Abstract
In this paper we introduce a class of descriptors for regular languages arising from an application of the Stone duality between finite Boolean algebras and finite sets. These descriptors, called classical fortresses, are object specified in classical propositional logic and capable to accept exactly regular languages. To prove this, we show that the languages accepted by classical fortresses and deterministic finite automata coincide. Classical fortresses, besides being propositional descriptors for regular languages, also turn out to be an efficient tool for providing alternative and intuitive proofs for the closure properties of regular languages.
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References
Blok, W., Pigozzi, D.: Algebraizable logics. Memoirs of The American Mathematical Society, vol. 77. American Mathematical Society (1989)
Brzozowski, J.A.: Canonical regular expressions and minimal state graphs for definite events. In: Mathematical Theory of Automata. MRI Symposia Series, vol. 12, pp. 529–561. Polytechnic Press, Polytechnic Institute of Brooklyn (1962)
Büchi, J.R.: Weak second-order arithmetic and finite automata. Zeitschrift für mathematische Logik und Grundlagen der Mathematik 6, 66–92 (1960)
Burris, S., Sankappanavar, H.P.: A course in Universal Algebra. Springer (1981)
Cintula, P., Hájek, P., Noguera, C.: Handbook of Mathematical Fuzzy Logic, vol. 2. College Publications (2011)
Elgot, C.C.: Decision problems of finite automata design and related arithmetics. Trans. Amer. Math. Soc. 98, 21–51 (1961)
Hansen, H.H., Panangaden, P., Rutten, J.J.M.M., Bonchi, F., Bonsangue, M.M., Silva, A.: Algebra-coalgebra duality in Brzozwski’s minimization algorithm. ACM Transactions on Computational Logic (to appear)
Gehrke, M.: Duality and recognition. In: Murlak, F., Sankowski, P. (eds.) MFCS 2011. LNCS, vol. 6907, pp. 3–18. Springer, Heidelberg (2011)
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)
Hopcroft, J.E., Motwani, R., Ullman, J.D.: Introduction to Automata Theory, Languages, and Computation, 2nd edn. Addison-Wesley (2000)
Johnstone, P.T.: Stone Spaces. Cambridge University Press (1982)
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 (1956)
Weyuker, E.J., Davis, M.E., Sigal, R.: Computability, complexity, and languages: Fundamentals of theoretical computer science. Academic Press, Boston (1994)
Trakhtenbrot, B.A.: Finite automata and the logic of oneplace predicates. Siberian Math. J. 3, 103–131 (1962)
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Aguzzoli, S., Diaconescu, D., Flaminio, T. (2014). A Logical Descriptor for Regular Languages via Stone Duality. In: Ciobanu, G., Méry, D. (eds) Theoretical Aspects of Computing – ICTAC 2014. ICTAC 2014. Lecture Notes in Computer Science, vol 8687. Springer, Cham. https://doi.org/10.1007/978-3-319-10882-7_3
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DOI: https://doi.org/10.1007/978-3-319-10882-7_3
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