Probabilistic Automata on Finite Words: Decidable and Undecidable Problems

Gimbert, Hugo; Oualhadj, Youssouf

doi:10.1007/978-3-642-14162-1_44

Cited by 73 publications

(113 citation statements)

References 13 publications

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“…(Actually, since ♯-acylic automata are already a subclass of simple automata, we do not consider them.) This implies that our decidability result extends the decidability results from [GO10,CT12]. 5.1.…”

Section: Examples and Subclasses Of Leaktight Automatasupporting

confidence: 82%

“…Our proof techniques totally depart from the ones used in [CSV11,CT12,GO10]. We make use of algebraic techniques and in particular Simon's factorization forest theorem, which was used successfully to prove the decidability of the boundedness problem for distance automata [Sim94], and extended models as desert automata and Bautomata [Kir05,Col09] Outline.…”

Section: Probabilistic Leaktight Automatamentioning

confidence: 99%

“…As a consequence, it is crucial to understand which decision problems are algorithmically tractable for probabilistic automata. From a language-theoretic perspective, several algorithmic properties of probabilistic automata are known: while language emptiness is undecidable [Paz71,Ber74,GO10], functional equivalence is decidable [Sch61,Tze92] as well as other properties [CMRR08].…”

Section: Introductionmentioning

confidence: 99%

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Deciding the Value 1 Problem for Probabilistic Leaktight Automata

Fijalkow

Gimbert

Oualhadj

2012

2012 27th Annual IEEE Symposium on Logic in Computer Science

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Abstract. The value 1 problem is a decision problem for probabilistic automata over finite words: given a probabilistic automaton, are there words accepted with probability arbitrarily close to 1? This problem was proved undecidable recently; to overcome this, several classes of probabilistic automata of different nature were proposed, for which the value 1 problem has been shown decidable. In this paper, we introduce yet another class of probabilistic automata, called leaktight automata, which strictly subsumes all classes of probabilistic automata whose value 1 problem is known to be decidable.We prove that for leaktight automata, the value 1 problem is decidable (in fact, PSPACEcomplete) by constructing a saturation algorithm based on the computation of a monoid abstracting the behaviours of the automaton. We rely on algebraic techniques developed by Simon to prove that this abstraction is complete. Furthermore, we adapt this saturation algorithm to decide whether an automaton is leaktight.Finally, we show a reduction allowing to extend our decidability results from finite words to infinite ones, implying that the value 1 problem for probabilistic leaktight parity automata is decidable. ACM CCS: [Theory of computation]:Models of computation-Probabilistic computation.

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Section: Examples and Subclasses Of Leaktight Automatasupporting

confidence: 82%

Section: Probabilistic Leaktight Automatamentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Deciding the Value 1 Problem for Probabilistic Leaktight Automata

Fijalkow

Gimbert

Oualhadj

2012

2012 27th Annual IEEE Symposium on Logic in Computer Science

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“…For example the emptiness, the threshold isolation and the value 1 problems are undecidable [11,2,8].…”

Section: Introductionmentioning

confidence: 99%

Two Recursively Inseparable Problems for Probabilistic Automata

Fijalkow

Gimbert

Horn

et al. 2014

Mathematical Foundations of Computer Science 2014

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Abstract. This paper introduces and investigates decision problems for numberless probabilistic automata, i.e. probabilistic automata where the support of each probabilistic transitions is specified, but the exact values of the probabilities are not. A numberless probabilistic automaton can be instantiated into a probabilistic automaton by specifying the exact values of the non-zero probabilistic transitions. We show that the two following properties of numberless probabilistic automata are recursively inseparable:• all instances of the numberless automaton have value 1, • no instance of the numberless automaton has value 1.

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“…Alternatively, they may make a probabilistic decision depending on the input letter, like (reactive) probabilistic automata [20]. The latter go back to Rabin [21] and are an object of ongoing research considering decision problems such as emptiness, language equivalence [24,28,9,10,18], and the value 1 problem [17].…”

Section: Introductionmentioning

confidence: 99%

A Probabilistic Kleene Theorem

Bollig

Gastin

Monmege

et al. 2012

Automated Technology for Verification and Analysis

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Abstract. We provide a Kleene Theorem for (Rabin) probabilistic automata over finite words. Probabilistic automata generalize deterministic finite automata and assign to a word an acceptance probability. We provide probabilistic expressions with probabilistic choice, guarded choice, concatenation, and a star operator. We prove that probabilistic expressions and probabilistic automata are expressively equivalent. Our result actually extends to two-way probabilistic automata with pebbles and corresponding expressions.

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Probabilistic Automata on Finite Words: Decidable and Undecidable Problems

Cited by 73 publications

References 13 publications

Deciding the Value 1 Problem for Probabilistic Leaktight Automata

Deciding the Value 1 Problem for Probabilistic Leaktight Automata

Two Recursively Inseparable Problems for Probabilistic Automata

A Probabilistic Kleene Theorem

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