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**Published in:** LIPIcs, Volume 202, 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)

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.

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)

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@InProceedings{bell_et_al:LIPIcs.MFCS.2021.15, author = {Bell, Paul C. and Semukhin, Pavel}, title = {{Decision Questions for Probabilistic Automata on Small Alphabets}}, booktitle = {46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)}, pages = {15:1--15:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-201-3}, ISSN = {1868-8969}, year = {2021}, volume = {202}, editor = {Bonchi, Filippo and Puglisi, Simon J.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2021.15}, URN = {urn:nbn:de:0030-drops-144559}, doi = {10.4230/LIPIcs.MFCS.2021.15}, annote = {Keywords: Probabilistic finite automata, unary alphabet, emptiness problem, bounded ambiguity} }

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**Published in:** LIPIcs, Volume 171, 31st International Conference on Concurrency Theory (CONCUR 2020)

We show the surprising result that the cutpoint isolation problem is decidable for probabilistic finite automata where input words are taken from a letter-bounded context-free language. A context-free language ℒ is letter-bounded when ℒ ⊆ a₁^* a₂^* ⋯ a_𝓁^* for some finite 𝓁 > 0 where each letter is distinct. A cutpoint is isolated when it cannot be approached arbitrarily closely. The decidability of this problem is in marked contrast to the situation for the (strict) emptiness problem for PFA which is undecidable under the even more severe restrictions of PFA with polynomial ambiguity, commutative matrices and input over a letter-bounded language as well as to the injectivity problem which is undecidable for PFA over letter-bounded languages. We provide a constructive nondeterministic algorithm to solve the cutpoint isolation problem, which holds even when the PFA is exponentially ambiguous. We also show that the problem is at least NP-hard and use our decision procedure to solve several related problems.

Paul C. Bell and Pavel Semukhin. Decidability of Cutpoint Isolation for Probabilistic Finite Automata on Letter-Bounded Inputs. In 31st International Conference on Concurrency Theory (CONCUR 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 171, pp. 22:1-22:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{bell_et_al:LIPIcs.CONCUR.2020.22, author = {Bell, Paul C. and Semukhin, Pavel}, title = {{Decidability of Cutpoint Isolation for Probabilistic Finite Automata on Letter-Bounded Inputs}}, booktitle = {31st International Conference on Concurrency Theory (CONCUR 2020)}, pages = {22:1--22:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-160-3}, ISSN = {1868-8969}, year = {2020}, volume = {171}, editor = {Konnov, Igor and Kov\'{a}cs, Laura}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CONCUR.2020.22}, URN = {urn:nbn:de:0030-drops-128344}, doi = {10.4230/LIPIcs.CONCUR.2020.22}, annote = {Keywords: Probabilistic finite automata, cutpoint isolation problem, letter-bounded context-free languages} }

Document

**Published in:** LIPIcs, Volume 138, 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)

We consider notions of freeness and ambiguity for the acceptance probability of Moore-Crutchfield Measure Once Quantum Finite Automata (MO-QFA). We study the distribution of acceptance probabilities of such MO-QFA, which is partly motivated by similar freeness problems for matrix semigroups and other computational models. We show that determining if the acceptance probabilities of all possible input words are unique is undecidable for 32 state MO-QFA, even when all unitary matrices and the projection matrix are rational and the initial configuration is defined over real algebraic numbers. We utilize properties of the skew field of quaternions, free rotation groups, representations of tuples of rationals as a linear sum of radicals and a reduction of the mixed modification Post’s correspondence problem.

Paul C. Bell and Mika Hirvensalo. Acceptance Ambiguity for Quantum Automata. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 70:1-70:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{bell_et_al:LIPIcs.MFCS.2019.70, author = {Bell, Paul C. and Hirvensalo, Mika}, title = {{Acceptance Ambiguity for Quantum Automata}}, booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)}, pages = {70:1--70:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-117-7}, ISSN = {1868-8969}, year = {2019}, volume = {138}, editor = {Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.70}, URN = {urn:nbn:de:0030-drops-110149}, doi = {10.4230/LIPIcs.MFCS.2019.70}, annote = {Keywords: Quantum finite automata, matrix freeness, undecidability, Post’s correspondence problem, quaternions} }

Document

**Published in:** LIPIcs, Volume 138, 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)

We consider the following variant of the Mortality Problem: given k x k matrices A_1, A_2, ...,A_{t}, does there exist nonnegative integers m_1, m_2, ...,m_t such that the product A_1^{m_1} A_2^{m_2} * ... * A_{t}^{m_{t}} is equal to the zero matrix? It is known that this problem is decidable when t <= 2 for matrices over algebraic numbers but becomes undecidable for sufficiently large t and k even for integral matrices.
In this paper, we prove the first decidability results for t>2. We show as one of our central results that for t=3 this problem in any dimension is Turing equivalent to the well-known Skolem problem for linear recurrence sequences. Our proof relies on the Primary Decomposition Theorem for matrices that was not used to show decidability results in matrix semigroups before. As a corollary we obtain that the above problem is decidable for t=3 and k <= 3 for matrices over algebraic numbers and for t=3 and k=4 for matrices over real algebraic numbers. Another consequence is that the set of triples (m_1,m_2,m_3) for which the equation A_1^{m_1} A_2^{m_2} A_3^{m_3} equals the zero matrix is equal to a finite union of direct products of semilinear sets.
For t=4 we show that the solution set can be non-semilinear, and thus it seems unlikely that there is a direct connection to the Skolem problem. However we prove that the problem is still decidable for upper-triangular 2 x 2 rational matrices by employing powerful tools from transcendence theory such as Baker’s theorem and S-unit equations.

Paul C. Bell, Igor Potapov, and Pavel Semukhin. On the Mortality Problem: From Multiplicative Matrix Equations to Linear Recurrence Sequences and Beyond. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 83:1-83:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{bell_et_al:LIPIcs.MFCS.2019.83, author = {Bell, Paul C. and Potapov, Igor and Semukhin, Pavel}, title = {{On the Mortality Problem: From Multiplicative Matrix Equations to Linear Recurrence Sequences and Beyond}}, booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)}, pages = {83:1--83:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-117-7}, ISSN = {1868-8969}, year = {2019}, volume = {138}, editor = {Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.83}, URN = {urn:nbn:de:0030-drops-110279}, doi = {10.4230/LIPIcs.MFCS.2019.83}, annote = {Keywords: Linear recurrence sequences, Skolem’s problem, mortality problem, matrix equations, primary decomposition theorem, Baker’s theorem} }

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Track B: Automata, Logic, Semantics, and Theory of Programming

**Published in:** LIPIcs, Volume 132, 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)

We consider the computability and complexity of decision questions for Probabilistic Finite Automata (PFA) with sub-exponential ambiguity. We show that the emptiness problem for non-strict cut-points of polynomially ambiguous PFA remains undecidable even when the input word is over a bounded language and all PFA transition matrices are commutative. In doing so, we introduce a new technique based upon the Turakainen construction of a PFA from a Weighted Finite Automata which can be used to generate PFA of lower dimensions and of subexponential ambiguity. We also study freeness/injectivity problems for polynomially ambiguous PFA and study the border of decidability and tractability for various cases.

Paul C. Bell. Polynomially Ambiguous Probabilistic Automata on Restricted Languages (Track B: Automata, Logic, Semantics, and Theory of Programming). In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 105:1-105:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{bell:LIPIcs.ICALP.2019.105, author = {Bell, Paul C.}, title = {{Polynomially Ambiguous Probabilistic Automata on Restricted Languages}}, booktitle = {46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)}, pages = {105:1--105:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-109-2}, ISSN = {1868-8969}, year = {2019}, volume = {132}, editor = {Baier, Christel and Chatzigiannakis, Ioannis and Flocchini, Paola and Leonardi, Stefano}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2019.105}, URN = {urn:nbn:de:0030-drops-106814}, doi = {10.4230/LIPIcs.ICALP.2019.105}, annote = {Keywords: Probabilistic finite automata, ambiguity, undecidability, bounded language, emptiness} }