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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

  1. Kristína Čevorová. Kleene star on unary regular languages. In Descriptional Complexity of Formal Systems: 15th International Workshop, DCFS 2013, London, ON, Canada, July 22-25, 2013. Proceedings 15, volume 8031 of LNCS, pages 277-288. Springer, 2013. Google Scholar
  2. Marek Chrobak. Finite automata and unary languages. Theoretical Computer Science, 47:149-158, 1986. Google Scholar
  3. Thomas Colcombet. Unambiguity in automata theory. In Seventeenth International Workshop on Descriptional Complexity of Formal Systems, volume 9118 of LNCS, pages 3-18. Springer, 2015. Google Scholar
  4. Maciej Dębski. State complexity of complementing unambiguous automata, Master’s thesis, University of Warsaw, 2017. Google Scholar
  5. Henning Fernau and Andreas Krebs. Problems on finite automata and the exponential time hypothesis. Algorithms, 10(1):24, 2017. Google Scholar
  6. Mika Göös, Stefan Kiefer, and Weiqiang Yuan. Lower bounds for unambiguous automata via communication complexity. In Forty Ninth International Colloquium on Automata, Languages and Programming, ICALP 2022, July 4-8, 2022, Paris, France, volume 229 of LIPIcs, pages 126:1-13, 2022. See also the technical report on URL: http://www.arxiv.org/abs/2109.09155.
  7. Markus Holzer and Martin Kutrib. Unary language operations and their nondeterministic state complexity. In International Conference on Developments in Language Theory, volume 2540 of LNCS, pages 162-172. Springer, 2002. Google Scholar
  8. Markus Holzer and Martin Kutrib. Descriptional and computational complexity of finite automata - A survey. Information and Computation, 209(3):456-470, 2011. Google Scholar
  9. John E Hopcroft, Rajeev Motwani, and Jeffrey D Ullman. Introduction to Automata Theory, Languages and Computation. Addison Wesley, 3rd edition, 2007. Google Scholar
  10. Russell Impagliazzo and Ramamohan Paturi. On the complexity of k-SAT. Journal of Computer and System Sciences, 62(2):367-375, 2001. Google Scholar
  11. Russell Impagliazzo, Ramamohan Paturi, and Francis Zane. Which problems have strongly exponential complexity? Journal of Computer and System Sciences, 63(4):512-530, 2001. Google Scholar
  12. Emil Indzhev and Stefan Kiefer. On complementing unambiguous automata and graphs with many cliques and cocliques. Information Processing Letters, 177:106270:1-5, 2022. Google Scholar
  13. Jozef Jirásek Jr, Galina Jirásková, and Juraj Šebej. Operations on unambiguous finite automata. International Journal of Foundations of Computer Science, 29(05):861-876, 2018. Google Scholar
  14. Galina Jirásková and Alexander Okhotin. State complexity of unambiguous operations on deterministic finite automata. In Twentieth International Conference on Descriptional Complexity of Formal Systems, volume 10952 of LNCS, pages 188-199. Springer, 2018. Google Scholar
  15. Alexander Okhotin. Unambiguous finite automata over a unary alphabet. Information and Computation, 212:15-36, 2012. Google Scholar
  16. Giovanni Pighizzini. Unary language concatenation and its state complexity. In Implementation and Application of Automata: 5th International Conference, CIAA 2000 London, Ontario, Canada, July 24-25, 2000 Revised Papers 5, volume 2088 of LNCS, pages 252-262. Springer, 2001. Google Scholar
  17. Giovanni Pighizzini and Jeffrey Shallit. Unary language operations, state complexity and Jacobsthal’s function. International Journal of Foundations of Computer Science, 13(01):145-159, 2002. Google Scholar
  18. Michael Raskin. A superpolynomial lower bound for the size of non-deterministic complement of an unambiguous automaton. In Fortyfifth International Colloquium on Automata, Languages and Programming, ICALP 2018, volume 107 of LIPIcs, pages 138:1-11, 2018. Google Scholar
  19. Walter J Savitch. Relationships between nondeterministic and deterministic tape complexities. Journal of computer and system sciences, 4(2):177-192, 1970. Google Scholar
  20. Richard Edwin Stearns and Harry B Hunt III. On the equivalence and containment problems for unambiguous regular expressions, regular grammars and finite automata. SIAM Journal on Computing, 14(3):598-611, 1985. Google Scholar
  21. Christopher Tan. Characteristics and computation of minimal automata, UROP, National University of Singapore, 2022. Google Scholar
  22. Sheng Yu, Qingyu Zhuang, and Kai Salomaa. The state complexities of some basic operations on regular languages. Theoretical Computer Science, 125(2):315-328, 1994. Google Scholar
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