Decision Questions for Probabilistic Automata on Small Alphabets

Authors Paul C. Bell , Pavel Semukhin



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Paul C. Bell
  • Department of Computer Science, Liverpool John Moores University, UK
Pavel Semukhin
  • Department of Computer Science, University of Oxford, UK

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Paul C. Bell and Pavel Semukhin. Decision Questions for Probabilistic Automata on Small Alphabets. In 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 202, pp. 15:1-15:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
https://doi.org/10.4230/LIPIcs.MFCS.2021.15

Abstract

We study the emptiness and λ-reachability problems for unary and binary Probabilistic Finite Automata (PFA) and characterise the complexity of these problems in terms of the degree of ambiguity of the automaton and the size of its alphabet. Our main result is that emptiness and λ-reachability are solvable in EXPTIME for polynomially ambiguous unary PFA and if, in addition, the transition matrix is over {0, 1}, we show they are in NP. In contrast to the Skolem-hardness of the λ-reachability and emptiness problems for exponentially ambiguous unary PFA, we show that these problems are NP-hard even for finitely ambiguous unary PFA. For binary polynomially ambiguous PFA with commuting transition matrices, we prove NP-hardness of the λ-reachability (dimension 9), nonstrict emptiness (dimension 37) and strict emptiness (dimension 40) problems.

Subject Classification

ACM Subject Classification
  • Theory of computation → Formal languages and automata theory
  • Theory of computation → Computability
  • Theory of computation → Probabilistic computation
Keywords
  • Probabilistic finite automata
  • unary alphabet
  • emptiness problem
  • bounded ambiguity

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