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Carathéodory Extensions of Subclasses of Regular Languages

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Developments in Language Theory (DLT 2021)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 12811))

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Abstract

A language \(L\) is said to be regular measurable if there exists an infinite sequence of regular languages that “converges” to \(L\). In [13], the author showed that, while many complex context-free languages are regular measurable, the set of all primitive words and certain deterministic context-free languages are regular immeasurable. This paper investigates general properties of measurability, including closure properties, decidability and different characterisation. Further, for a suitable subclass \({\mathcal C}\) of regular languages, we show that the class of all \({\mathcal C}\)-measurable regular languages has a good algebraic structure.

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Notes

  1. 1.

    Here \(p\) can be \(0\) and we call a singleton \(\{q\}\) arithmetic progression in this case.

References

  1. Adámek, J., Milius, S., Myers, R.S.R., Urbat, H.: Generalized Eilenberg theorem I: local varieties of languages. In: Muscholl, A. (ed.) FoSSaCS 2014. LNCS, vol. 8412, pp. 366–380. Springer, Heidelberg (2014). https://doi.org/10.1007/978-3-642-54830-7_24

    Chapter  MATH  Google Scholar 

  2. Berstel, J., Perrin, D., Reutenauer, C.: Codes and Automata (Encyclopedia of Mathematics and its Applications). Cambridge University Press, Cambridge (2009)

    Google Scholar 

  3. Buck, R.C.: The measure theoretic approach to density. Am. J. Math. 68(4), 560–580 (1946)

    Article  MathSciNet  Google Scholar 

  4. Dömösi, P., Horváth, S., Ito, M.: On the connection between formal languages and primitive words. In: First Session on Scientific Communication, pp. 59–67 (1991)

    Google Scholar 

  5. Eilenberg, S., Tilson, B.: Automata, languages and machines. In: Pure and Applied Mathematics, vol. B. Academic Press, New-York (1976)

    Google Scholar 

  6. Gehrke, M., Grigorieff, S., Pin, J.É.: Duality and equational theory of regular languages. In: Aceto, L., Damgård, I., Goldberg, L.A., Halldórsson, M.M., Ingólfsdóttir, A., Walukiewicz, I. (eds.) ICALP 2008. LNCS, vol. 5126, pp. 246–257. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-70583-3_21

    Chapter  MATH  Google Scholar 

  7. Greibach, S.A.: A note on undecidable properties of formal languages. Math. Syst. Theor. 2, 1–6 (1968)

    Article  MathSciNet  Google Scholar 

  8. Lawson, M.V.: Finite automata. Birkhäuser (2005)

    Google Scholar 

  9. Pin, J.E.: Mathematical foundations of automata theory (draft)

    Google Scholar 

  10. Salomaa, A., Soittola, M.: Automata Theoretic Aspects of Formal Power Series. Springer, New York (1978). https://doi.org/10.1007/978-1-4612-6264-0

    Book  MATH  Google Scholar 

  11. Schützenberger, M.P.: On finite monoids having only trivial subgroups. Inf. Control 8(2), 190–194 (1965)

    Article  MathSciNet  Google Scholar 

  12. Sin’ya, R.: An automata theoretic approach to the zero-one law for regular languages. In: Games, Automata, Logics and Formal Verification, pp. 172–185 (2015)

    Google Scholar 

  13. Sin’ya, R.: Asymptotic approximation by regular languages. In: Bureš, T., et al. (eds.) SOFSEM 2021. LNCS, vol. 12607, pp. 74–88. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-67731-2_6

    Chapter  Google Scholar 

  14. Sin’ya, R.: Carathéodory extensions of subclasses of regular languages (full version) (2021). http://www.math.akita-u.ac.jp/~ryoma/misc/caratheodory.pdf

  15. Tao, T.: An Introduction to Measure Theory (Graduate Studies in Mathematics). American Mathematical Society, Providence (2013)

    Google Scholar 

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Acknowledgements

This work was supported by JSPS KAKENHI Grant Number JP19K14582.

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Correspondence to Ryoma Sin’ya .

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Sin’ya, R. (2021). Carathéodory Extensions of Subclasses of Regular Languages. In: Moreira, N., Reis, R. (eds) Developments in Language Theory. DLT 2021. Lecture Notes in Computer Science(), vol 12811. Springer, Cham. https://doi.org/10.1007/978-3-030-81508-0_29

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  • DOI: https://doi.org/10.1007/978-3-030-81508-0_29

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  • Online ISBN: 978-3-030-81508-0

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