ICALP 2008: Automata, Languages and Programming pp 246-257

# Duality and Equational Theory of Regular Languages

• Mai Gehrke
• Serge Grigorieff
• Jean-Éric Pin
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5126)

## Abstract

This paper presents a new result in the equational theory of regular languages, which emerged from lively discussions between the authors about Stone and Priestley duality. Let us call lattice of languages a class of regular languages closed under finite intersection and finite union. The main results of this paper (Theorems 5.2 and 6.1) can be summarized in a nutshell as follows:

A set of regular languages is a lattice of languages if and only if it can be defined by a set of profinite equations.

The product on profinite words is the dual of the residuation operations on regular languages.

In their more general form, our equations are of the form uv, where u and v are profinite words. The first result not only subsumes Eilenberg-Reiterman’s theory of varieties and their subsequent extensions, but it shows for instance that any class of regular languages defined by a fragment of logic closed under conjunctions and disjunctions (first order, monadic second order, temporal, etc.) admits an equational description. In particular, the celebrated McNaughton-Schützenberger characterisation of first order definable languages by the aperiodicity condition xω = xω + 1, far from being an isolated statement, now appears as an elegant instance of a very general result.

## Preview

### References

1. 1.
Adams, M.E.: The Frattini sublattice of a distributive lattice. Alg. Univ. 3, 216–228 (1973)
2. 2.
Almeida, J.: Residually finite congruences and quasi-regular subsets in uniform algebras. Partugaliæ Mathematica 46, 313–328 (1989)
3. 3.
Almeida, J.: Finite semigroups and universal algebra. World Scientific Publishing Co. Inc., River Edge (1994)
4. 4.
Almeida, J.: Profinite semigroups and applications. In: Structural theory of automata, semigroups, and universal algebra. NATO Sci. Ser. II Math. Phys. Chem., vol. 207, pp. 1–45. Springer, Dordrecht (2005); Notes taken by Alfredo Costa
5. 5.
Almeida, J., Volkov, M.V.: Profinite identities for finite semigroups whose subgroups belong to a given pseudovariety. J. Algebra Appl. 2(2), 137–163 (2003)
6. 6.
Birkhoff, G.: On the structure of abstract algebras. Proc. Cambridge Phil. Soc. 31, 433–454 (1935)
7. 7.
Eilenberg, S.: Automata, languages, and machines, vol. B. Academic Press [Harcourt Brace Jovanovich Publishers], New York (1976)
8. 8.
Ésik, Z.: Extended temporal logic on finite words and wreath products of monoids with distinguished generators. In: Ito, M., Toyama, M. (eds.) DLT 2002. LNCS, vol. 2450, pp. 43–58. Springer, Heidelberg (2003)
9. 9.
Goldblatt, R.: Varieties of complex algebras. Ann. Pure App. Logic 44, 173–242 (1989)
10. 10.
Kunc, M.: Equational description of pseudovarieties of homomorphisms. Theoretical Informatics and Applications 37, 243–254 (2003)
11. 11.
Pin, J.-E.: A variety theorem without complementation. Russian Mathematics (Iz. VUZ) 39, 80–90 (1995)
12. 12.
Pin, J.-É., Straubing, H.: Some results on $$\mathcal C$$-varieties. Theoret. Informatics Appl. 39, 239–262 (2005)
13. 13.
Pin, J.-É., Weil, P.: A Reiterman theorem for pseudovarieties of finite first-order structures. Algebra Universalis 35, 577–595 (1996)
14. 14.
Pippenger, N.: Regular languages and Stone duality. Theory Comput. Syst. 30(2), 121–134 (1997)
15. 15.
Polák, L.: Syntactic semiring of a language. In: Sgall, J., Pultr, A., Kolman, P. (eds.) MFCS 2001. LNCS, vol. 2136, pp. 611–620. Springer, Heidelberg (2001)
16. 16.
Priestley, H.A.: Representation of distributive lattices by means of ordered Stone spaces. Bull. London Math. Soc. 2, 186–190 (1970)
17. 17.
Reilly, N.R., Zhang, S.: Decomposition of the lattice of pseudovarieties of finite semigroups induced by bands. Algebra Universalis 44(3-4), 217–239 (2000)
18. 18.
Reiterman, J.: The Birkhoff theorem for finite algebras. Algebra Universalis 14(1), 1–10 (1982)
19. 19.
Stone, M.: The theory of representations for Boolean algebras. Trans. Amer. Math. Soc. 40, 37–111 (1936)
20. 20.
Stone, M.H.: Applications of the theory of Boolean rings to general topology. Trans. Amer. Math. Soc. 41(3), 375–481 (1937)
21. 21.
Straubing, H.: On logical descriptions of regular languages. In: Rajsbaum, S. (ed.) LATIN 2002. LNCS, vol. 2286, pp. 528–538. Springer, Heidelberg (2002)
22. 22.
Weil, P.: Profinite methods in semigroup theory. Int. J. Alg. Comput. 12, 137–178 (2002)
23. 23.
Yu, S.: Regular languages. In: Rozenberg, G., Salomaa, A. (eds.) Handbook of language theory, ch. 2, vol. 1, pp. 679–746. Springer, Heidelberg (1997)Google Scholar