Solving Equational Problems Efficiently

  • Reinhard Pichler
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1632)


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.


Logic Programming Free Variable Transformation Rule Tree Representation Predicate Symbol 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    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).Google Scholar
  2. 2.
    H. Comon, C. Delor: Equational Formulae with Membership Constraints, Journal of Information and Computation, Vol 112, pp. 167–216 (1994).zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    H. Comon, P. Lescanne: Equational Problems and Disunification, Journal of Symbolic Computation, Vol 7, pp. 371–425 (1989).MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    R. Caferra, N. Peltier: Extending semantic Resolution via automated Model Building: applications, Proceedings of IJCAI’95, Morgan Kaufmann, (1995)Google Scholar
  5. 5.
    R. Caferra, N. Zabel: Extending Resolution for Model Construction, in Proceedings of Logics in AI-JELIA’90, LNAI 478, pp. 153–169, Springer (1991).Google Scholar
  6. 6.
    C. Fermüller, A. Leitsch: Hyperresolution and Automated Model Building, Journal of Logic and Computation, Vol 6 No 2, pp.173–230 (1996).zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    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).Google Scholar
  8. 8.
    J.-L. Lassez, K. Marriott: Explicit Representation of Terms defined by Counter Examples, Journal of Automated Reasoning, Vol 3, pp. 301–317 (1987).zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    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).Google Scholar
  10. 10.
    D. Lugiez: A Deduction Procedure for First Order Programs, in Proceedings of ICLP’89, pp. 585–599, Lisbon (1989).Google Scholar
  11. 11.
    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).Google Scholar
  12. 12.
    A. Martelli, U. Montanari: An efficient unification algorithm, ACM Transactions on Programming Languages and Systems, Vol 4 No 2, pp. 258–282 (1982).zbMATHCrossRefGoogle Scholar
  13. 13.
    R. Pichler: Algorithms on Atomic Representations of Herbrand Models, in Proceedings of Logics in AI-JELIA’98, LNAI 1489, pp. 199–215, Springer (1998).Google Scholar
  14. 14.
    J.A. Robinson: A machine oriented logic based on the resolution principle, Journal of the ACM, Vol 12, No 1, pp. 23–41 (1965).zbMATHCrossRefGoogle Scholar
  15. 15.
    T. Sato, F. Motoyoshi: A complete Top-down Interpreter for First Order Programs, in Proceedings of ILPS’91, pp. 35–53, (1991).Google Scholar
  16. 16.
    S. Vorobyov: An Improved Lower Bound for the Elementary Theories of Trees, in Proceedings of CADE-13, LNAI 1104, pp. 275–287, Springer (1996).Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • Reinhard Pichler
    • 1
  1. 1.Technische Universität WienGermany

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