Skip to main content
SpringerLink
Log in

Regular Language Representations in the Constructive Type Theory of Coq

  • Published:
Journal of Automated Reasoning Aims and scope Submit manuscript

Abstract

We explore the theory of regular language representations in the constructive type theory of Coq. We cover various forms of automata (deterministic, nondeterministic, one-way, two-way), regular expressions, and the logic WS1S. We give translations between all representations, show decidability results, and provide operations for various closure properties. Our results include a constructive decidability proof for the logic WS1S, a constructive refinement of the Myhill-Nerode characterization of regularity, and translations from two-way automata to one-way automata with verified upper bounds for the increase in size. All results are verified with an accompanying Coq development of about 3000 lines.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2

Similar content being viewed by others

Notes

  1. Here, abstract means that no witness can be extracted from the existence proof. The function \(\mathsf {xchoose}_X\) computes a witness by enumerating elements of X. The proof argument is only used to guarantee termination.

  2. The language \(L := \{\,a^nb^m\,\big |\,\text {The} n-\text {th Turing machine holds within m steps on input} \varepsilon \,\}\) is decidable, but the image of L under the homomorphism mapping a to a and b to \(\varepsilon \) is undecidable.

  3. In [24], \({\hat{\delta }}\,\) is defined recursively starting from the right end of the word. In Coq, structural recursion is more natural and the impact on the proofs appears to be minimal.

  4. Wu et al. [42] derive the closure of regular expressions under complement by proving the Myhill–Nerode theorem using regular expressions. The proof is significantly more complex than the automata constructions.

  5. That the transitive closure of a decidable relation is decidable is established in the Ssreflect libraries using depth-first search.

References

  1. Abadi, M., Cardelli, L., Curien, P., Lévy, J.: Explicit substitutions. J. Funct. Program. 1(4), 375–416 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  2. Berghofer, S., Reiter, M.: Formalizing the logic-automaton connection. In: Berghofer, S., Nipkow, T., Urban, C., Wenzel, M. (eds.) Theorem Proving in Higher Order Logics (TPHOLs 2009). LNCS, vol. 5674, pp. 147–163. Springer (2009)

  3. Blazy, S., Paulin-Mohring, C., Pichardie, D. (eds.): Interactive Theorem Proving—4th International Conference, ITP 2013, Rennes, France, July 22–26, 2013. Proceedings, LNCS, vol. 7998. Springer (2013)

  4. Braibant, T., Pous, D.: Deciding kleene algebras in Coq. Log. Methods. Comput. Sci. 8(1:16), 1–42 (2012)

    MathSciNet  MATH  Google Scholar 

  5. Brzozowski, J.A.: Derivatives of regular expressions. J. ACM 11(4), 481–494 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  6. Büchi, J.R.: Weak second-order arithmetic and finite automata. Zeitschr. f. math. Logic und Grundladen d. Math. 6, 66–92 (1960)

    Article  MathSciNet  MATH  Google Scholar 

  7. Cohen, C.: Pragmatic quotient types in Coq. In: Blazy et al. [3], pp. 213–228

  8. Constable, R.L., Jackson, P.B., Naumov, P., Uribe, J.C.: Constructively formalizing automata theory. In: Plotkin, G.D., Stirling, C., Tofte, M. (eds.) Proof, Language, and Interaction, pp. 213–238. The MIT Press, Cambridge (2000)

    Google Scholar 

  9. Coquand, T., Siles, V.: A decision procedure for regular expression equivalence in type theory. In: Jouannaud, J.P., Shao, Z. (eds.) Certified Programs and Proofs (CPP 2011). LNCS, vol. 7086, pp. 119–134. Springer (2011)

  10. Doczkal, C., Kaiser, J.O., Smolka, G.: Coq development accompanying this paper (Online Resource 1), https://github.com/chdoc/coq-reglang

  11. Doczkal, C., Kaiser, J., Smolka, G.: A constructive theory of regular languages in Coq. In: Gonthier, G., Norrish, M. (eds.) Certified Programs and Proofs (CPP 2013). LNCS, vol. 8307, pp. 82–97. Springer (2013)

  12. Doczkal, C., Smolka, G.: Two-way automata in Coq. In: Blanchette, J.C., Merz, S. (eds.) Interactive Theorem Proving (ITP 2016). LNCS, vol. 9807, pp. 151–166. Springer (2016)

  13. Elgot, C.C.: Decision problems of finite automata design and related arithmetics. Trans. Am. Math. Soc. 98, 21–51 (1961)

    Article  MathSciNet  MATH  Google Scholar 

  14. Filliâtre, J.C.: Finite automata theory in Coq: A constructive proof of kleene’s theorem. Tech. Rep. 97-04, LIP - ENS Lyon (1997)

  15. Gelade, W., Neven, F.: Succinctness of the complement and intersection of regular expressions. ACM Trans. Comput. Logic 13(1), 4:1–4:19 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  16. Gonthier, G., Mahboubi, A., Rideau, L., Tassi, E., Théry, L.: A modular formalisation of finite group theory. In: Schneider, K., Brandt, J. (eds.) Theorem Proving in Higher Order Logics (TPHOLs 2007). LNCS, vol. 4732, pp. 86–101. Springer (2007)

  17. Grädel, E., Thomas, W., Wilke, T. (eds.): Automata, logics, and infinite games: a guide to current research [outcome of a Dagstuhl seminar, February 2001], LNCS, vol. 2500. Springer (2002)

  18. Hedberg, M.: A coherence theorem for Martin-Löf’s type theory. J. Funct. Program. 8(4), 413–436 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  19. Hofmann, M., Lange, M.: Automatentheorie und Logik. eXamen.press, Springer, Berlin (2011)

    Book  MATH  Google Scholar 

  20. Hopcroft, J.E., Motwani, R., Ullman, J.D.: Introduction to Automata Theory, Languages, and Computation - international edition (2. ed). Addison-Wesley, Boston (2001)

    MATH  Google Scholar 

  21. Johnsonbaugh, R., Miller, D.P.: Converses of pumping lemmas. In: Austing, R.H., Cassel, L.N., Miller, J.E., Joyce, D.T. (eds.) Proceedings of the 21st SIGCSE Technical Symposium on Computer Science Education, 1990, Washington, DC, USA, 1990. pp. 27–30. ACM (1990)

  22. Khoussainov, B., Nerode, A.: Automata Theory and its Applications. Springer, Berlin (2012)

    MATH  Google Scholar 

  23. Kleene, S.C.: Representation of events in nerve nets and finite automata. In: Shannon, C.E., McCarthy, J. (eds.) Automata Studies, pp. 3–42. Princeton University Press, Princeton (1956)

    Google Scholar 

  24. Kozen, D.: Automata and computability. Undergraduate texts in computer science. Springer, Berlin (1997)

    Google Scholar 

  25. Krauss, A., Nipkow, T.: Proof pearl: Regular expression equivalence and relation algebra. J. Autom. Reason. 49(1), 95–106 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  26. Moreira, N., Pereira, D., de Sousa, S.M.: Deciding kleene algebra terms equivalence in Coq. J. Log. Algebr. Methods Program. 84(3), 377–401 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  27. Nipkow, T.: Verified lexical analysis. In: Grundy, J., Newey, M.C. (eds.) Theorem Proving in Higher Order Logics (TPHOLs ’98). LNCS, vol. 1479, pp. 1–15. Springer (1998)

  28. Paulson, L.C.: A formalisation of finite automata using hereditarily finite sets. In: Felty, A.P., Middeldorp, A. (eds.) Automated deduction (CADE-25). LNCS, vol. 9195, pp. 231–245. Springer (2015)

  29. Pighizzini, G.: Two-way finite automata: old and recent results. Fundam. Inform. 126(2–3), 225–246 (2013)

    MathSciNet  MATH  Google Scholar 

  30. Pous, D.: Kleene algebra with tests and coq tools for while programs. In: Blazy et al. [3], pp. 180–196

  31. Rabin, M.O., Scott, D.: Finite automata and their decision problems. IBM J. Res. Dev. 3(2), 114–125 (1959)

    Article  MathSciNet  MATH  Google Scholar 

  32. Reinhardt, K.: The complexity of translating logic to finite automata. In: Grädel et al. [17], pp. 231–238

  33. Rosenberg, A.L.: State. In: Goldreich, O., Rosenberg, A.L., Selman, A.L. (eds.) Theoretical computer science, essays in memory of Shimon even. LNCS, vol. 3895, pp. 375–398. Springer (2006)

  34. Shepherdson, J.: The reduction of two-way automata to one-way automata. IBM J. Res. Dev. 3, 198–200 (1959)

    Article  MathSciNet  MATH  Google Scholar 

  35. The Coq Proof Assistant: http://coq.inria.fr

  36. The Mathematical Components Project: http://math-comp.github.io/math-comp/

  37. Trakhtenbrot, B.A.: Finite automata and the logic of monadic predicates. Dokl. Akad. Nauk SSSR 140, 326–329 (1961)

    Google Scholar 

  38. Traytel, D., Nipkow, T.: Verified decision procedures for MSO on words based on derivatives of regular expressions. J. Funct. Program. 25, 1–30 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  39. Vardi, M.Y.: A note on the reduction of two-way automata to one-way automata. Inf. Process. Lett. 30(5), 261–264 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  40. Vardi, M.Y.: Endmarkers can make a difference. Inf. Process. Lett. 35(3), 145–148 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  41. Wu, C., Zhang, X., Urban, C.: A formalisation of the Myhill-Nerode theorem based on regular expressions (proof pearl). In: van Eekelen, M.C.J.D., Geuvers, H., Schmaltz, J., Wiedijk, F. (eds.) Interactive Theorem Proving (ITP 2011). LNCS, vol. 6898, pp. 341–356. Springer (2011)

  42. Wu, C., Zhang, X., Urban, C.: A formalisation of the Myhill-Nerode theorem based on regular expressions. J. Autom. Reason. 52(4), 451–480 (2014)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

We thank Jan-Oliver Kaiser, who was involved in our previous work on one-way automata and also in some of the early experiments with two-way automata. We also thank Damien Pous and the anonymous reviewers for helpful comments.

Author information

Authors and Affiliations

  1. Univ Lyon, CNRS, ENS de Lyon, UCB Lyon 1, LIP, Lyon, France

    Christian Doczkal

  2. Saarland University, Saarbrücken, Germany

    Gert Smolka

Authors
  1. Christian Doczkal
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Gert Smolka
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Christian Doczkal.

Additional information

C. Doczkal: This author has been funded by the European Research Council (ERC) under the European Union’s Horizon 2020 programme (CoVeCe, Grant agreement No 678157). This work was supported by the LABEX MILYON (ANR-10-LABX-0070) of Université de Lyon, within the program “Investissements d’Avenir” (ANR-11-IDEX-0007) operated by the French National Research Agency (ANR).

Electronic supplementary material

Below is the link to the electronic supplementary material.

Supplementary material 1 (zip 69 KB)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Doczkal, C., Smolka, G. Regular Language Representations in the Constructive Type Theory of Coq. J Autom Reasoning 61, 521–553 (2018). https://doi.org/10.1007/s10817-018-9460-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10817-018-9460-x

Keywords

Search

Navigation