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Density of Rational Languages Under Shift Invariant Measures

Authors Valérie Berthé , Herman Goulet-Ouellet , Dominique Perrin

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ACM Subject Classification
  • Theory of computation → Regular languages
Keywords
  • Automata theory
  • Symbolic dynamics
  • Semigroup theory
  • Ergodic theory

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Abstract

We study density of rational languages under shift invariant probability measures on spaces of two-sided infinite words, which generalizes the classical notion of density studied in formal languages and automata theory. The density for a language is defined as the limit in average (if it exists) of the probability that a word of a given length belongs to the language. We establish the existence of densities for all rational languages under all shift invariant measures. We also give explicit formulas under certain conditions, in particular when the language is aperiodic. Our approach combines tools and ideas from semigroup theory and ergodic theory.

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Valérie Berthé, Herman Goulet-Ouellet, and Dominique Perrin. Density of Rational Languages Under Shift Invariant Measures. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 143:1-143:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ICALP.2025.143

Author Details

Valérie Berthé
  • Université Paris Cité, IRIF, CNRS, France
Herman Goulet-Ouellet
  • Faculty of Information Technology, Czech Technical University in Prague, Czech Republic
Dominique Perrin
  • LIGM, Université Gustave Eiffel, Marne-la-Vallée, France

Funding

  • Berthé, Valérie: The first author was supported by the Agence Nationale de la Recherche through the projects "SymDynAr" (ANR-23-CE40-0024) and "IZES" (ANR-22-CE40-0011), as well as by the 2024 ERC Synergy Project DynAMiCs (101167561).
  • Goulet-Ouellet, Herman: The second author was supported by the CTU Global Postdoc Fellowship program.
  • Perrin, Dominique: The third author was supported by the Agence Nationale de la Recherche through the project "IZES" (ANR-22-CE40-0011).

References

  1. Jean-Paul Allouche and Jeffrey Shallit. Automatic Sequences. Cambridge University Press, 2003. URL: https://doi.org/10.1017/cbo9780511546563.
  2. Jorge Almeida. Profinite semigroups and applications. In V. Kudryavtsev and I. G. Rosenberg, editors, Structural Theory of Automata, Semigroups, and Universal Algebra, volume 207 of NATO Science Series II: Mathematics, Physics and Chemistry, pages 1-45. Springer, 2005. URL: https://doi.org/10.1007/1-4020-3817-8_1.
  3. Jorge Almeida, Alfredo Costa, Revekka Kyriakoglou, and Dominique Perrin. On the group of a rational maximal bifix code. Forum Math., 32(3):553-576, 2020. URL: https://doi.org/10.1515/forum-2018-0270.
  4. Marie-Pierre Béal, Dominique Perrin, and Antonio Restivo. Decidable problems in substitution shifts. J. Comput. System Sci., 143, 2024. URL: https://doi.org/10.1016/j.jcss.2024.103529.
  5. Jean Berstel. Sur la densité asymptotique de langages formels. In M. Nivat, editor, Automata Languages and Programming, pages 345-358. North-Holland, 1972. Google Scholar
  6. Jean Berstel, Clelia De Felice, Dominique Perrin, Christophe Reutenauer, and Giuseppina Rindone. Bifix codes and Sturmian words. J. Algebra, 369:146-202, 2012. Google Scholar
  7. Jean Berstel, Dominique Perrin, and Christophe Reutenauer. Codes and Automata. Cambridge University Press, 2009. Google Scholar
  8. Valérie Berthé. Fréquences des facteurs des suites sturmiennes. Theor. Comput. Sci., 165(2):295-309, 1996. URL: https://doi.org/10.1016/0304-3975(95)00224-3.
  9. Valérie Berthé, Herman Goulet-Ouellet, Carl-Fredrik Nyberg-Brodda, Dominique Perrin, and Karl Petersen. Density of group languages in shift spaces, 2024. URL: https://doi.org/10.48550/arXiv.2403.17892.
  10. Manuel Bodirsky, Tobias Gärtner, Timo von Oertzen, and Jan Schwinghammer. Efficiently computing the density of regular languages. In LATIN 2004: Theoretical Informatics, pages 262-270. Springer Berlin Heidelberg, 2004. URL: https://doi.org/10.1007/978-3-540-24698-5_30.
  11. Mike Boyle and Karl Petersen. Hidden Markov processes in the context of symbolic dynamics, pages 5-71. Cambridge University Press, 2011. URL: https://doi.org/10.1017/cbo9780511819407.002.
  12. Olivier Carton and Wolfgang Thomas. The monadic theory of morphic infinite words and generalizations. Inf. Comput., 176(1):51-65, 2002. URL: https://doi.org/10.1006/inco.2001.3139.
  13. Thomas Colcombet. Green’s relations and their use in automata theory. In Adrian-Horia Dediu, Shunsuke Inenaga, and Carlos Martín-Vide, editors, Language and Automata Theory and Applications, volume 6638 of Lecture Notes in Computer Science, pages 1-21. Springer, 2011. URL: https://doi.org/10.1007/978-3-642-21254-3_1.
  14. Fabien Durand and Dominique Perrin. Dimension Groups and Dynamical Systems. Cambridge University Press, 2022. URL: https://doi.org/10.1017/9781108976039.
  15. Samuel Eilenberg. Automata, Languages and Machines, vol. A. Pure and applied mathematics. Academic Press, 1974. Google Scholar
  16. Samuel Eilenberg. Automata, Languages and Machines, vol. B. Pure and applied mathematics. Academic Press, 1976. Google Scholar
  17. N. Pytheas Fogg. Substitutions in dynamics, arithmetics and combinatorics, volume 1794 of Lecture Notes in Mathematics. Springer-Verlag, 2002. Google Scholar
  18. Pierre A. Grillet. Semigroups: An Introduction to the Structure Theory. Marcel Dekker, 1995. URL: https://doi.org/10.4324/9780203739938.
  19. G. Hansel and D. Perrin. Codes and Bernoulli partitions. Math. Syst. Theory, 16(1):133-157, 1983. URL: https://doi.org/10.1007/BF01744574.
  20. G. Hansel and D. Perrin. Rational probability measures. Theor. Comput. Sci., 65(2):171-188, 1989. URL: https://doi.org/10.1016/0304-3975(89)90042-x.
  21. John M. Howie. Fundamentals of semigroup theory. Oxford science publications. Oxford University Press, 1995. URL: https://doi.org/10.1093/oso/9780198511946.001.0001.
  22. Toshihiro Koga. On the density of regular languages. Fundam. Inform., 168(1):45-49, 2019. URL: https://doi.org/10.3233/fi-2019-1823.
  23. Jakub Kozik. Conditional densities of regular languages. Electron. Notes Theor. Comput. Sci., 140:67-79, 2005. URL: https://doi.org/10.1016/j.entcs.2005.06.023.
  24. G. Lallement. Semigroups and Combinatorial Applications. Wiley, 1979. Google Scholar
  25. M. Lothaire. Combinatorics on Words. Cambridge University Press, 1997. URL: https://doi.org/10.1017/cbo9780511566097.
  26. James F. Lynch. Convergence laws for random words. Australas. J. Combin., 7:145-156, 1993. URL: http://ajc.maths.uq.edu.au/pdf/7/ocr-ajc-v7-p145.pdf.
  27. Pierre Michel. Stricte ergodicité d'ensembles minimaux de substitution. C. R. Acad. Sci., 278:811-813, 1974. Google Scholar
  28. Dominique Perrin. Codes and automata in minimal sets. In Florin Manea and Dirk Nowotka, editors, Combinatorics on Words, volume 9304 of Lecture Notes in Computer Science, pages 35-46. Springer, 2015. URL: https://doi.org/10.1007/978-3-319-23660-5_4.
  29. Dominique Perrin and Paul E. Schupp. Automata on the integers, recurrence distinguishability, and the equivalence and decidability of monadic theories. In Proceedings of the First Annual IEEE Symposium on Logic in Computer Science (LICS 1986), pages 301-304. IEEE Computer Society Press, 1986. Google Scholar
  30. Dominique Perrin and Jean Éric Pin. Infinite Words: Automata, Semigroups, Logic and Games. Elsevier, 2004. Google Scholar
  31. Karl E. Petersen. Ergodic Theory. Cambridge University Press, 1983. URL: https://doi.org/10.1017/cbo9780511608728.
  32. Robert R. Phelps. Lectures on Choquet’s theorem. Lecture Notes in Mathematics. Springer Berlin Heidelberg, 2001. URL: https://doi.org/10.1007/b76887.
  33. Jean-Éric Pin. Varieties of formal languages. Springer New York, 1986. Google Scholar
  34. M. Queffélec. Substitution Dynamical Systems: Spectral Analysis, volume 1294 of Lecture Notes in Mathematics. Springer-Verlag, 2010. URL: https://doi.org/10.1007/978-3-642-11212-6.
  35. Ville Salo. Decidability and universality of quasiminimal subshifts. J. Comput. Syst. Sci., 89:288-314, 2017. URL: https://doi.org/10.1016/j.jcss.2017.05.017.
  36. Arto Salomaa and Matti Soittola. Automata-Theoretic Aspects of Formal Power Series. Springer New York, 1978. URL: https://doi.org/10.1007/978-1-4612-6264-0.
  37. M. P. Schützenberger. Sur certains sous-monoïdes libres. Bull. Soc. Math. Fr., 93:209-223, 1965. URL: https://doi.org/10.24033/bsmf.1623.
  38. Marcel Paul Schützenberger. On finite monoids having only trivial subgroups. Inform. Control, 8(2):190-194, 1965. URL: https://doi.org/10.1016/s0019-9958(65)90108-7.
  39. Ryoma Sin’ya. An automata theoretic approach to the zero-one law for regular languages: Algorithmic and logical aspects. Electron. Proc. Theor. Comput. Sci., 193:172-185, 2015. URL: https://doi.org/10.4204/eptcs.193.13.
  40. William A. Veech. Strict ergodicity in zero dimensional dynamical systems and the Kronecker-Weyl theorem mod 2. Trans. Amer. Math. Soc., 140:1, 1969. URL: https://doi.org/10.2307/1995120.
  41. William A. Veech. Finite group extensions of irrational rotations. Israel J. Math., 21(2):240-259, 1975. URL: https://doi.org/10.1007/BF02760801.
  42. Peter Walters. An Introduction to Ergodic Theory, volume 79 of Graduate Texts in Mathematics. Springer, 1982. Google Scholar
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