The Vlue 1 Prolem for Proilistic Automt Bruxelles Nthnël Fijlkow LIAFA, Université Denis Diderot - Pris 7, Frnce Institute of Informtics, Wrsw University, Polnd nth@lif.univ-pris-diderot.fr June 20th, 2014
Proilistic utomt (Rin, 1963), 1 2, 0.4, 0.6 1 3 P A : A [0, 1] P A (w) is the proility tht run for w ends up in F
The vlue 1 prolem 2 This tlk is out the vlue 1 prolem: INPUT:Aproilistic utomton OUTPUT: for ll ε > 0, there exists w A,P A (w) 1 ε. In other words, define vl(a) = sup w A P A (w), is vl(a) = 1?
The vlue 1 prolem 2 This tlk is out the vlue 1 prolem: INPUT:Aproilistic utomton OUTPUT: for ll ε > 0, there exists w A,P A (w) 1 ε. In other words, define vl(a) = sup w A P A (w), is vl(a) = 1? It is undecidle (Gimert nd Oulhdj, 2010). But to wht extent?
Ojective 3 Construct n lgorithm to decide the vlue 1 prolem, which is often correct.
Ojective 3 Construct n lgorithm to decide the vlue 1 prolem, which is often correct. Quntify how often.
Ojective 3 Construct n lgorithm to decide the vlue 1 prolem, which is often correct. Quntify how often. Argue tht you cnnot do more often thn tht.
Outline 3 1 Theory A first ttempt: get rid of numericl vlues A second ttempt: the Mrkov Monoid Algorithm On the optimlity of the Mrkov Monoid Algorithm 2 Prctice: ACMÉ
Outline 3 1 Theory A first ttempt: get rid of numericl vlues A second ttempt: the Mrkov Monoid Algorithm On the optimlity of the Mrkov Monoid Algorithm 2 Prctice: ACMÉ
Outline 3 1 Theory A first ttempt: get rid of numericl vlues A second ttempt: the Mrkov Monoid Algorithm On the optimlity of the Mrkov Monoid Algorithm 2 Prctice: ACMÉ
At first thought 4 Does the undecidility come from the numericl vlues?
At first thought 4 Does the undecidility come from the numericl vlues? Consider numerless proilistic utomt: 1 2 Two decision prolems: for ll, vl(a[ ]) = 1, there exists, such tht vl(a[ ]) = 1.
No future (in this direction) 5 Theorem (F., Horn, Gimert nd Oulhdj) There is no lgorithm such tht: On input A ( non-deterministic utomton), if for ll, vl(a[ ]) = 1 then YES, if for ll, vl(a[ ]) < 1 then NO, nything in the other cses.
Outline 5 1 Theory A first ttempt: get rid of numericl vlues A second ttempt: the Mrkov Monoid Algorithm On the optimlity of the Mrkov Monoid Algorithm 2 Prctice: ACMÉ
An exmple 6, 0 1 F,
An exmple 6, 0 1 F, 0 1 F
An exmple 6, 0 1 F, 0 1 F 0 1 F
An exmple 6, 0 1 F, 0 1 F 0 1 F
An exmple 6, 0 1 F, 0 1 F 0 1 F 0 1 F
An exmple 6, 0 1 F, 0 1 F ( ) 0 1 F
An exmple 6, 0 1 F, 0 1 F 0 1 F 0 1 F 0 1 F ( ) 0 1 F
Stiliztion monoids (Colcomet) 7 This is n lgeric structure with two opertions: inry composition stiliztion, denoted.
Boolen mtrices representtions 8, 0 1 F, = 1 0 0 0 0 1 1 0 0 0 1 0 0 0 0 1 = 1 0 0 0 1 0 0 0 0 1 1 0 0 0 0 1 I u F = 1 if nd only if P A (u) > 0
Defining stiliztion 9 1 2 = ( 1 1 0 1 In, the stte 1 is trnsient nd the stte 2 is recurrent. )
Defining stiliztion 9 1 2 = ( 1 1 0 1 ) = ( 0 1 0 1 In, the stte 1 is trnsient nd the stte 2 is recurrent. )
Defining stiliztion 9 1 2 = ( 1 1 0 1 ) = ( 0 1 0 1 In, the stte 1 is trnsient nd the stte 2 is recurrent. ) { M 1 if M(s, t) = 1 nd t recurrent in M, (s, t) = 0 otherwise.
The Mrkov Monoid Algorithm 10 Compute monoid inside the finite monoid M Q Q ({0, 1},, ). Compute for A: { 1 if PA (s t) > 0, (s, t) = 0 otherwise. Close under product nd stiliztion.
The Mrkov Monoid Algorithm 10 Compute monoid inside the finite monoid M Q Q ({0, 1},, ). Compute for A: { 1 if PA (s t) > 0, (s, t) = 0 otherwise. 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.
Correctness 11 Theorem If there exists mtrix M such tht then A hs vlue 1. t Q, M(s 0, t) = 1 t F
Correctness 11 Theorem If there exists mtrix M such tht then A hs vlue 1. t Q, M(s 0, t) = 1 t F But the vlue 1 prolem is undecidle, so...
No completeness,, 12 L 2 R 2, 1 x 0, x L 1 R 1, x, 1 2, 1 2 Left nd right prts re symmetric, so for ll M:, 1 x M(0, L 2 ) = 1 M(0, R 2 ) = 1. Yet: it hs vlue 1 if nd only if x > 1 2.
A lek, 13 2 1 3
A lek, 13 2 2 1 1 3 3
A lek, 13 2 2 1 ε 1 3 3 There is lek from 1 to 2.
A lek, 13 2 2 1 1 3 3 There is lek from 1 to 2. Definition An utomton A is lektight if it hs no lek.
Lektight utomt 14 Theorem (F.,Gimert nd Oulhdj 2012) The lgorithm is complete for lektight utomt. Hence, the vlue 1 prolem is decidle for lektight utomt. The proof relies on Simon s fctoriztion forest theorem.
Other decidle suclsses: in 2012 15 simple structurlly simple lektight -cyclic hierrchicl deterministic
Other decidle suclsses: tody 16 (F.,Gimert,Kelmendi nd Oulhdj 2013) lektight
Corollry 17 So fr, the Mrkov Monoid Algorithm is the most correct lgorithm known to solve the vlue 1 prolem.
Corollry 17 So fr, the Mrkov Monoid Algorithm is the most correct lgorithm known to solve the vlue 1 prolem. But for how long?
Outline 17 1 Theory A first ttempt: get rid of numericl vlues A second ttempt: the Mrkov Monoid Algorithm On the optimlity of the Mrkov Monoid Algorithm 2 Prctice: ACMÉ
Wht it misses: different convergence speeds,, 18 L 2 R 2, 1 4 0, 3 4 L 1 R 1, 1 2, 1 2, 3 4 lim n P A (( n ) f(n)) = 1 if nd only if lim n f(n), 1 4 ( ) 3 n =, 4
Wht it misses: different convergence speeds,, 18 L 2 R 2, 1 4 0, 3 4 L 1 R 1, 1 2, 1 2, 3 4 lim n P A (( n ) f(n)) = 1 if nd only if lim n f(n), 1 4 ( ) 3 n =, 4 so f(n) = 2 n works ut f(n) = n does not.
A chrcteriztion 19 Ã is the spce of prostochstic words. A = Ã [0] Ã [1] Ã [2] Ã. Lemm The following re equivlent: The vlue 1 prolem over finite words, The emptiness prolem over prostochstic words.
A chrcteriztion 19 à is the spce of prostochstic words. A = à [0] à [1] à [2] Ã. Lemm The following re equivlent: Theorem The vlue 1 prolem over finite words, The emptiness prolem over prostochstic words. 1 The Mrkov Monoid Algorithm nswers YES if nd only if there exists x à [1] ccepted y A, 2 The following prolem is undecidle: determine whether there exists x à [2] ccepted y A.
Prostochstic words 20 Definition (u n ) n N converges if for every A, the limit lim n P A (u n ) exists.
Prostochstic words 20 Definition (u n ) n N converges if for every A, the limit lim n P A (u n ) exists. Definition Two (converging) sequences (u n ) n N nd (v n ) n N re equivlent if for every A, lim n P A (u n ) > 0 lim n P A (v n ) > 0.
Prostochstic words 20 Definition (u n ) n N converges if for every A, the limit lim n P A (u n ) exists. Definition Two (converging) sequences (u n ) n N nd (v n ) n N re equivlent if for every A, lim n P A (u n ) > 0 lim n P A (v n ) > 0. Definition A prostochstic word is n equivlence clss of converging sequences.
The ω opertors 21 Definition Let u e converging sequence. u ω 1 is the converging sequence (u n! n ) n N.
The ω opertors 21 Definition Let u e converging sequence. u ω 1 is the converging sequence (u n! n ) n N. Definition Let u e converging sequence. u ω k is the converging sequence (u (n!)k n ) n N.
The ω opertors 21 Definition Let u e converging sequence. u ω 1 is the converging sequence (u n! n ) n N. Definition Let u e converging sequence. u ω k is the converging sequence (u (n!)k n ) n N. Exmple The prostochstic words ( ω 1) ω 1 nd ( ω 1) ω 2 re not equl.
An equivlent chrcteriztion 22 Theorem The Mrkov Monoid Algorithm nswers YES if nd only if there exists regulr sequence (u n ) n N of finite words such tht lim n P A (u n ) = 1. The regulr sequences re descried y the following grmmr: u = u u (u n n) n N.
Corollry 23 In some sense, the Mrkov Monoid Algorithm is the most correct lgorithm to solve the vlue 1 prolem.
Outline 23 1 Theory A first ttempt: get rid of numericl vlues A second ttempt: the Mrkov Monoid Algorithm On the optimlity of the Mrkov Monoid Algorithm 2 Prctice: ACMÉ
ACMÉ 24 The tool ACME (Automt with Counters, Monoids nd Equivlence) hs een written in OCml y Nthnël Fijlkow nd Denis Kupererg. ********************************** Sttistics on the Mrkov Monoid Algorithm: ********************************** Automt tht re lektight nd do not hve vlue 1: 540 Automt tht re lektight nd hve vlue 1: 133 Automt tht re not lektight nd my hve vlue 1: 17 Automt tht re not lektight nd hve vlue 1: 310
The end. 25 Thnk you for your ttention!