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Languages Given by Finite Automata over the Unary Alphabet

Authors Wojciech Czerwiński, Maciej Dębski, Tomasz Gogasz, Gordon Hoi, Sanjay Jain, Michał Skrzypczak , Frank Stephan , Christopher Tan

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Author Details

Wojciech Czerwiński
  • Institute of Informatics, Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Poland
Maciej Dębski
  • Warsaw, Poland
Tomasz Gogasz
  • Institute of Informatics, Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Poland
Gordon Hoi
  • School of Informatics and IT, Temasek Polytechnic, Singapore, Singapore
Sanjay Jain
  • School of Computing, National University of Singapore, Singapore
Michał Skrzypczak
  • Institute of Informatics, Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Poland
Frank Stephan
  • Department of Mathematics and School of Computing, National University of Singapore, Singapore
Christopher Tan
  • Department of Mathematics, National University of Singapore, Singapore

Funding

  • Czerwiński, Wojciech: supported by the ERC grant INFSYS, agreement no. 950398.
  • Jain, Sanjay: supported by the Singapore Ministry of Education Academic Research Fund Tier 2 grant MOE2019-T2-2-121 / R146-000-304-112 as well as NUS Provost Chair grant E-252-00-0021-01.
  • Skrzypczak, Michał: supported by the National Science Centre, Poland (grant number 2021/41/B/ST6/03914).
  • Stephan, Frank: supported by the Singapore Ministry of Education Academic Research Fund Tier 2 grant MOE2019-T2-2-121 / R146-000-304-112.

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Wojciech Czerwiński, Maciej Dębski, Tomasz Gogasz, Gordon Hoi, Sanjay Jain, Michał Skrzypczak, Frank Stephan, and Christopher Tan. Languages Given by Finite Automata over the Unary Alphabet. In 43rd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 284, pp. 22:1-22:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.FSTTCS.2023.22

Abstract

This paper studies the complexity of operations on finite automata and the complexity of their decision problems when the alphabet is unary and n the number of states of the finite automata considered. The following main results are obtained: 1) Equality and inclusion of NFAs can be decided within time 2^O((n log n)^{1/3}); previous upper bound 2^O((n log n)^{1/2}) was by Chrobak (1986) via DFA conversion. 2) The state complexity of operations of UFAs (unambiguous finite automata) increases for complementation and union at most by quasipolynomial; however, for concatenation of two n-state UFAs, the worst case is an UFA of at least 2^Ω(n^{1/6}) states. Previously the upper bounds for complementation and union were exponential-type and this lower bound for concatenation is new.

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
Keywords
  • Nondeterministic Finite Automata
  • Unambiguous Finite Automata
  • Upper Bounds on Runtime
  • Conditional Lower Bounds
  • Languages over the Unary Alphabet

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References

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