Abstract
We provide an algebraic characterization of the expressive power of various naturally defined logics on finite trees. These logics are described in terms of Lindström quantifiers, and particular cases include first-order logic and modular logic. The algebraic characterization we give is expressed in terms of a new algebraic structure, finitary preclones, and uses a generalization of the block product operation.
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References
Almeida, J.: On pseudovarieties, varieties of languages, filters of congruences, pseudoidentities and related topics. Algebra Universalis 27, 333–350 (1990)
Arnold, A., Dauchet, M.: Theorie des magmoides. I. and II. (in French), RAIRO Theoretical Informatics and Applications, 12(1978), 235–257, 3(1979), 135–154.
Bloom, S.L., Ésik, Z.: Iteration Theories. Springer, Heidelberg (1993)
Büchi, J.R.: Weak second-order arithmetic and finite automata. Z. Math. Logik Grundlagen Math. 6, 66–92 (1960)
Cohen, J., Pin, J.-E., Perrin, D.: On the expressive power of temporal logic. J. Computer and System Sciences 46, 271–294 (1993)
Courcelle, B.: The monadic second-order logic of graphs, I. Recognizable sets of finite graphs. Information and Computation 85, 12–75 (1990)
Denecke, K., Wismath, S.L.: Universal Algebra and Applications in Theoretical Computer Science. Chapman and Hall, Boca Raton (2002)
Diekert, V.: Combinatorics on Traces. LNCS, vol. 454. Springer, Heidelberg (1990)
Doner, J.: Tree acceptors and some of their applications. J. Comput. System Sci. 4, 406–451 (1970)
Ebbinghaus, H.-D., Flum, J.: Finite Model Theory. Springer, Heidelberg (1995)
Eilenberg, S.: Automata, Languages, and Machines, vol. A and B. Academic Press, London (1976 and 1976)
Eilenberg, S., Wright, J.B.: Automata in general algebras. Information and Control 11, 452–470 (1967)
Elgot, C.C.: Decision problems of finite automata design and related arithmetics. Trans. Amer. Math. Soc. 98, 21–51 (1961)
Ésik, Z.: A variety theorem for trees and theories. Publicationes Mathematicae 54, 711–762 (1999)
Ésik, Z., Larsen, K.G.: Regular languages definable by Lindström quantifiers. Theoretical Informatics and Applications (to appear)
Gabbay, D.M., Pnueli, A., Shelah, S., Stavi, J.: On the temporal analysis of fairness. In: proc.12th ACM Symp. Principles of Programming Languages, Las Vegas, pp. 163–173 (1980)
Heuter, U.: First-order properties of trees, star-free expressions, and aperiodicity. In: Cori, R., Wirsing, M. (eds.) STACS 1988. LNCS, vol. 294, pp. 136–148. Springer, Heidelberg (1988)
Kamp, J.A.: Tense logic and the theory of linear order, Ph. D. Thesis, UCLA (1968)
Lindström, P.: First order predicate logic with generalized quantifiers. Theoria 32, 186–195 (1966)
MacLane, S.: Categories for the Working Mathematician. Springer, Heidelberg (1971)
McNaughton, R., Papert, S.: Counter-Free Automata. MIT Press, Cambridge (1971)
Mezei, J., Wright, J.B.: Algebraic automata and context-free sets. Information and Control 11, 3–29 (1967)
Potthoff, A.: Modulo counting quantifiers over finite trees. In: Raoult, J.-C. (ed.) CAAP 1992. LNCS, vol. 581, Springer, Heidelberg (1992)
Potthoff, A.: First order logic on finite trees. In: Mosses, P.D., Schwartzbach, M.I., Nielsen, M. (eds.) CAAP 1995, FASE 1995, and TAPSOFT 1995. LNCS, vol. 915, Springer, Heidelberg (1995)
Rhodes, J., Tilson, B.: The kernel of monoid morphisms. J. Pure and Appl. Alg. 62, 227–268 (1989)
Schützenberger, M.P.: On finite monoids having only trivial subgroups. Information and Control 8, 190–194 (1965)
Steinby, M.: General varieties of tree languages. Theoret. Comput. Sci. 205, 1–43 (1998)
Straubing, H.: Finite Automata, Formal Logic, and Circuit Complexity. Birkhauser Boston, Inc., Boston (1994)
Straubing, H., Therien, D., Thomas, W.: Regular languages defined with generalized quantifiers. Information and Computation 118, 289–301 (1995)
Thatcher, J.W., Wright, J.B.: Generalized finite automata theory with an application to a decision problem of second-order logic. Math. Systems Theory 2, 57–81 (1968)
Wilke, T.: An algebraic characterization of frontier testable tree languages. Theoret. Comput. Sci. 154, 85–106 (1996)
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Ésik, Z., Weil, P. (2003). On Logically Defined Recognizable Tree Languages. In: Pandya, P.K., Radhakrishnan, J. (eds) FST TCS 2003: Foundations of Software Technology and Theoretical Computer Science. FSTTCS 2003. Lecture Notes in Computer Science, vol 2914. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-24597-1_17
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DOI: https://doi.org/10.1007/978-3-540-24597-1_17
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