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Nonuniform Deterministic Finite Automata over Finite Algebraic Structures

Authors Paweł M. Idziak, Piotr Kawałek , Jacek Krzaczkowski

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ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
  • Theory of computation → Complexity classes
Keywords
  • program satisfiability
  • circuit equivalence
  • identity checking

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Abstract

Nonuniform deterministic finite automata (NUDFA) over monoids were invented by Barrington in [Barrington, 1985] to study boundaries of nonuniform constant-memory computation. Later, results on these automata helped to identify interesting classes of groups for which equation satisfiability problem (PolSat) is solvable in (probabilistic) polynomial time [Mikael Goldmann and Alexander Russell, 2002; Idziak et al., 2022]. Based on these results, we present a full characterization of groups, for which the identity checking problem (called PolEqv) has a probabilistic polynomial-time algorithm. We also go beyond groups, and propose how to generalise the notion of NUDFA to arbitrary finite algebraic structures. We study satisfiability of these automata in this more general setting. As a consequence, we present a full description of finite algebras from congruence modular varieties for which testing circuit equivalence CEqv can be solved by a probabilistic polynomial-time procedure. In our proofs we use two computational complexity assumptions: randomized Expotential Time Hypothesis and Constant Degree Hypothesis.

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Paweł M. Idziak, Piotr Kawałek, and Jacek Krzaczkowski. Nonuniform Deterministic Finite Automata over Finite Algebraic Structures. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 161:1-161:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ICALP.2025.161

Author Details

Paweł M. Idziak
  • Institute of Theoretical Computer Science, Jagiellonian University, Krakow, Poland
Piotr Kawałek
  • Institute of Discrete Mathematics and Geometry, TU Wien, Austria
  • Institute of Theoretical Computer Science, Jagiellonian University, Krakow, Poland
Jacek Krzaczkowski
  • Institute of Computer Science and Mathematics, University of Maria Curie-Skłodowska, Lublin, Poland

Funding

  • Kawałek, Piotr: This research was funded in whole or in part by National Science Centre, Poland #2021/41/N/ST6/03907. For the purpose of Open Access, the author has applied a CC-BY public copyright licence to any Author Accepted Manuscript (AAM) version arising from this submission. Funded by the European Union (ERC, POCOCOP, 101071674). Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Research Council Executive Agency. Neither the European Union nor the granting authority can be held responsible for them.
  • Krzaczkowski, Jacek: This research was funded in whole or in part by National Science Centre, Poland #2022/45/B/ST6/02229. For the purpose of Open Access, the author has applied a CC-BY public copyright licence to any Author Accepted Manuscript (AAM) version arising from this submission.

Acknowledgements

The second and third authors would like to express their deep gratitude to Paweł Idziak, who passed away during the preparation of this paper. It was thanks to him that we met and collaborated for many years on problems related to satisfiability and equivalence in finite algebraic structures. His contributions to Algebra, Logic, and Computer Science will be remembered by many. Rest in peace, Paweł.

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