Abstract
Equational problems (i.e.: first-order formulae with quantifier prefix ∃* ∀*, whose only predicate symbol is syntactic equality) are an important tool in many areas of automated deduction, e.g.: restricting the set of ground instances of a clause via equational constraints allows the definition of stronger redundancy criteria and hence, in general, of more efficient theorem provers. Moreover, also the inference rules themselves can be restricted via constraints. In automated model building, equational problems play an important role both in the definition of an appropriate model representation and in the evaluation of clauses in such models. Also, many problems in the area of logic programming can be reduced to equational problem solving.
The goal of this work is a complexity analysis of the satisfiability problem of equational problems in CNF over an infinite Herbrand universe. The main result will be a proof of the NP-completeness (and, in particular, of the NP-membership) of this problem.
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References
F. Baader, J.H. Siekmann: Unification Theory, in Handbook of Logic in Artificial Intelligence and Logic Programming, D.M. Gabbay, C.J. Hogger, J.A. Robinson (eds.), Oxford University Press (1994).
H. Comon, C. Delor: Equational Formulae with Membership Constraints, Journal of Information and Computation, Vol 112, pp. 167–216 (1994).
H. Comon, P. Lescanne: Equational Problems and Disunification, Journal of Symbolic Computation, Vol 7, pp. 371–425 (1989).
R. Caferra, N. Peltier: Extending semantic Resolution via automated Model Building: applications, Proceedings of IJCAI’95, Morgan Kaufmann, (1995)
R. Caferra, N. Zabel: Extending Resolution for Model Construction, in Proceedings of Logics in AI-JELIA’90, LNAI 478, pp. 153–169, Springer (1991).
C. Fermüller, A. Leitsch: Hyperresolution and Automated Model Building, Journal of Logic and Computation, Vol 6 No 2, pp.173–230 (1996).
G. Gottlob, R. Pichler: Working with ARMs: Complexity Results on Atomic Representations of Herbrand Models, to appear in Proceedings of LICS’99, IEEE Computer Society Press, (1999).
J.-L. Lassez, K. Marriott: Explicit Representation of Terms defined by Counter Examples, Journal of Automated Reasoning, Vol 3, pp. 301–317 (1987).
J.-L. Lassez, M. Maher, K. Marriott: Elimination of Negation in Term Algebras, in Proceedings of MFCS’91, LNCS 520, pp. 1–16, Springer (1991).
D. Lugiez: A Deduction Procedure for First Order Programs, in Proceedings of ICLP’89, pp. 585–599, Lisbon (1989).
M. Maher: Complete Axiomatizations of the Algebras of Finite, Rational and Infinite Trees, in Proceedings of LICS’88, pp. 348–357, IEEE Computer Society Press, (1988).
A. Martelli, U. Montanari: An efficient unification algorithm, ACM Transactions on Programming Languages and Systems, Vol 4 No 2, pp. 258–282 (1982).
R. Pichler: Algorithms on Atomic Representations of Herbrand Models, in Proceedings of Logics in AI-JELIA’98, LNAI 1489, pp. 199–215, Springer (1998).
J.A. Robinson: A machine oriented logic based on the resolution principle, Journal of the ACM, Vol 12, No 1, pp. 23–41 (1965).
T. Sato, F. Motoyoshi: A complete Top-down Interpreter for First Order Programs, in Proceedings of ILPS’91, pp. 35–53, (1991).
S. Vorobyov: An Improved Lower Bound for the Elementary Theories of Trees, in Proceedings of CADE-13, LNAI 1104, pp. 275–287, Springer (1996).
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Pichler, R. (1999). Solving Equational Problems Efficiently. In: Automated Deduction — CADE-16. CADE 1999. Lecture Notes in Computer Science(), vol 1632. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-48660-7_7
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DOI: https://doi.org/10.1007/3-540-48660-7_7
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