Decision Questions for Probabilistic Automata on Small Alphabets
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.
Probabilistic finite automata
unary alphabet
emptiness problem
bounded ambiguity
Theory of computation~Formal languages and automata theory
Theory of computation~Computability
Theory of computation~Probabilistic computation
15:1-15:17
Regular Paper
https://arxiv.org/abs/2105.10293
Paul C.
Bell
Paul C. Bell
Department of Computer Science, Liverpool John Moores University, UK
https://orcid.org/0000-0003-2620-635X
Pavel
Semukhin
Pavel Semukhin
Department of Computer Science, University of Oxford, UK
https://orcid.org/0000-0002-7547-6391
10.4230/LIPIcs.MFCS.2021.15
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Paul C. Bell and Pavel Semukhin
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