Abstract
There are many examples in the research literature of families of regular languages defined by purely model-theoretic means (that is, in terms of the kinds of formulas of predicate logic used to define them) that can be characterized algebraically (that is, in terms of the syntactic monoids or syntactic morphisms of the languages). In fact the existence of such algebraic characterizations appears to be the rule. The present paper gives an explanation of the phenomenon: A generalization of Eilenberg’s variety theorem is proved, and then applied to logic. We find that a very wide assortment of families of regular languages defined in model-theoretic terms form varieties in this new sense, and that consequently membership in the family depends only on the syntactic morphism of the language.
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References
J. Almeida, Finite Semigroups and Universal Algebra, World Scientific, Singapore, 1994.
D. Mix Barrington, K. Compton, H. Straubing, and D. Thérien, “Regular Languages in NC1”, J. Comp. Syst. Sci. 44 (1992) 478–499.
J. Brzozowski and I. Simon, “Characterizations of Locally Testable Events”, Discrete Math. 4 (1973) 243–271.
J. R. Büchi, “Weak Second-order Arithmetic and Finite Automata”, Z. Math. Logik Grundl. Math. 6 66–92 (1960).
S. Eilenberg, Automata, Languages and Machines, vol. B, Academic Press, New York, 1976.
R. McNaughton and S. Papert, Counter-Free Automata, MIT Press, Cambridge, Massachusetts, 1971.
J. E. Pin, Varieties of Formal Languages, Plenum, London, 1986.
J.-E. Pin, A variety theorem without complementation, Russian Mathematics (Izvestija vuzov.Matematika) 39 (1995), 80–90.
J.-E. Pin, H. Straubing and D. Thrien, “New results on the generalized star-height problem”, Information and Computation 101 219–250 (1992).
H. Straubing, Finite Automata, Formal Languages, and Circuit Complexity, Birkhäuser, Boston, 1994.
H. Straubing and D. Thérien, “Regular Languages Defined by Generalized First-Order Sentences with a Bounded Number of Bound Variables”, in STACS 2001, Springer, Berlin 551–562 (2001) (Lecture Notes in Computer Science 2010.)
H. Straubing, D. Thérien, and W. Thomas, “Regular Languages Defined by Generalized Quantifiers”, Information and Computation 118 289–301 (1995).
D. Thérien and T. Wilke, “Over Words, Two Variables are as Powerful as One Quantifier Alternation,” Proc. 30th ACM Symposium on the Theory of Computing 256–263 (1998).
W. Thomas, “Classifying Regular Events in Symbolic Logic”, J. Computer and System Sciences 25 (1982) 360–376.
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Straubing, H. (2002). On Logical Descriptions of Regular Languages. In: Rajsbaum, S. (eds) LATIN 2002: Theoretical Informatics. LATIN 2002. Lecture Notes in Computer Science, vol 2286. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45995-2_46
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DOI: https://doi.org/10.1007/3-540-45995-2_46
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