Boolean Automata and Atoms of Regular Languages

Author Hellis Tamm

Thumbnail PDF


  • Filesize: 0.54 MB
  • 13 pages

Document Identifiers

Author Details

Hellis Tamm
  • Department of Software Science, Tallinn University of Technology, Estonia


The author is indebted to late Janusz Brzozowski for suggesting the topic and for collaboration during the early stages of this work.

Cite AsGet BibTex

Hellis Tamm. Boolean Automata and Atoms of Regular Languages. In 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 202, pp. 86:1-86:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


We examine the role that atoms of regular languages play in boolean automata. We observe that the size of a minimal boolean automaton of a regular language is directly related to the number of atoms of the language. We present a method to construct minimal boolean automata, using the atoms of a given regular language. The "illegal" cover problem of the Kameda-Weiner method for NFA minimization implies that using the union operation only to construct an automaton from a cover - as is the case with NFAs -, is not sufficient. We show that by using the union and the intersection operations (without the complementation operation), it is possible to construct boolean automata accepting a given language, for a given maximal cover.

Subject Classification

ACM Subject Classification
  • Theory of computation → Formal languages and automata theory
  • Boolean automaton
  • Regular language
  • Atoms


  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    PDF Downloads


  1. Dana Angluin, Sarah Eisenstat, and Dana Fisman. Learning regular languages via alternating automata. In Qiang Yang and Michael J. Wooldridge, editors, Proceedings of the Twenty-Fourth International Joint Conference on Artificial Intelligence, IJCAI 2015, Buenos Aires, Argentina, July 25-31, 2015, pages 3308-3314. AAAI Press, 2015. Google Scholar
  2. Sebastian Berndt, Maciej Liskiewicz, Matthias Lutter, and Rüdiger Reischuk. Learning residual alternating automata. In Satinder P. Singh and Shaul Markovitch, editors, Proceedings of the Thirty-First AAAI Conference on Artificial Intelligence, February 4-9, 2017, San Francisco, California, USA, pages 1749-1755. AAAI Press, 2017. Google Scholar
  3. Jean-Camille Birget. Intersection and union of regular languages and state complexity. Inf. Process. Lett., 43(4):185-190, 1992. Google Scholar
  4. Janusz A. Brzozowski. Canonical regular expressions and minimal state graphs for definite events. In Proc. Symp. on Mathematical Theory of Automata, volume 12 of MRI Symposia Series, pages 529-561. Polytechnic Press, Polytechnic Institute of Brooklyn, N.Y., 1963. Google Scholar
  5. Janusz A. Brzozowski and Ernst L. Leiss. On equations for regular languages, finite automata, and sequential networks. Theor. Comput. Sci., 10:19-35, 1980. Google Scholar
  6. Janusz A. Brzozowski and Hellis Tamm. Theory of átomata. Theor. Comput. Sci., 539:13-27, 2014. Google Scholar
  7. Ashok K. Chandra, Dexter C. Kozen, and Larry J. Stockmeyer. Alternation. J. ACM, 28(1):114-133, 1981. Google Scholar
  8. Loris D'Antoni, Zachary Kincaid, and Fang Wang. A symbolic decision procedure for symbolic alternating finite automata. In Proceedings of the Thirty-Fourth Conference on the Mathematical Foundations of Programming Semantics, MFPS 2018, Dalhousie University, Halifax, Canada, June 6-9, 2018, volume 341 of Electronic Notes in Theoretical Computer Science, pages 79-99. Elsevier, 2018. Google Scholar
  9. Abdelaziz Fellah, Helmut Jürgensen, and Sheng Yu. Constructions for alternating finite automata. Int. J. of Comput. Math., 35(1-4):117-132, 1990. Google Scholar
  10. Ian Glaister and Jeffrey O. Shallit. A lower bound technique for the size of nondeterministic finite automata. Inf. Process. Lett., 59(2):75-77, 1996. Google Scholar
  11. Hermann Gruber and Markus Holzer. Finding lower bounds for nondeterministic state complexity is hard. In Proc. of DLT 2006, volume 4036 of Lecture Notes in Computer Science, pages 363-374. Springer, 2006. Google Scholar
  12. Szabolcs Iván. Complexity of atoms, combinatorially. Inf. Process. Lett., 116(5):356-360, 2016. Google Scholar
  13. Tsunehiko Kameda and Peter Weiner. On the state minimization of nondeterministic finite automata. IEEE Trans. Computers, 19(7):617-627, 1970. Google Scholar
  14. Dexter Kozen. On parallelism in Turing machines. In 17th Annual Symposium on Foundations of Computer Science, Houston, Texas, USA, 25-27 October 1976, pages 89-97, 1976. Google Scholar
  15. Dexter Kozen. Theory of Computation. Texts in Computer Science. Springer, 2006. Google Scholar
  16. Ernst L. Leiss. Succinct representation of regular languages by boolean automata. Theor. Comput. Sci., 13:323-330, 1981. Google Scholar
  17. Ernst L. Leiss. Succinct representation of regular languages by boolean automata II. Theor. Comput. Sci., 38:133-136, 1985. Google Scholar
  18. Oliver Matz and Andreas Potthoff. Computing small finite nondeterministic automata. In U. H. Engberg, K. G. Larsen, and A. Skou, editors, Proceedings of the Workshop on Tools and Algorithms for Construction and Analysis of Systems, BRICS Note Series, pages 74-88. BRICS, 1995. Google Scholar
  19. Michael O. Rabin and Dana S. Scott. Finite automata and their decision problems. IBM J. Res. Dev., 3(2):114-125, 1959. Google Scholar
  20. Caleb Stanford, Margus Veanes, and Nikolaj Bjørner. Symbolic boolean derivatives for efficiently solving extended regular expression constraints. In Stephen N. Freund and Eran Yahav, editors, PLDI '21: 42nd ACM SIGPLAN International Conference on Programming Language Design and Implementation, Virtual Event, Canada, June 20-25, 20211, pages 620-635. ACM, 2021. Google Scholar
  21. Hellis Tamm. New interpretation and generalization of the Kameda-Weiner method. In 43rd Int. Colloq. on Automata, Languages, and Programming (ICALP 2016), volume 55 of Leibniz Int. Proc. in Informatics (LIPIcs), pages 116:1-116:12, Dagstuhl, Germany, 2016. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. Google Scholar
  22. Hellis Tamm and Brink van der Merwe. Lower bound methods for the size of nondeterministic finite automata revisited. In Language and Automata Theory and Applications - 11th International Conference, LATA 2017, Umeå, Sweden, March 6-9, 2017, Proceedings, volume 10168 of Lecture Notes in Computer Science, pages 261-272, 2017. Google Scholar
  23. Brink van der Merwe, Hellis Tamm, and Lynette van Zijl. Minimal DFA for symmetric difference NFA. In Descriptional Complexity of Formal Systems - 14th International Workshop, DCFS 2012, Braga, Portugal, July 23-25, 2012. Proceedings, volume 7386 of Lecture Notes in Computer Science, pages 307-318. Springer, 2012. Google Scholar
  24. Lynette van Zijl. On binary ⊕-nfas and succinct descriptions of regular languages. Theor. Comput. Sci., 328(1-2):161-170, 2004. Google Scholar
Questions / Remarks / Feedback

Feedback for Dagstuhl Publishing

Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail