monotone deterministic rl-automata don’t need auxiliary symbols supported by grants from dfg and...

Monotone Deterministic RL-Automata Don’t Need Auxiliary Symbols

Supported by grants from DFG and the Grant Agency of the Czech Republic

Tomasz Jurdziński

University of WrocławPoland

Martin Plátek

Charles University PragueCzech Republic

Friedrich OttoFrantišek MrázUniversität KasselGermany

7.7.2005 DLT 2005 - Palermo

Overview

motivation restarting automata

definition, subclasses right-, left- monotonicity

previous results about (nondeterministic and deterministic) restarting automata with left/right monotonicity

monotone deterministic two-way restarting automata do not need auxiliary symbols characterization by deterministic transducers and DCFL

conclusions

7.7.2005 DLT 2005 - Palermo

Restarting automata - Motivation

model for the analysis by reduction a stepwise simplification of an input sentence while preserving

its (non)correctness until a simple sentence is got or an error is found

each simplification (reduction) means a rewriting a limited part of the sentence by a shorter one

rich taxonomy of constrains for various models of analyzers and parsers

in particular constrains on word-order – applications for free word-order languages (many Slavonic languages)

7.7.2005 DLT 2005 - Palermo

Motivation

(right) monotonicty during a computation, places of rewriting do not increase their

distance from the right end of the word characterizations of CFL (by nondeterministic) and DCFL (by

deterministic) one-way restarting automata left monotonicty

during a computation, places of rewriting do not increase their distance from the left end of the word

nondeterministic left-monotone restarting automata – similar power as right-monotone ones

deterministic left-monotone restarting automata – different

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b a b b a a b a a b b a b

Restarting automaton

finite set of states

c $

q control unit

scanning window

input word

|

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Operations

MVR - move right

MVL - move left

rewrite by v

RESTART

ACCEPT

v

the initial state q0

7.7.2005 DLT 2005 - Palermo

Instructions

(1) (p,u) (q,MVR )

(3) (p,u) RESTART

(5) (p,u) (q,v )

k

input alphabet

working alphabetTwo alphabets

c ...| u ... $

p

(2) (p,u) (q,MVL )

(4) (p,u) ACCEPT

!!! |v |<|u | !!!

ACCEPT

Accepting configuration

c| $

Initial configurationc| $input word

q0 the initial state

input symbols only

Restarting configuration

$current word

q0 the initial state

working symbols allowed

c|

If there is no applicable instruction, the automaton rejects

7.7.2005 DLT 2005 - Palermo

• cycle – part of a computation between two restarting configurations contains exactly one

rewriting• tail – part of a computation between a restarting

configuration and a halting configuration

Cycles

W1

W2

W3

Wn-1

c|

c|

c|

c|

$

$

$

$

Wnc| $

cycles

tail

7.7.2005 DLT 2005 - Palermo

Subclasses of restarting automata

RLWW RRWW RWW

RLW RRW RW

RL RR R

no MVL

restart immediately after each rewrite

deleting only

no working symbols

back

7.7.2005 DLT 2005 - Palermo

(right-)mon-RLWW-automaton – all its computations starting on an input word are (right) monotone

monotonicity (right-monotonicity)

right monotone computation – the places of rewriting in cycles do not increase their distance to the right sentinel

c| $

c| $

c| $

c| $

c| $ tail not considered

initial configuration

c| $

7.7.2005 DLT 2005 - Palermo

left-monotonicity

left-monotone computation – the places of rewriting in cycles do not increase their distance from the left sentinel

left-mon-RLWW-automaton – all its computations starting on an input word are left-monotone

c| $

c| $

c| $

c| $

c| $ tail not considered

initial configuration

c| $

7.7.2005 DLT 2005 - Palermo

Known results for nondeterministic restarting automata

RLWW RRWW RWW

RLW RRW RW

RL RR R

CFL

nondeterministic right-mon-

RLWW RRWW RWW

RLW RRW RW

RL RR R

nondeterministic left-mon

CFL

X Y equivalence L (X) = L (Y)X Y proper inclusion L (X) L (Y)

7.7.2005 DLT 2005 - Palermo

Known results for deterministic restarting automata

L={ anbnc, anb2nd | n0 } L L (det-mon-RL) \ DCFL

L L (det-left-mon-RL) \ L (det-RRW)

deterministic (right-) mon-

RLWW RRWW RWW

RLW RRW RW

RL RR R

DCFL

deterministic left-mon-

RLWW RRWW RWW

RLW RRW RW

RL RR R

DCFLR

7.7.2005 DLT 2005 - Palermo

Mirror Symmetry

Lemma: For each X{ , W, WW }:

L (det-left-mon-RLX) = [L (det-mon-RLX)]R

deterministic right-mon-

RLWW RRWW RWW

RLW RRW RW

RL RR R

DCFL


RLWW RRWW RWW

RLW RRW RW

RL RR R

DCFLR

7.7.2005 DLT 2005 - Palermo

The main theorem

a) L (det-mon-RLWW) = L (det-mon-RLW) = L (det-mon-RL)

b) L (det-left-mon-RLWW) = L (det-left-mon-RLW) = L (det-left-mon-RL)

L (det-left-mon-RRWW)

=

7.7.2005 DLT 2005 - Palermo

q0

(q0 , caa) (q1, MVR )

W.l.o.g. an RRWW-automaton M scans whole tape before it restarts / accepts / rejects

Characterization of L (det-left-mon-RRWW)

a a b ba $bc | a a bb

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q1

(q1 , aaa) (q2, MVR )




q0

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q2

(q2 , aaa) (q3, MVR )




q0 q1

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q3

(q3 , aaa) (q4, MVR )




q0 q1 q2

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q4

(q4 , aab) (q5, MVR )




q0 q1 q2 q3

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q5

(q5 , abb) (q6, b )




q0 q1 q2 q3 q4

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c

q6

(q6 , bbb) (q7, MVR )



a a b ba $b | a b

q0 q1 q2 q3 q4 q5

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q7

(q7 , bb$) (q8, MVR )



q0 q1 q2 q3 q4 q5 q6

c a a b ba $b | a b

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q8

(q8 , b$) (q9, MVR )



q0 q1 q2 q3 q4 q5 q6

c a a b ba $b | a b

q7

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q9

(q9 , $) RESTART



q0 q1 q2 q3 q4 q5 q6

c a a b ba $b | a b

q7 q8

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q9

(q9 , $) RESTART


Trace

q0 q1 q2 q3 q4 q5 q6

c a a b ba $b | a b

q7 q8 q5 q5trace

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Computing the trace

q0 q1 q2 q3 q4 q5 q6

c a a b ba $b | a b

q5 q5

a a b ba ba a bb

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Computing the trace

q0 q1 q2 q3 q4 q5 q6 q5 q5

a a b ba ba a bb


q2 deterministic transducer GM

w =

GM (w) =

GM(L) = { GM(w) | w L }

Proposition:For each det-left-mon-RRWW-automaton M: GM(L(M)) DCFLR

7.7.2005 DLT 2005 - Palermo


Proposition:

For each det-left-mon-RRWW-automaton M: GM(L(M)) DCFLR.

Idea: knowing the trace, a deterministic RRWW-automaton could accept L(M) working from right to left

c| $

c| $

c| $

c| $

c| $

c| $

$

$

$

$

$

|c|c|c|c||

|c|c|c|c||

|c|c|c|c||

|c|c|c|c||

|c|c|c|c||

|c|c|c|c||

$

$

7.7.2005 DLT 2005 - Palermo

Known results for deterministic restarting automata


RLWW RRWW RWW

RLW RRW RW

RL RR R

DCFL


RLWW RRWW RWW

RLW RRW RW

RL RR R

DCFLR

7.7.2005 DLT 2005 - Palermo

Proposition: Let G be a deterministic transducer with input alphabet , and let L +. If G (L)

R DCFL, then there exists a det-left-mon-RL-automaton M that accepts the language L.

Idea: replace G by a deterministic transducer G ’

G ’ outputs one symbol for each letter read, and G ’ (L)

R DCFL it is possible to “compute” G ’ (w) from w “on-the-fly” by a two-

way finite state automaton A; let qi be the state of G ’ when it is

scanning w[i]; knowing qi the automaton A can compute

qi+1 - obvious

qi-1 - [Hopcroft, Ullman 1967]


7.7.2005 DLT 2005 - Palermo

[G’ (L)]R DCFL, i.e. [G ’ (L)]R can be accepted by a det-mon-R-automaton M1

a det-left-mon-RL-automaton M2 can accept G ’ (L) by

simulating M1

first scans x G ’ (L) from left to right and checks whether x could be output of G ’

then simulates M1 moving from right to left

a det-left-mon-RL-automaton M3 can accept L by simulating

M2 using A to compute G’ (L)


7.7.2005 DLT 2005 - Palermo

Conclusions

for deterministic two-way restarting automata which are right or left monotone we do not need auxiliary symbols, we do not need to rewrite symbols, deleting is enough


RLWW RRWW RWW

RLW RRW RW

RL RR R

DCFL


RLWW RRWW RWW

RLW RRW RW

RL RR R

DCFLR

7.7.2005 DLT 2005 - Palermo

LF

Conclusions

For modelling the analysis by reduction in the case of left/right monotone rewritings only we need not auxiliary symbols - it is possible to reduce a given sentence to a ‘simple’ form within the same language

the main result carries over to det-right-left-mononotone two-way restarting automata, but it was shown by a pumping techniques

LF

L ( R ) L ( R )

monotone deterministic rl-automata don’t need auxiliary symbols supported by grants from dfg and...

Documents

restarting configurations

places of rewriting

right monotonemtail

right end

left end

u restart5 p

rewriting tail

right sentinelright