 /2}M2/ #v +�b2bX
.2}M2 i?2 b�iBb7�+iBQM `2H�iBQM 7Q` _o@GhG ✏_o@GhG �b 7QHHQrbX
� ✏_o@GhG ' =
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>>>:
> B7 � ✏GhG3 ' = >? B7 � ✏GhG3 ' = ?>P B7 � ✏GhG3 ' = ? �M/ � ✏6GhG= >?P B7 � ✏GhG3 ' = ? �M/ � ✏6GhG= ?
>P Bb T`QMQmM+2/ ǵT`2bmK�#Hv i`m2Ƕ �M/ ?P Bb T`QMQmM+2/ ǵT`2bmK�#Hv 7�Hb2ǶX LQi2i?�i ?P �M/ >P �`2 T�`i Q7 i?2 b2K�MiB+ K2i�@H�M;m�;2- MQi i?2 7Q`KmH� bvMi�tX
1t�KTH2 kXkXRy U_o@GhG 2t�KTH2bVX G2i r, y, g 2 P �M/ ' = ⇤(g ! (¬rly)) #2 �M _o@GhG7Q`KmH�- i?2 b�K2 i`�{+ HB;?i 7Q`KmH� 7Q`KmH� 7`QK 1t�KTH2 kXRXNXh?2 i`�+2 ({g}, {r}) pBQH�i2b 'X h?2 i`�+2 ({g}) T`2bmK�#Hv pBQH�i2b 'X 6BM�HHv ({g}, {y}, {r})T`2bmK�#Hv b�iBb}2b i?2 i`�+2XG2i = ⇤⌃g- i?2 b2+QM/ T`QT2`iv BM 1t�KTH2 kXkXkXh?2 i`�+2 ({g}, {r}) T`2bmK�#Hv pBQH�i2b �b /Q2b ({g}, {y}, {r})X h?2 i`�+2 ({g}) T`2bmK@�#Hv b�iBb}2b X
kXj hBK2/ S`QTQbBiBQM�H h2KTQ`�H GQ;B+hBK2/ S`QTQbBiBQM�H h2KTQ`�H GQ;B+ UhShGV(�>N9) Bb �M 2ti2MbBQM Q7 GhG iQ 2M+Q/2 2t@THB+Bi iBK2 BMiQ i?2 HQ;B+X AM GhG- QMHv i?2 Q`/2` Q7 bi�i2b Bb `2�bQM2/ �#Qmi- BX2X- �i Bb#27Q`2 �i+1- #mi 7Q` K�Mv p2`B}+�iBQM T`Q#H2Kb Bi +�M #2 mb27mH iQ `2�bQM �#Qmi i?2 2tTHB+BiiBK2 �i r?B+? � bi�i2 Q++m`b- BX2X- �i Bb j iBK2 mMBib 2�`HB2` i?�M �i+1X 6Q` i?Bb- r2 /2}M2 �iBK2/ i`�+2 i?�i +QMi�BMb i?2 b2[m2M+2 Q7 iBK2b +Q``2bTQM/BM; iQ i?2 b2[m2M+2 Q7 bi�i2bX �[m�MiB}2` +�HH2/ � 7`22x2 QT2`�iQ` Bb �//2/ iQ i?2 HQ;B+- r?B+? #BM/b Q++m``2M+2b Q7 � p�`B�#H2iQ i?2 iBK2 �i r?B+? i?2 QT2`�iQ` Bb 2p�Hm�i2/X AMi2;2` +QKT�`BbQM Bb �//2/ iQ i?2 HQ;B+- iQ�HHQr 7Q` bT2+B}+�iBQMb bm+? �b, dzA7 p #2+QK2b i`m2- q rBHH #2+QK2 i`m2 rBi?BM 8 iBK2 mMBibǴXhShG Bb � ;2M2`�HBb�iBQM Q7 GhG- �HH T`QT2`iB2b 2tT`2bb2/ BM GhG �`2 2tT`2bbB#H2 BM hShG�b r2HHX
d
.2}MBiBQM kXjXR UhBK2/ h`�+2VX � hBK2/ h`�+2 ⇢ = (�, ⌧) Bb � T�B` Q7 � i`�+2 Ubi�i2b2[m2M+2V � 2 P+ �M/ � iBK2 b2[m2M+2 ⌧ 2 N+ r?B+? Bb � b2[m2M+2 Q7 MQM@M2;�iBp2BMi2;2`b i?�i Bb KQMQiQMB+�HHv MQM@/2+`2�bBM; U8i. ⌧i ⌧i+1V �M/ 2p2Mim�HHv �Hr�vbBM+`2�bBM; U8i.9j. ⌧i < ⌧jVX
�bbmK2 i?2`2 2tBbi �M BM}MBi2 bmTTHv Q7 iBK2 p�`B�#H2b �M/ H2i V = {x, y, z . . . } #2 i?2 b2iQ7 i?2b2 p�`B�#H2bX
.2}MBiBQM kXjXk UhShG avMi�tVX G2i a 2 P #2 � T`QTQbBiBQM�H p�`B�#H2- x 2 V #2 �iBK2 p�`B�#H2 �M/ c 2 N � +QMbi�MiXhShG i2`Kb ⇡ �M/ 7Q`KmH�b ' �`2 BM/m+iBp2Hv /2}M2/ �b 7QHHQrb,
⇡ ::= x+ c | c' ::= a | ⇡ ⇡ | ? | ¬' | ' _ ' | ' ^ ' | � ' | 'l' | x.'
q2 `2�/ x.' �b ǵ7`22x2 x BM 'ǶX
LQi�iBQM kXjXj U�##`2pB�iBQMbVX hShG ?�b i?2 b�K2 �##`2pB�iBQMb �b GhG- THmb �##`2@pB�iBQMb 7Q` +QKT�`BbQM QT2`�iBQMb #2ir22M i2`Kb bm+? �b ⇡1 < ⇡2 := ⇡1 ⇡2 ^¬(⇡2 ⇡1)- �M/ bQ QM �b 2tT2+i2/ 7Q` >,�,=, 6=X
.2}MBiBQM kXjX9 UhShG a2K�MiB+bVX G2i ⇢ = (�, ⌧) #2 � iBK2/ i`�+2- x 2 V - c 2 N-�M/ H2i E : V ! N #2 �M UBMBiB�HHv 2KTivV 2p�Hm�iBQM Q7 iBK2 p�`B�#H2b iQ iBK2b- rBi?E(x+c) = E(x)+c- E(c) = c �M/ E [x := c] /2MQiBM; i?2 2p�Hm�iBQM r?B+? �;`22b rBi? E7Q` V \ {x} �M/ K�Tb x iQ cX hShG mb2b i?2 b�K2 b2K�MiB+b �b GhG U.2}MBiBQM kXRXeV7Q` i?Qb2 QT2`�iQ`b r?B+? Qp2`H�T rBi? GhG- 2t+2Ti i?2 `2H�iBQM Bb /2}M2/ Qp2` � iBK2/i`�+2 `�i?2` i?�M � bi�M/�`/ i`�+2X 6Q` i?Qb2 +�b2b- bm#biBimi2 Q++m``2M+2b Q7 � ✏GhG · · ·rBi? ⇢, E ✏hShG · · · X h?2 `2K�BM/2` Q7 i?2 /2}MBiBQM Bb �b 7QHHQrb,
⇢, E ✏hShG ⇡1 ⇡2 Bz E(⇡1) E(⇡2)⇢, E ✏hShG x.' Bz ⇢, E [x := ⌧0] ✏ '
� 7mHH /2}MBiBQM Bb ;Bp2M �b .2}MBiBQM dXRXj
1t�KTH2 kXjX8 UhShG 2t�KTH2bVX G2i r, y, g 2 P �M/
' = ⇤ x1. (g ! (¬r l x2. (y ^ x2 x1 + 10)))
#2 � hShG 7Q`KmH�- Qm` i`�{+ HB;?i 2t�KTH2- #mi rBi? i?2 �//2/ `2[mB`2K2Mi i?�i v2HHQr�Hr�vb Q++m`b rBi?BM Ry iBK2 bi2Tb 7`QK ;`22MXh?2 iBK2/ i`�+2 ((1, {g}), (2, {r}), . . . ) pBQH�i2b '- �b /Q2b ((1, {g}), (12, {y}), (15, {r}), . . . )-�M/ �HbQ ((1, {g}), (2, {g}), . . . )Xh?2 iBK2/ i`�+2 ((0, {g}), (10, {y}), (15, {r}), (20, {g}), (30, {y}), (35, {r}), . . . ) b�iBb}2b '- �b/Q2b ((100, {}), (200, {}), . . . ) Up�+mQmbHvVX
kX9 6BMBi2 hShGA1@hShG�b BM A1@GhG- BM A1@hShG r2 �bbmK2 i?�i i?2 }M�H bi�i2 Q7 � i`�+2 `2T2�ib 7Q`2p2`X AǶK MQibm`2 B7 i?Bb ?�b #22M bim/B2/ #27Q`2X AM *?�B �M/ a+?HBM;HQz (*aRj) � `mMiBK2 p2`B}+�iBQM�H;Q`Bi?K 7Q` hShG Bb BMi`Q/m+2/- r?B+? HQQFb HBF2 A1@hShG- #mi Bi Bb MQi +H2�` B7 i?�iǶbr?�i Bb mb2/X
3
AM A1@hShG- i?2 QMHv /Bz2`2M+2 7`QK hShG Bb i?2 b2K�MiB+b BM i?2 }M�H bi�i2X h?2 bvMi�t`2K�BMb i?2 b�K2 �b BM hShGX
.2}MBiBQM kX9XR UA1@hShG b2K�MiB+bVX .2}M2 i?2 b2K�MiB+ `2H�iBQM ✏A1@hShG Q7 A1@hShG �b 7QHHQrbX *QTv i?2 b2K�MiB+b 7`QK i?2 hShG b2K�MiB+b U.2}MBiBQM kXjX9V- �M/`2/2}M2 i?2 ǵM2tiǶ � �M/ ǵmMiBHǶ l QT2`�iQ`b �b 7QHHQrb,
⇢, E ✏ �' =
8><
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> B7 ⇢1, E ✏ ' �M/ ⇢1 6= ✏
> B7 ⇢, E ✏ ' �M/ ⇢1 = ✏
? Qi?2`rBb2
⇢, E ✏ 'l =
8><
>:
> B7 ⇢i, E ✏ 7Q` bQK2 i > 0 �M/ ⇢j , E ✏ ' 7Q` �HH 0 j < i
> B7 ⇢, E ✏ �M/ ⇢1 = ✏
? Qi?2`rBb2
� 7mHH /2}MBiBQM Bb ;Bp2M �b .2}MBiBQM dXRX9
_2K�`F kX9XkX A1@hShG ?�b bQK2 BMi2`2biBM; +QMb2[m2M+2b r?2M �TTHB2/ iQ iBK2/7Q`KmH�2X 6Q` 2t�KTH2- i?2 7Q`KmH� ⇤x. ⌃y. y > x Bb �Hr�vb 7�Hb2- bBM+2 i?2 iBK2 Q7 i?2}M�H bi�i2 Bb `2T2�i2/ 7Q`2p2`X A bmbT2+i �MQi?2` bQHmiBQM Bb /2bB`�#H2X� bQHmiBQM iQ i?Bb T`Q#H2K +QmH/ #2 iQ bBKTHB7v i?2 7Q`KmH� BM i?2 H�bi bi�i2 iQ QMHv?�p2 iBK2 +QMbi`�BMib `2K�BMBM;- r?B+? +�M #2 /QM2 bBM+2 r2 FMQr i?�i i?2 bi�i2 rBHH`2K�BM i?2 b�K2 BM �M BM}MBi2 2ti2MbBQMX PM2 rQmH/ i?2M T�bb i?2 iBK2 +QMbi`�BMib iQ� [m�MiB}2` 2HBKBM�iBQM T`Q+2/m`2 BM Q`/2` iQ K�F2 � /2+BbBQMX
1t�KTH2 kX9Xj UA1@hShG 2t�KTH2bVX G2i r, y, g 2 P �M/
' = ⇤ x1. (g ! (¬r l x2. (y ^ x2 x1 + 10)))
#2 �M A1@hShG 7Q`KmH�- Qm` i`�{+ HB;?i 2t�KTH2 7`QK 1t�KTH2 kXjX8Xh?2 iBK2/ i`�+2 ((1, {g}), (2, {r})) pBQH�i2b '- �b /Q2b ((1, {g}), (12, {y}), (15, {r}))- �M/ �HbQ((1, {g}), (2, {g}))Xh?2 iBK2/ i`�+2 ((0, {g}), (10, {y}), (15, {r})) b�iBb}2b '- �b /Q2b ((100, {})) Up�+mQmbHvVX
_o@hShGh?2 7QHHQrBM; }MBi2 b2K�MiB+b 7Q` hShG- r?B+? A M�K2 _o@hShG- Bb � +QK#BM�iBQM Q7i?2 _o@GhG 7Qm` p�Hm2/ HQ;B+ �M/ hShGX A ?�p2 MQi b22M i?Bb +QK#BM�iBQM 2Hb2r?2`2 BMi?2 HBi2`�im`2- �M/ i?mb �bbmK2 Bi Bb M2rX h?2 +HQb2bi A ?�p2 b22M Bb i?2 i?`22 p�Hm2/hGhG3 (�`�Yy8- TX e) r?B+? Bb #�b2/ QM GhG3 �M/ hGhG- � /Bz2`2Mi iBK2/ i2KTQ`�H HQ;B+Xh?2 #`Q�/ B/2� Bb i?�i B7 _o@GhG Bb }MBi2 GhG Y mM+2`i�BMiv �M/ hShG Bb GhG Y 2tTHB+BiiBK2- i?2M _o@hShG Bb }MBi2 GhG Y mM+2`i�BMiv Y 2tTHB+Bi iBK2X
.2}MBiBQM kX9X9 U_o@hShG avMi�tVX .2}M2 i?2 7Q`KmH�b Q7 _o@hShG �b,
⇡ ::= x+ c | c' ::= a | ⇡ ⇡ | ? | ¬' | ' _ ' | ' ^ ' | �+ ' | �� ' | 'l' | x.'
_o@hShG mb2b i?2 b�K2 �##`2pB�iBQMb �b hShG ULQi�iBQM kXjXjVX
N
.2}MBiBQM kX9X8 U_o@hShG a2K�MiB+bVX A rBHH ;Bp2 i?2 b2K�MiB+b #�b2/ QM hShG- BM�M �M�HQ;Qmb r�v iQ ?Qr _o@GhG r�b ;Bp2MX G2i E : V ! N #2 �M UBMBiB�HHv 2KTivV2p�Hm�iBQM Q7 iBK2 p�`B�#H2b iQ iBK2b �b BM hShG U.2}MBiBQM kXjX9VX h?2 MQi�iBQM ⇢!`272`b iQ i?2 +QM+�i2M�iBQM Q7 i?2 BM}MBi2 iBK2/ i`�+2 ! iQ i?2 }MBi2 iBK2/ i`�+2 ⇢XA rBHH /2}M2 irQ b2K�MiB+ `2H�iBQMb BM Q`/2` iQ i?2M /2}M2 i?2 _o@hShG b2K�MiB+bXh?2b2 �`2 2bb2MiB�HHv hShG3 �M/ 6hShG `2bT2+iBp2HvXG2i ✏T3 #2 ;Bp2M #v,
⇢, E ✏T3 ' =
8><
>:
> B7 ⇢!, E ✏hShG ' 7Q` �HH ! 2 (P+,N+)
? B7 ⇢!, E ✏hShG ' 7Q` �HH ! 2 (P+,N+)
? Qi?2`rBb2
G2i ✏FT #2 /2}M2/ #v i�FBM; hShG b2K�MiB+b �M/ �//BM;,
⇢, E ✏FT �+' Bz ⇢1, E ✏TPTL ' �M/ ⇢1 6= ✏
⇢, E ✏FT ��' Bz ⇢1, E ✏TPTL ' Q` ⇢1 = ✏
6BM�HHv- /2}M2 ✏_o@hShG- i?2 b�iBb7�+iBQM `2H�iBQM 7Q` _o@hShG- #v i?2 7QHHQrBM;,
⇢ ✏_o@hShG ' =
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>>>:
> B7 ⇢, E ✏T3 ' = >? B7 ⇢, E ✏T3 ' = ?>P B7 ⇢, E ✏T3 ' = ? �M/ ⇢, E ✏FT= >?P B7 ⇢, E ✏T3 ' = ? �M/ ⇢, E ✏FT= ?
1t�KTH2 kX9Xe U_o@hShG 2t�KTH2bVX G2i r, y, g 2 P �M/
' = ⇤ x1. (g ! (¬r l x2. (y ^ x2 x1 + 10)))
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⇤(g ! (¬rly)) ){g} ⇤(g ! (¬rly)) ^ (¬rly) UdV){r} ? U3V
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RR
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⇤x.(sent ! ⌃y.(received ^ y x+ 3))
){sent}⇤x.(sent ! ⌃y.(received ^ y x+ 3))
^ ⌃y.(received ^ y 4) UNV){received}(��⇤x.(sent ! ⌃y.(received ^ y x+ 3))
^�+⌃y.(received ^ y 4))
_��⇤x.(sent ! ⌃y.(received ^ y x+ 3)) URyV⌘>P URRV
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Rk
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abstract class Formula {
...
def /\(f:Formula) = And(this,f)def \/(f:Formula) = Or(this,f)def ->(f:Formula) = Or(this.compl,f)def unary_!() = Neg(this)def U(f:Formula) = Until(this,f)
}
object Formula {
def X(f:Formula) = Next(f)
def N(f:Formula) = WeakNext(f)
def F(f:Formula) = Future(f)
def G(f:Formula) = Globally(f)
...
}
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case class And(f1: Formula, f2: Formula) extends BinOp("∧") {...}
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case class Atom[A](name: String, interpretation: A => Boolean) extends Formula
...
def apply[A](state: (Int,A)): Formula = this match {
if (interpretation(state._2)) True
else False
}
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?�b �M interpretation 7mM+iBQM UT�bb2/ iQ i?2 +QMbi`m+iQ`V r?B+? i�F2b �Mv ivT2 �b BMTmiU2X;X- � ai`BM;- � GBbi- 2i+V �M/ QmiTmib � #QQH2�M- �M/ i?2 apply 7mM+iBQM K�Tb i?�i #QQH2�MiQ True Q` False- r?B+? �`2 Const Q#D2+ib BM Kv HQ;B+- MQi i?2 a+�H� #QQH2�Mb true �M/ falseX
abstract class Const(name: String, n: Int) extends Formula with Ordered[Const] {...}
case object False extends Const("⊥",0)
case object PFalse extends Const("⊥ᴾ",1)
case object PTrue extends Const("⊤ᴾ",2)
case object True extends Const("⊤",3)
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implicit def TimeVarSymbolNoOffset(x: Symbol) = TimeVar(x,0)
implicit class TimeVarSymbol(x: Symbol) {
def ~(f: Formula) = Freeze(x,f) // e.g. 'x ~ (a /\ b) is a Formula
Rj
def +(t: Int) = TimeVar(x,t) // e.g. 'x + 15 is a TimeTermdef <=(t2: TimeTerm) = LTEQ(TimeVar(x,0),t2)
def ~=(t2: TimeTerm) = LTEQ(TimeVar(x,0),t2) /\ LTEQ(t2,TimeVar(x,0))
def <(t2: TimeTerm) = LTEQ(TimeVar(x,0),t2) /\ !LTEQ(t2,TimeVar(x,0))
def >(t2: TimeTerm) = !LTEQ(TimeVar(x,0),t2)
def >=(t2: TimeTerm) = LTEQ(t2,TimeVar(x,0))
}
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case class Freeze(x: Symbol, f: Formula) extends Formula {...}
case class LTEQ(t1: TimeTerm, t2: TimeTerm) extends Formula {...}
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#`2�Fb /QrM BMTmi 7Q`KmH� BMiQ /BbDmM+iBp2 MQ`K�H 7Q`K U.L6V +H�mb2b U� b2i Q7 /BbDmM+iBQMb#2ir22M b2ib Q7 +QMDmM+iBQMbVX Ai i?2M mb2b Kv bBKTHB}+�iBQM K2i?Q/b �M/ 2tT�MbBQM H�rbUG2KK� kXkX8V iQ 2tT�M/ i?2 7Q`KmH� U�M/ bm#7Q`KmH�VX PM+2 � 7Q`KmH� Bb 2Bi?2` bBKTHB}2/iQ h`m2f6�Hb2 Q` ?�b ǵM2tiǶ U�+,��V BM 7`QMi Q7 Bi- Bi Bb /QM2X PM+2 �HH 7Q`KmH�2 �`2 /QM2-i?2 `2bmHiBM; +H�mb2 b2i Bb `2im`M2/Xh?2 Qi?2` T�`i Q7 ǵT`Q;`2bbBQMǶ Bb i?2 `2KQp�H Q7 ǵM2tiǶ QT2`�iQ`b- r?B+? Bb /QM2 #v i?2 unX
7mM+iBQMX h?Bb 7mM+iBQM bBKTHv K�Tb �+' �M/ ��' iQ 'Xh?2 bBKTHB}+�iBQM Bb T`Q+2bb2/ #v Kv apply �M/ simplify K2i?Q/bX h?2 /Bz2`2M+2 #2ir22Mi?2b2 irQ K2i?Q/b Bb i?�i apply /Q2b bBKTHB7v #�b2/ QM i?2 +m``2Mi bi�i2 Ui?2 bi�i2 Bb �TTHB2/iQ i?2 7Q`KmH�V- r?BH2 simplify QMHv mb2b i?2 +m``2Mi iBK2X "Qi? K2i?Q/b +?2+F 7Q` bQK2#�bB+ i�miQHQ;B2b HBF2 a _ ¬aX h?2 `2�bQM 7Q` i?2 /Bz2`2M+2 #2ir22M i?2 irQ bBKTHB}+�iBQMK2i?Q/b Bb i?�i A QMHv r�Mi iQ bBKTHB7v T`QTQbBiBQM�H bm#7Q`KmH�2 Ubm#7Q`KmH�2 MQi T�`i Q7� i2KTQ`�H bm#7Q`KmH�V #�b2/ QM i?2 +m``2Mi bi�i2 BMi2`T`2i�iBQMXq?2M � i2KTQ`�H bm#7Q`KmH� Bb 2M+QmMi2`2/- apply +�HHb simplify QM BiX h?2 `2�bQM simplify
?�b � iBK2 T�`�K2i2`- Bb iQ +?2+F B7 �Mv iBK2 +QMbi`�BMib BM i?2 bm#7Q`KmH� rBHH �Hr�vb #2 i`m2Q` 7�Hb2X A 7QmM/ i?Bb bBKTHB}+�iBQM i2+?MB[m2 iQ #2 � bB;MB}+�Mi T2`7Q`K�M+2 BKT`Qp2K2MiQp2` TH�BM `2r`BiBM;- �Hi?Qm;? A /Q MQi ?�p2 K2�bm`2K2Mib Q7 i?Bb BKT`Qp2K2MiXJv p2`B}2` i�F2b � iBK2/ i`�+2 �M/ � 7Q`KmH� �M/ `2+m`bBp2Hv T�bb2b Bi iQ i?2 expand �M/unX 7mM+iBQMb /2b+`B#2/ �#Qp2- mMiBH i?2 }M�H bi�i2 Bb `2�+?2/X AM i?2 }M�H bi�i2 i?2 }M�HBb2K2i?Q/ 7Q` i?2 +?Qb2M HQ;B+ Bb +�HH2/X A rBHH MQr /2b+`B#2 i?2b2 K2i?Q/b b2T�`�i2HvX
R9
A1@hShG6Q` i?2 AM}MBi2 1ti2MbBQM b2K�MiB+b- r2 �bbmK2 i?2 }M�H bi�i2 Bb `2T2�i2/ 7Q`2p2`X "2+�mb2Q7 i?Bb r2 +�M bi`BT �HH i2KTQ`�H [m�MiB}2`b BX2X- ⇤',⌃', l',�' �HH #2+QK2 'X q?BH2 KvBKTH2K2Mi�iBQM ?�b ǵbi`QM; M2tiǶ �+ �M/ ǵr2�F M2tiǶ �� BM Bi- BM A1@hShG i?2v �`2 #Qi?i`2�i2/ �b B7 i?2v r2`2 Dmbi i?2 `2;mH�` ǵM2tiǶ �X6Q` i?2 7`22x2 QT2`�iQ`- i?2 iBK2 Q7 i?2 }M�H bi�i2 Bb bm#biBimi2/ BMiQ i?2 Q++m``2M+2b Q7 Bib#QmM/ p�`B�#H2X �Mv `2K�BMBM; iBK2 +QKT�`BbQMb rBi? 7`22 p�`B�#H2b �`2 i`2�i2/ �b � iB;?iHv#QmM/ 7`22x2 [m�MiB}2` 2X;X- x + 2 < 5 Bb i`2�i2/ HBF2 x.(x + 2 < 5)X hBK2 +QKT�`BbQMbrBi?Qmi 7`22 p�`B�#H2b �`2 bBKTHv 2p�Hm�i2/Xh?2 `2K�BMBM; QT2`�iQ`b �`2 bBKTHB}2/ #v apply �b MQ`K�HX1t�KTH2 9XkXR UA1@hShG p2`B}+�iBQM 2t�KTH2VXh?2 7QHHQrBM; b?Qrb �M 2t�KTH2 Q7 p2`B7vBM; i?2 T`QT2`iv ⌃y.((y x + 3) ^ a) �;�BMbi i?2i`�+2 ((2, ✏), (5, (a))) �M/ i?2M ((2, ✏))X Ai i�F2b TH�+2 BM i?2 a+�H� BMi2`T`2i2`,
a: settlr.Atom[List[String]] = a
scala> val p = 'x ~ F('y~ (('y <= 'x + 3) /\ a))
p: settlr.Freeze = 'x. ◇ 'y. (('y <= 'x + 3) ∧ a)
scala> val trace = List( (2,List()), (5,List("a")))
trace: List[(Int, List[String])] = List((0,List()), (1,List(a)))
scala> verifyIETrace(p,trace)
Verifying: 'x. ◇ 'y. (('y <= 'x + 3) ∧ a)
Trace: [(2, List()),(5, List(a))]
Forumula Progressed: ◇ 'y. (('y <= 5) ∧ a)
Remaining Trace: [(5, List(a))]
finalise: ◇ 'y. (('y <= 5) ∧ a) @ time: 5, state: List(a)
Finalise result: ⊤
Satisfies trace.
res0: settlr.Formula = ⊤
scala> val trace2 = List( (2,List()))
trace2: List[(Int, List[Nothing])] = List((2,List()))
scala> verifyIETrace(p,trace2)
Verifying: 'x. ◇ 'y. (('y <= 'x + 3) ∧ a)
Trace: [(2, List())]
finalise: 'x. ◇ 'y. (('y <= 'x + 3) ∧ a) @ time: 2, state: List()
Finalise result: ⊥
Violates trace.
res1: settlr.Formula = ⊥
_o@hShGJv BKTH2K2Mi�iBQM Bb �M �TT`QtBK�iBQM Q7 _o@hShG- bBM+2 A /Q MQi +?2+F i?2 p�Hm2b Q7�HH bm{t2b Q7 i?2 i`�+2- BMbi2�/ A mb2 bBKTHB}+�iBQM K2i?Q/bX A7 iBK2 T2`KBii2/ A rQmH/ mb2� i�#H2�m T`Q+2/m`2 BM i?2 }M�H bi�i2 �b A /2b+`B#2/ BM .2+BbBQM S`Q+2/m`2 jXRXjX Jv �T@T`QtBK�iBQM b?QmH/ MQi ;Bp2 �Mv BM+Q``2+i `2bmHib Q7 ǵb�iBb}2/fpBQH�i2/Ƕ- #mi K�v BM+Q``2+iHv;Bp2 ǵT`2bmK�#Hv b�iBb}2/Ƕ r?2M Bi b?QmH/ ;Bp2 ǵb�iBb}2/Ƕ �M/ ǵT`2bmK�#Hv pBQH�i2/Ƕ r?2MBi b?QmH/ ;Bp2 ǵpBQH�i2/ǶX Jv BKTH2K2Mi�iBQM +�MMQi /2i2+i i?2b2 BM+Q``2+i +�b2bX h?Bb Bb`2;`2ii�#H2 �M/ rQmH/ #2 � T`BQ`Biv B7 i?2`2 r2`2 KQ`2 iBK2X
R8
AM i?2 _o@hShG finalise K2i?Q/- i?2 7Q`KmH� Bb `2r`Bii2M U#mi MQi T`Q;`2bb2/ rBi? unXV#27Q`2 }M�HBbBM;X h?Bb K2�Mb i?2 `2bmHiBM; 7Q`KmH� Bb K�/2 mT Q7 +QMDmM+iBQMb �M/fQ`/BbDmM+iBQMb Q7 ǵM2tiǶ bm#7Q`KmH�2X h?Bb `2r`Bi2 Bb M2+2bb�`v #2+�mb2 i?2 b2K�MiB+b �`2 MQi/2}M2/ 7Q` ⇤,⌃,l BM i?2 }M�H bi�i2 U/2HB#2`�i2HvVXh?2M �HH Q++m``2M+2b Q7 ǵr2�F M2tiǶ U��V �`2 `2TH�+2/ rBi? ǵT`2bmK�#Hv i`m2Ƕ U>P V �M/Q++m``2M+2b Q7 ǵbi`QM; M2tiǶ U�+V �`2 `2TH�+2/ rBi? ǵT`2bmK�#Hv 7�Hb2Ƕ U?P VX LQr i?2+QMDmM+iBQMb �M/ /BbDmM+iBQMb +�M #2 2p�Hm�i2/- r?2`2 ǵ�M/Ƕ Bb i?2 KBMBKmK Q7 i?2 +QMDmM+ib�M/ ǵQ`Ƕ Bb i?2 K�tBKmK Q7 i?2 /BbDmM+ib- rBi? i?2 Q`/2`BM; > > >P > ?P > ?X1t�KTH2 9XkXk U_o@hShG p2`B}+�iBQM 2t�KTH2bVXh?2 7QHHQrBM; 2t�KTH2 Bb � +QMiBMm�iBQM Q7 i?2 1t�KTH2 9XkXR 2t�KTH2 b?QrBM; ?Qr p2`B}@+�iBQM `2bmHib BM _o@hShG /Bz2`X
Verifying: 'x. ◇ 'y. (('y <= 'x + 3) ∧ a)
Trace: [(2, List()),(5, List(a))]
Forumula Progressed: ◇ 'y. (('y <= 5) ∧ a)
Remaining Trace: [(5, List(a))]
finalise: ◇ 'y. (('y <= 5) ∧ a) @ time: 5, state: List(a)
Result is {{}} by simplification
Finalise result: ⊤
Satisfies trace.
res2: settlr.Formula = ⊤
scala> verifyRVTrace(p,trace2)
Verifying: 'x. ◇ 'y. (('y <= 'x + 3) ∧ a)
Trace: [(2, List())]
finalise: 'x. ◇ 'y. (('y <= 'x + 3) ∧ a) @ time: 2, state: List()
Finalise result: ⊥ᴾ
Presumably violates trace.
res3: settlr.Formula = ⊥ᴾ
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8 6mim`2 qQ`FaBM+2 Kv BKTH2K2Mi�iBQM Q7 i?2 _o@hShG `mMiBK2 p2`B}+�iBQM T`Q+2/m`2 Bb BM+QKTH2i2-Kv }`bi bi2T BM �Mv 7mim`2 rQ`F rQmH/ #2 iQ +Q``2+iHv �M/ +QKTH2i2Hv BKTH2K2Mi .2+BbBQMS`Q+2/m`2 jXRXjX A rBHH MQr /2b+`B#2 bQK2 Qi?2` BMi2`2biBM; BKT`Qp2K2MibX
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Research School of Computer Science Form updated Jun 2018
INDEPENDENT STUDY CONTRACT PROJECTS
Note: Enrolment is subject to approval by the course convenor SECTION A (Students and Supervisors)
UniID: u6060697
SURNAME: Cox FIRST NAMES: Alexander Jared
PROJECT SUPERVISOR (may be external): Peter Baumgartner
FORMAL SUPERVISOR (if different, must be an RSSCS academic): ___________________________________
COURSE CODE, TITLE AND UNITS: COMP4560 - Advanced Computing Project - 12 Units
COMMENCING SEMESTER/YEAR Semester 2 - 2019
PROJECT TITLE:
Runtime Monitoring for Situational Awareness using Linear Temporal Logic
LEARNING OBJECTIVES:
1. apply implementation skills and knowledge of temporal logics and model checking techniques.
2. deepen knowledge of temporal logics and domain specific languages through undertaking the project.
3. learn the Scala programming language and techniques for embedding domain specific languages.
4. learn runtime verification and monitoring skills and algorithms, and apply them to the project.
5. learn project management and communication skills, and apply these to the project.
PROJECT DESCRIPTION: The aim of this project is to implement a runtime monitor for checking properties of a situational awareness system. The properties will be expressed in a finite-trace variant of Linear Temporal Logic (LTL). The implementation will be written in Scala. First, I will implement a shallow-embedding LTL as a Domain Specific Language. I will then write a runtime monitor which checks a trace (a sequence of states) against specified LTL properties, e.g. "The tomatoes are eventually delivered", or "The tomatoes are in transit for less than 5 hours". I will need to implement a variant of LTL for finite traces (the usual LTL semantics are for infinite traces). I will start with a simple variant and then may implement more complicated variant, such as 3-valued semantics (e.g. true/false/inconclusive). If time permits, I will extend the logic to a probabilistic variant. The aim will be to get my implementation to work with an existing system for situational awareness developed at Data61 called "fusemate", but this is not essential.
Research School of Computer Science Form updated Jun 2018
ASSESSMENT (as per the project course’s rules web page, with any differences noted below).
Assessed project components: % of mark Due date Evaluated by:
Report: style: Research Report (e.g. research report, software description...,)
45 (min 45,
default 60)
(examiner )
Artefact: kind: Implementation in Scala (e.g. software, user interface, robot...,)
45 (max 45,
default 30)
Peter Baumgartner
(supervisor)
Presentation:
10 (course convenor)
MEETING DATES (IF KNOWN): STUDENT DECLARATION: I agree to fulfil the above defined contract:
………………………………………………….. ……………………….. Signature Date SECTION B (Supervisor): I am willing to supervise and support this project. I have checked the student's academic record and believe this student can complete the project. I nominate the following examiner, and have obtained their consent to review the report (via signature below or attached email) ………………..………………..……………….. ……………………….. Signature Date Examiner: Name: ………………………………………… Signature …………………… (Nominated examiners may be subject to change on request by the supervisor or course convenor) REQUIRED DEPARTMENT RESOURCES: SECTION C (Course convenor approval) ………………………………………………….. ……………………….. Signature Date
14/6/2019
Patrik Haslum