Deciding the value 1 problem for probabilistic leaktight automata

Transcription

1 Deciding the vlue 1 prolem for proilistic lektight utomt Nthnël Fijlkow, joint work with Hugo Gimert nd Youssouf Oulhdj LIAFA, Université Pris 7, Frnce, University of Wrsw, Polnd. LICS, Durovnik, Croti June 26th 2012

2 Proilistic utomt (Rin, 1963) 1, 3,.4, P A : A [0, 1]

3 The vlue 1 prolem 2 The vlue 1 prolem is, given proilistic utomton A: re there words ccepted y A with ritrrily high proility?

4 The vlue 1 prolem 2 The vlue 1 prolem is, given proilistic utomton A: re there words ccepted y A with ritrrily high proility? Define vl(a) = sup w P A (w). An equivlent formultion of this prolem is: vl(a)? = 1.

5 The vlue 1 prolem 2 The vlue 1 prolem is, given proilistic utomton A: re there words ccepted y A with ritrrily high proility? Define vl(a) = sup w P A (w). An equivlent formultion of this prolem is: vl(a)? = 1. Theorem (Gimert, Oulhdj, 2010) The vlue 1 prolem is undecidle.

6 Our ojective 3 Decide the vlue 1 prolem for suclss of proilistic utomt, y lgeric nd non-numericl mens.

7 Our ojective 3 Decide the vlue 1 prolem for suclss of proilistic utomt, y lgeric nd non-numericl mens. lgeric: focus on the utomton structure, non-numericl: strct wy the vlues.

8 Our ojective 3 Decide the vlue 1 prolem for suclss of proilistic utomt, y lgeric nd non-numericl mens. lgeric: focus on the utomton structure, non-numericl: strct wy the vlues. Hence we consider non-deterministic utomt: 1 2

9 Weighted utomt using lger (Schützenerger) F, = = I F = 1 if nd only if P A () > 0

10 The stiliztion opertion = ( ) In, the stte 1 is trnsient nd the stte 2 is recurrent.

11 The stiliztion opertion = ( ) = ( ) In, the stte 1 is trnsient nd the stte 2 is recurrent.

12 The stiliztion opertion = ( ) = ( ) In, the stte 1 is trnsient nd the stte 2 is recurrent. M = lim n M n

13 A sturtion lgorithm 6 Compute monoid inside the finite monoid M Q Q ({0, 1},+, ). Compute for A Close under product nd stiliztion.

14 A sturtion lgorithm 6 Compute monoid inside the finite monoid M Q Q ({0, 1},+, ). Compute for A Close under product nd stiliztion. If there exists mtrix M such tht t Q, M(s 0, t) = 1 t F then A hs vlue 1, otherwise A does not hve vlue 1.

15 An exmple 0 1 F, 7

16 An exmple 0 1 F, F

17 An exmple 0 1 F, F 0 1 F

18 An exmple 0 1 F, F 0 1 F

19 An exmple 0 1 F, F 0 1 F 0 1 F

20 An exmple 0 1 F, F ( ) 0 1 F

21 An exmple 0 1 F, F 0 1 F 0 1 F 0 1 F ( ) 0 1 F

22 Correct, ut not complete 8 Theorem (Correctness) If the lgorithm nswers A hs vlue 1 then A hs vlue 1.

23 Correct, ut not complete 8 Theorem (Correctness) If the lgorithm nswers A hs vlue 1 then A hs vlue 1. But the vlue 1 prolem is undecidle, so the converse cnnot hold!

24 Completeness in the sence of leks 9 Definition An utomton A is lektight if it hs no lek. Theorem (Completeness) If A is lektight nd hs vlue 1, then the lgorithm nswers A hs vlue 1. The proof relies on Simon s fctoriztion forest theorem.

25 A lek 10, 3 1 2

26 A lek 10,

27 A lek 10, ε There is lek from 1 to 3.

28 Conclusion nd perspectives 11 We defined suclss of proilistic utomt which susumes ll suclsses of proilistic utomt whose vlue 1 prolem is known to e decidle,

29 Conclusion nd perspectives 11 We defined suclss of proilistic utomt which susumes ll suclsses of proilistic utomt whose vlue 1 prolem is known to e decidle, We defined n lgeric lgorithm for the vlue 1 prolem nd proved its completeness for the clss of lektight utomt.

30 Conclusion nd perspectives 11 We defined suclss of proilistic utomt which susumes ll suclsses of proilistic utomt whose vlue 1 prolem is known to e decidle, We defined n lgeric lgorithm for the vlue 1 prolem nd proved its completeness for the clss of lektight utomt. Wht does this lgorithm ctully compute?

31 Conclusion nd perspectives 11 We defined suclss of proilistic utomt which susumes ll suclsses of proilistic utomt whose vlue 1 prolem is known to e decidle, We defined n lgeric lgorithm for the vlue 1 prolem nd proved its completeness for the clss of lektight utomt. Wht does this lgorithm ctully compute? Cn we use similr lgorithms for other semirings?