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International Conference on Automated Deduction

CADE 1988: 9th International Conference on Automated Deduction pp 517–526Cite as

Solving disequations in equational theories

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Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 310))

Abstract

Disunification is the problem to solve a system < s i = t i : 1 ≤ i ≤ n, p j ∩ q j : 1 ≤j ≤ m〉 of equations and disequations. Solutions are substitutions for the variables of the problem that make the two terms of each equation equal, but let those of the disequations different. We investigate this in the case, where equality is defined by an equational theory E. We show how E-disunification can be reduced to E-unification, that is solving equations only, and give a disunification algorithm for equational theories provided there is a unification algorithm. In fact this result shows that for theories, where the solutions of all unification problems can be represented by finitely many substitutions, there is also a finite representation of the solutions of all disunification problems. As an application we discuss how AC1-disunification can be used as a method to represent many AC-unifiers by a few AC1-disunifiers.

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References

  • W. Buntine: A Theory of Equations, Inequations, and Solutions for Logic Programming. New South Wales Institute of Technology, 1986.

    Google Scholar 

  • H.-J. Bürckert: Lazy Theory Unification in PROLOG: An Extension of the Warren Abstract Machine. Proc. of 10th German Workshop on Art. Intelligence, Springer, 1986, p. 277–288.

    Google Scholar 

  • H.-J. Bürckert: Matching — A Special Case of Unification? SEKI-Report, Universität Kaiserslautern, 1987.

    Google Scholar 

  • H.-J. Bürckert: Solving Disequations in Equational Theories. SEKI-Report, Universität Kaiserslautern, 1987.

    Google Scholar 

  • H.-J. Bürckert, A. Herold & M. Schmidt-Schauß: On Equational Theories, Unification, and Decidability. Proc. of 2nd Conf. on Rewriting Techniques and Applications, Springer, LNCS 256, 1987, p. 204–215; to appear also in J. of Symb. Comp., Special Issue on Unification (ed. C. Kirchner), 1987.

    Google Scholar 

  • H.-J. Bürckert, A. Herold, J. Siekmann, M. Stickel & M. Tepp: Opening the AC-Unification Race. In preparation, 1988

    Google Scholar 

  • W. Büttner: Unification in the Datastructure Multisets. J. of Automated Reasoning, Vol. 2, No. 1, 1986, p. 75–88.

    Google Scholar 

  • S. Burris, H.P. Sankappanavar: A Course in Universal Algebra. Springer, 1979.

    Google Scholar 

  • C.-L. Chang & R.C.-T. Lee: Symbolic Logic and Theorem Proving. Academic Press, 1973.

    Google Scholar 

  • A. Colmerauer: Equations and Inequations on Finite and Infinite Trees. Proc. of Conf. on Fifth Gen. Comp. Syst., ICOT, 1984, p. 85–99.

    Google Scholar 

  • H. Comon: Sufficient Completeness, Term Rewriting Systems, and Anti-unification. Proc. of Conf. on Automated Deduction, Springer LNCS 230, 1986, p. 128–140.

    Google Scholar 

  • H. Comon: Private communications. 1987/88.

    Google Scholar 

  • H. Comon: Unification et Disunification. Théorie et Applications. Thesis (in French), Université de Grenoble, 1988.

    Google Scholar 

  • H. Comon & P. Lescanne: Equational Problems and Disunification. Université de Grenoble and Centre de Rech.en Inform. de Nancy, 1988.

    Google Scholar 

  • F. Fages: Associative-Commutative Unification. Proc. of 7th Conf. on Automated Deduction, Springer, LNCS 170, 1984, p. 194–208.

    Google Scholar 

  • F. Fages & G. Huet: Complete Sets of Unifiers and Matchers in Equational Theories. Proc. of CAAP'83, Springer, LNCS 159, 1983, p. 205–220; see also J. of Theoret. Comp. Sci. 43, 1986, p. 189–200.

    Google Scholar 

  • A. Fortenbacher: An Algebraic Approach to Unification under Associativity and Commutativity. Proc. of Conf. on Rewriting Techniques and Applications, Springer, LNCS 202, 1985, p. 381–397.

    Google Scholar 

  • J. Gallier & S. Raatz: SLD-Resolution Methods for Horn Clauses with Equality Based on E-Unification. Proc. of Int. Symp. on Logic Programming, 1986

    Google Scholar 

  • J.A. Goguen & J. Meseguer: EQLOG — Equality, Types, and Generic Modules for Logic Programming. In: Logic Programming: Functions, Relations, and Equations. Prentice Hall, 1986, p. 295–363.

    Google Scholar 

  • G. Grätzer: Universal Algebra. Springer, 1979.

    Google Scholar 

  • A. Herold: Combination of Unification Algorithms. Proc. of 8th Conf. on Automated Deduction, Springer, LNCS 230, 1986.

    Google Scholar 

  • A. Herold: Combination of Unification Algorithms in Equational Theories. Dissertation, Universität Kaiserslautern, 1987.

    Google Scholar 

  • A. Herold & J.H. Siekmann: Unification in Abelian Semigroups. MEMO-SEKI, Universität Kaiserslautern, 1986.

    Google Scholar 

  • A. Herold, J.H. Siekmann & M.E. Stickel: Benchmarks for AC-Unification. Private communication, 1987.

    Google Scholar 

  • G. Huet & D.C. Oppen: Equations and Rewrite Rules: A Survey. In: Formal Languages: Perspectives and Open Problems. (ed. R. Book), Academic Press, 1980.

    Google Scholar 

  • J.M. Hullot: Compilation des Formes Canoniques dans des Théories Equationelles. Thэse (in French), Université de Paris-Sud, 1980

    Google Scholar 

  • J. Jaffar, J.-L. Lassez & M. Maher: Logic Programming Language Scheme. In: Logic Programming: Functions, Relations, Equations. (eds. D. DeGroot, G. Lindstrom), Prentice Hall, 1986.

    Google Scholar 

  • J.P. Jouannaud & H. Kirchner: Completion of a Set of Rules Modulo a Set of Equations. Proc. of 11th ACM Conf. on Principles of Programming Languages, 1984.

    Google Scholar 

  • C. Kirchner: Methodes et Outils de Conception Systematique d'Algorithmes d'Unification dans les Théories Equationnelles. Thèse de Doctorat d'Etat (in French), Université de Nancy, 1985.

    Google Scholar 

  • C. Kirchner & H. Kirchenr. Implementation of a General Completion Procedure Parametrized by Built-in Theories and Strategies. Proc. of EUROCAL Conf., 1985.

    Google Scholar 

  • C. Kirchner & P. Lescanne: Solving Disequations. Proc. IEEE 2nd Symp. on Logic in Comp. Sci., 1987.

    Google Scholar 

  • D. Lankford & R.M. Ballantyne: Decision Procedures for Simple Equational Theories with Commutative-Associative Axioms: Complete Sets of Commutative-Associative Reductions. Internal Report, University of Texas, Austin, 1977.

    Google Scholar 

  • J.-L. Lassez, M.J. Maher & K. Marriot: Unification Revisited. Technical Report, IBM Yorktown Heights, 1987.

    Google Scholar 

  • M. Livesey & J.H. Siekmann: Unification of AC-Terms (Bags) and ACI-Terms (Sets). Int. Report, Essex University, 1975, and Universität Karlsruhe, 1976.

    Google Scholar 

  • H.J. Ohlbach: Link Inheritance in Abstract Clause Graphs. J. of Automated Reasoning, Vol 3, No 1, 1987, p. 1–34.

    Article  Google Scholar 

  • G.E. Peterson & M.E. Stickel: Complete Sets of Reductions for Equational Theories with Complete Unification Algorithms. JACM, Vol 28, No 2, 1981, p. 322–364.

    Article  Google Scholar 

  • G. Plotkin: Building in Equational Theories. Machine Intelligence 7, 1972, p. 73–90.

    Google Scholar 

  • J.A. Robinson: A Machine Oriented Logic Based on the Resolution Principle. JACM, Vol 12, No 1, 1965, p. 23–41.

    Article  Google Scholar 

  • M. Schmidt-Schauß: Combination of Unification Algorithms in Arbitrary Disjoint Equational Theories. SEKI-Report, Universität Kaiserslautern, 1987, also in this proceedings.

    Google Scholar 

  • J.H. Siekmann: Unification and Matching Problems. PH.D. Thesis, Essex University, 1978.

    Google Scholar 

  • J.H. Siekmann: Unification Theory. A Survey. J. of Symb. Comp., Special Issue on Unification (ed. C. Kirchner), 1987.

    Google Scholar 

  • J.H. Siekmann & P. Szabo: The Undecidability of the DA-unification Problem. SEKI-Report SR-86-19, Universität Kaiserslautern, 1986.

    Google Scholar 

  • G. Smolka, W. Nutt, J.A. Goguen & J. Meseguer: Order-Sorted Equational Computation. SEKI-Report, Universität Kaiserslautern, 1987.

    Google Scholar 

  • M.E. Stickel: A Complete Unification Algorithm for Associative-Commutative Functions. Proc. of 4th Int. Joint Conf. on Art. Intelligence, Tblisi, 1975, p. 71–82.

    Google Scholar 

  • M.E. Stickel: Unification Algorithms for Artificial Intelligence. Ph. D. Thesis, Carnegie-Mellon University, 1976.

    Google Scholar 

  • M.E. Stickel: A Unification Algorithm for Associative-Commutative Functions. JACM, Vol 28, No 3, 1981, p. 423–434.

    Article  Google Scholar 

  • M.E. Stickel: A Case Study of Theorem Proving by the Knuth-Bendix Method Discovering that X 3 = X implies Ring Commutativity. Proc. of 7th Conf. on Automated Deduction, Springer, LNCS 170, 1984, p. 248–258.

    Google Scholar 

  • M.E. Stickel: A Comparison of the Variable-Abstraction and Constant-Abstraction Methods for Associative-Commutative Unification. J. of Automated Reasoning, Vol 3, No 3, 1987, p. 285–289.

    Google Scholar 

  • P. Szabo: Unifikationstheorie erster Ordnung. (In German), Dissertation, Universität Karlsruhe, 1982.

    Google Scholar 

  • W. Taylor: Equational Logic. Houston Journal of Mathematics 5, 1979.

    Google Scholar 

  • L. Wos, R. Overbeek, E. Lusk, J. Boyle: Automated Reasoning — Introduction and Applications. Prentice Hall, 1984.

    Google Scholar 

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Authors and Affiliations

  1. FB Informatik, Universität Kaiserslautern, Postfach 3049, D-6750, Kaiserslautern, FR Germany

    Hans-Jürgen Bürckert

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  1. Hans-Jürgen Bürckert
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Ewing Lusk Ross Overbeek

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© 1988 Springer-Verlag Berlin Heidelberg

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Bürckert, HJ. (1988). Solving disequations in equational theories. In: Lusk, E., Overbeek, R. (eds) 9th International Conference on Automated Deduction. CADE 1988. Lecture Notes in Computer Science, vol 310. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0012853

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  • DOI: https://doi.org/10.1007/BFb0012853

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-19343-2

  • Online ISBN: 978-3-540-39216-3

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