Abstract
The fact that one can associate a finite monoid with universal properties to each language recognised by an automaton is central to the solution of many practical and theoretical problems in automata theory. It is particularly useful, via the advanced theory initiated by Eilenberg and Reiterman, in separating various complexity classes and, in some cases it leads to decidability of such classes. In joint work with Jean-Éric Pin and Serge Grigorieff we have shown that this theory may be seen as a special case of Stone duality for Boolean algebras extended to a duality between Boolean algebras with additional operations and Stone spaces equipped with Kripke style relations. This is a duality which also plays a fundamental role in other parts of the foundations of computer science, including in modal logic and in domain theory. In this talk I will give a general introduction to Stone duality and explain what this has to do with the connection between regular languages and monoids.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Abramsky, S.: Domain Theory in Logical Form. Ann. Pure Appl. Logic 51, 1–77 (1991)
Davey, B.A., Priestley, H.A.: Introduction to Lattices and Order, 2nd edn. Cambridge University Press, Cambridge (2002)
Eilenberg, S.: Automata, languages, and machines, vol. B. Academic Press (Harcourt Brace Jovanovich Publishers), New York (1976)
Esakia, L.L.: Topological Kripke models. Soviet Math. Dokl. 15, 147–151 (1974)
Fagin, R., Halpern, J.Y., Moses, Y., Vardi, M.Y.: Reasoning About Knowledge. MIT Press, Cambridge (1995)
Gehrke, M.: Stone duality and the recognisable languages over an algebra. In: Kurz, A., Lenisa, M., Tarlecki, A. (eds.) CALCO 2009. LNCS, vol. 5728, pp. 236–250. Springer, Heidelberg (2009)
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, Part II. LNCS, vol. 5126, pp. 246–257. Springer, Heidelberg (2008)
Goldblatt, R.: Varieties of complex algebras. Ann. Pure App. Logic 44, 173–242 (1989)
Pin, J.-E.: Profinite methods in automata theory. In: Albers, S., Marion, J.-Y. (eds.) 26th International Symposium on Theoretical Aspects of Computer Science, STACS 2009, Internationales Begegnungs- und Forschungszentrum für Informatik (IBFI), Schloss Dagstuhl, Germany, pp. 31–50 (2009)
Pippenger, N.: Regular languages and Stone duality. Theory Comput. Syst. 30(2), 121–134 (1997)
Reiterman, J.: The Birkhoff theorem for finite algebras. Algebra Universalis 14(1), 1–10 (1982)
Schützenberger, M.P.: On finite monoids having only trivial subgroups. Inform. Control 8, 190–194 (1965)
Stone, M.: The theory of representations for Boolean algebras. Trans. Amer. Math. Soc. 40, 37–111 (1936)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2011 Springer-Verlag GmbH Berlin Heidelberg
About this paper
Cite this paper
Gehrke, M. (2011). Duality and Recognition. In: Murlak, F., Sankowski, P. (eds) Mathematical Foundations of Computer Science 2011. MFCS 2011. Lecture Notes in Computer Science, vol 6907. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-22993-0_3
Download citation
DOI: https://doi.org/10.1007/978-3-642-22993-0_3
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-22992-3
Online ISBN: 978-3-642-22993-0
eBook Packages: Computer ScienceComputer Science (R0)