Decomposing Finite Languages

Author Daniel Alexander Spenner

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Daniel Alexander Spenner
  • Technische UniversitΓ€t Dortmund, Germany


I want to thank Thomas Schwentick for his advice and encouragement.

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Daniel Alexander Spenner. Decomposing Finite Languages. In 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 272, pp. 83:1-83:14, Schloss Dagstuhl – Leibniz-Zentrum fΓΌr Informatik (2023)


The paper completely characterizes the primality of acyclic DFAs, where a DFA π’œ is prime if there do not exist DFAs π’œ_1,… ,π’œ_t with β„’(π’œ) = β‹‚_{i=1}^t β„’(π’œ_i) such that each π’œ_i has strictly less states than the minimal DFA recognizing the same language as π’œ. A regular language is prime if its minimal DFA is prime. Thus, this result also characterizes the primality of finite languages. Further, the NL-completeness of the corresponding decision problem Prime-DFA_fin is proven. The paper also characterizes the primality of acyclic DFAs under two different notions of compositionality, union and union-intersection compositionality. Additionally, the paper introduces the notion of S-primality, where a DFA π’œ is S-prime if there do not exist DFAs π’œβ‚,… ,π’œ_t with β„’(π’œ) = β‹‚_{i=1}^t β„’(π’œ_i) such that each π’œ_i has strictly less states than π’œ itself. It is proven that the problem of deciding S-primality for a given DFA is NL-hard. To do so, the NL-completeness of 2Minimal-DFA, the basic problem of deciding minimality for a DFA with at most two letters, is proven.

Subject Classification

ACM Subject Classification
  • Theory of computation β†’ Regular languages
  • Theory of computation β†’ Problems, reductions and completeness
  • Deterministic finite automaton (DFA)
  • Regular languages
  • Finite languages
  • Decomposition
  • Primality
  • Minimality


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  1. Christel Baier and Joost-Pieter Katoen. Principles of model checking. MIT Press, 2008. URL:
  2. Sang Cho and Dung T. Huynh. The parallel complexity of finite-state automata problems. Inf. Comput., 97(1):1-22, 1992. URL:
  3. Willem P. de Roever, Hans Langmaack, and Amir Pnueli, editors. Compositionality: The Significant Difference, International Symposium, COMPOS'97, Bad Malente, Germany, September 8-12, 1997. Revised Lectures, volume 1536 of Lecture Notes in Computer Science. Springer, 1998. URL:
  4. Henning Fernau and Markus Holzer. Personal communication. Google Scholar
  5. Peter Gazi and Branislav Rovan. Assisted problem solving and decompositions of finite automata. In Viliam Geffert, Juhani KarhumΓ€ki, Alberto Bertoni, Bart Preneel, Pavol NΓ‘vrat, and MΓ‘ria BielikovΓ‘, editors, SOFSEM 2008: Theory and Practice of Computer Science, 34th Conference on Current Trends in Theory and Practice of Computer Science, NovΓ½ Smokovec, Slovakia, January 19-25, 2008, Proceedings, volume 4910 of Lecture Notes in Computer Science, pages 292-303. Springer, 2008. URL:
  6. E. Mark Gold. Complexity of automaton identification from given data. Inf. Control., 37(3):302-320, 1978. URL:
  7. Neil Immerman. Descriptive complexity. Graduate texts in computer science. Springer, 1999. URL:
  8. IsmaΓ«l Jecker, Orna Kupferman, and Nicolas Mazzocchi. Unary prime languages. In Javier Esparza and Daniel KrΓ‘l', editors, 45th International Symposium on Mathematical Foundations of Computer Science, MFCS 2020, August 24-28, 2020, Prague, Czech Republic, volume 170 of LIPIcs, pages 51:1-51:12. Schloss Dagstuhl - Leibniz-Zentrum fΓΌr Informatik, 2020. URL:
  9. IsmaΓ«l Jecker, Nicolas Mazzocchi, and Petra Wolf. Decomposing permutation automata. In Serge Haddad and Daniele Varacca, editors, 32nd International Conference on Concurrency Theory, CONCUR 2021, August 24-27, 2021, Virtual Conference, volume 203 of LIPIcs, pages 18:1-18:19. Schloss Dagstuhl - Leibniz-Zentrum fΓΌr Informatik, 2021. URL:
  10. Orna Kupferman and Jonathan Mosheiff. Prime languages. Inf. Comput., 240:90-107, 2015. URL:
  11. Niklas Lauffer, Beyazit Yalcinkaya, Marcell Vazquez-Chanlatte, Ameesh Shah, and Sanjit A. Seshia. Learning deterministic finite automata decompositions from examples and demonstrations. In Alberto Griggio and Neha Rungta, editors, 22nd Formal Methods in Computer-Aided Design, FMCAD 2022, Trento, Italy, October 17-21, 2022, pages 1-6. IEEE, 2022. URL:
  12. Wim Martens, Matthias Niewerth, and Thomas Schwentick. Schema design for XML repositories: complexity and tractability. In Jan Paredaens and Dirk Van Gucht, editors, Proceedings of the Twenty-Ninth ACM SIGMOD-SIGACT-SIGART Symposium on Principles of Database Systems, PODS 2010, June 6-11, 2010, Indianapolis, Indiana, USA, pages 239-250. ACM, 2010. URL:
  13. Alexandru Mateescu, Arto Salomaa, and Sheng Yu. Factorizations of languages and commutativity conditions. Acta Cybern., 15(3):339-351, 2002. URL:
  14. Arto Salomaa and Sheng Yu. On the decomposition of finite languages. In Grzegorz Rozenberg and Wolfgang Thomas, editors, Developments in Language Theory, Foundations, Applications, and Perspectives, Aachen, Germany, 6-9 July 1999, pages 22-31. World Scientific, 1999. URL:
  15. Philip Sieder. A lower bound for primality of finite languages. CoRR, abs/1902.06253, 2019. URL:
  16. Stavros Tripakis. Compositionality in the science of system design. Proc. IEEE, 104(5):960-972, 2016. URL:
  17. Moshe Y. Vardi and Pierre Wolper. An automata-theoretic approach to automatic program verification (preliminary report). In Proceedings of the Symposium on Logic in Computer Science (LICS '86), Cambridge, Massachusetts, USA, June 16-18, 1986, pages 332-344. IEEE Computer Society, 1986. URL:
  18. Wojciech Wieczorek. An algorithm for the decomposition of finite languages. Log. J. IGPL, 18(3):355-366, 2010. URL:
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