Abstract
This paper introduces an abstract notion of fragments of monadic second-order logic. This concept is based on purely syntactic closure properties. We show that over finite words, every logical fragment defines a lattice of languages with certain closure properties. Among these closure properties are residuals and inverse \(\mathcal C\)-morphisms. Here, depending on certain closure properties of the fragment, \(\mathcal C\) is the family of arbitrary, non-erasing, length-preserving, length-multiplying, or lengthreducing morphisms. In particular, definability in a certain fragment can often be characterized in terms of the syntactic morphism. This work extends a result of Straubing in which he investigated certain restrictions of first-order formulae.
As motivating examples, we present (1) a fragment which captures the stutter-invariant part of piecewise-testable languages and (2) an acyclic fragment of Σ2. As it turns out, the latter has the same expressive power as two-variable first-order logic FO2.
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References
Büchi, J.R.: Weak second-order arithmetic and finite automata. Z. Math. Logik Grundlagen Math. 6, 66–92 (1960)
Chaubard, L., Pin, J.-É., Straubing, H.: Actions, wreath products of \(\mathcal{C}\)-varieties and concatenation product. Theor. Comput. Sci. 356, 73–89 (2006)
Chaubard, L., Pin, J.-É., Straubing, H.: First order formulas with modular predicates. In: LICS 2006, pp. 211–220. IEEE Computer Society (2006)
Diekert, V., Gastin, P.: First-order definable languages. In: Flum, J., Grädel, E., Wilke, T. (eds.) Logic and Automata: History and Perspectives. Texts in Logic and Games, pp. 261–306. Amsterdam University Press (2008)
Diekert, V., Gastin, P., Kufleitner, M.: A survey on small fragments of first-order logic over finite words. Int. J. Found. Comput. Sci. 19(3), 513–548 (2008)
Eilenberg, S.: Automata, Languages, and Machines, vol. B. Academic Press (1976)
Elgot, C.C.: Decision problems of finite automata design and related arithmetics. Trans. Amer. Math. Soc. 98, 21–51 (1961)
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)
Kamp, J.A.W.: Tense Logic and the Theory of Linear Order. PhD thesis, University of California (1968)
Kufleitner, M., Lauser, A.: Lattices of logical fragments over words. CoRR, abs/1202.3355 (2012), http://arxiv.org/abs/1202.3355
Kunc, M.: Equational description of pseudovarieties of homomorphisms. Theor. Inform. Appl. 37, 243–254 (2003)
McNaughton, R., Papert, S.: Counter-Free Automata. The MIT Press (1971)
Pin, J.-É.: A variety theorem without complementation. Russian Mathematics (Iz. VUZ) 39, 80–90 (1995)
Pin, J.-É.: Expressive power of existential first-order sentences of Büchi’s sequential calculus. Discrete Math. 291(1-3), 155–174 (2005)
Pin, J.-É., Straubing, H.: Some results on \(\mathcal{C}\)-varieties. Theor. Inform. Appl. 39, 239–262 (2005)
Schützenberger, M.P.: On finite monoids having only trivial subgroups. Inf. Control 8, 190–194 (1965)
Simon, I.: Piecewise Testable Events. In: Brakhage, H. (ed.) GI-Fachtagung 1975. LNCS, vol. 33, pp. 214–222. Springer, Heidelberg (1975)
Straubing, H.: Finite Automata, Formal Logic, and Circuit Complexity. Birkhäuser (1994)
Straubing, H.: On Logical Descriptions of Regular Languages. In: Rajsbaum, S. (ed.) LATIN 2002. LNCS, vol. 2286, pp. 528–538. Springer, Heidelberg (2002)
Straubing, H., Thérien, D., Thomas, W.: Regular languages defined with generalized quantifiers. Inf. Comput. 118(2), 289–301 (1995)
Tesson, P., Thérien, D.: Diamonds are forever: The variety DA. In: Gomes, G., et al. (eds.) Semigroups, Algorithms, Automata and Languages 2001, pp. 475–500. World Scientific (2002)
Tesson, P., Thérien, D.: Logic meets algebra: The case of regular languages. Log. Methods Comput. Sci. 3(1), 1–37 (2007)
Thérien, D., Wilke, T.: Over words, two variables are as powerful as one quantifier alternation. In: STOC 1998, pp. 234–240. ACM Press (1998)
Thomas, W.: Classifying regular events in symbolic logic. J. Comput. Syst. Sci. 25, 360–376 (1982)
Trakhtenbrot, B.A.: Finite automata and logic of monadic predicates. Dokl. Akad. Nauk SSSR 140, 326–329 (1961) (in Russian)
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Kufleitner, M., Lauser, A. (2012). Lattices of Logical Fragments over Words. In: Czumaj, A., Mehlhorn, K., Pitts, A., Wattenhofer, R. (eds) Automata, Languages, and Programming. ICALP 2012. Lecture Notes in Computer Science, vol 7392. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-31585-5_27
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