Abstract
We consider the theoryT IT of infinite terms. The axioms for the theoryT IT are analogous to the Mal'cev's axioms for the theoryT IT of finite terms whose models are the locally free algebras. Recently Maher [Ma] has proved that the theoryT IT in a finite non singular signature plus the Domain Closure Axiom is complete. We give a description of all the complete extension ofT IT from which an effective decision procedure forT IT is obtained. Our approach considers formulas built up with syntactic terms containing pointers. Using such a technique, the analysis of the theoryT IT can be carried out in analogy with Mal'cev's analysis ofT FT. Our results follow from an effective quantifier elimination procedure for formulas with pointers.
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[Be] Belegradek, O.V.: Theory of models of locally free algebras. Tr. Inst. Mat.8, 3–25 (1988)
[BP] Bouscaren, E., Poizat, B.: Des belles paires aux beaux uples. J. Symb. Logic53(2), 434–442 (1988)
[Cl] Clark, K.L.: Negation as failure. In: Gallaire, H., Minker, J. (eds.) Logic and databases. New York: Plenum Press 1978
[Col] Colmerauer, A.: Equations and inequations on finite and infinite trees. FGCS'84 Proc., pp. 85–99 (1984)
[CL] Comon, H., Lescanne, P.: Equational problems and disunification. J. Sym. Comput.7, 375–426 (1989)
[Com] Comon, H.: Disunification: a Survey. Preprint Draft Version 1.5, 1990
[Co] Courcelle, B.: Fundamentals properties of infinite trees. Theor. Comput. Sci.25(2), 95–169 (1985)
[Fi] Fitting, M.R.: A Kripke-Kleene semantics for logic programs. J. Logic Program2, 295–312 (1985)
[Gu] Guessarian, I.: Algebraic semantics. In: Goos, G., Hartmanis, J. (ed.) (Lect. Notes Comput. Sci., vol. 99, pp. XI, 158) Berlin Heidelberg New York: Springer 1981
[JLM] Jaffar, J., Lassez, J.-L., Maher, M.J.: Prolog-II as an instance of the logic programming language scheme. In: Wirsing, M. (ed.) Formal descriptions of programming Concepts III, pp. 275–299. Amsterdam: North-Holland 1987
[Ku] Kunen, K.: Negation in logic programming. J. Logic Program4, 289–308 (1987)
[Ll] Lloyd, J.W.: Foundation of logic programming. Berlin Heidelberg New York: Springer 1984
[Ma] Maher, M.J.: Complete axiomatizations of the algebras of finite, infinite, and rational trees. Proc. 3rd IEEE Symp. Logic in Computer Science, pp. 348–357. Edinburgh 1988
[Mal1] Mal'cev, A.I.: On the elementary theories of locally free algebras. Sov. Math. Dokl.138, 768–771 (1961)
[Mal2] Mal'cev, A.I.: Axiomatizable classes of locally free algebras of various types. In: Wells, B.F. (ed.) The metamatematics of algebraic systems: collected papers, Studies in Logic, pp. 262–281. Amsterdam: North-Holland 1971
[MMP] Mancarella, P., Martini, S., Pedreschi, D.: Complete logic programs with domain-closure-axiom. J. Logic Program5, 263–276 (1988)
[Op] Oppen, D.C.: Reasoning about recursively defined data structures. J. ACM27(3), 403–411 (1980)
[Ra] Rabin, M.O.: Decidable theories. In: Barwise, J. (ed.) Handbook of Mathematical Logic, Studies in Logic, vol. 90, pp. 595–630. Amsterdam: North-Holland 1977
[Sh] Shepherdson, J.C.: Introduction to the theory of logic programming. In: Drake, F.R., Truss, J.K. (ed.) Logic Colloquium '86, Studies in Logic, vol. 124, pp. 277–318. Amsterdam: North-Holland 1988
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Marongiu, G., Tulipani, S. Quantifier elimination for infinite terms. Arch Math Logic 31, 1–17 (1991). https://doi.org/10.1007/BF01370691
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DOI: https://doi.org/10.1007/BF01370691