Combination of unification algorithms

  • Alexander Herold
Unification Theory
Part of the Lecture Notes in Computer Science book series (LNCS, volume 230)


Unification in equational theories, i.e. solving equations in varieties, is a basic operation in many applications of Computer Science, particularly in Automated Deduction [Si 84]. A combination of unification algorithms for regular finitary collapse free equational theories with disjoint function symbols is presented. The idea is first to replace certain subterms by constants and to unify this constant abstraction and then to handle the replaced subterms in a recursive step. Total correctness is shown, i.e. the algorithm terminates and yields a correct and complete set of unifiers provided the special algorithms do.


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  1. [Bi 35]
    Birkhoff, G., ‘On the Structure of Abstract Algebra', Proc. Cambridge Phil. Soc., Vol. 31, 433–454, (1935)Google Scholar
  2. [BS 81]
    Burris, S. and Sankappanavar, H.P., ‘A Course in Universal Algebra', Springer-Verlag, (1981)Google Scholar
  3. [Fa 84]
    Fages, F., ‘Associative-Commutative Unification', in Proc. of 7th CADE (ed. R.E.Shostak). Springer-Verlag, LNCS 170, 194–208, (1984)Google Scholar
  4. [FH 83]
    Fages, F. and Huet, G., ‘Unification and Matching in Equational Theories', Proc. of CAAP'83 (ed. G. Ausiello and M. Protasi), Springer-Verlag, LNCS 159, 205–220, (1983)Google Scholar
  5. [GT 78]
    Goguen, J. A., Thatcher, J.W. and Wagner, E. G., ‘An Initial Algebra Approach to the Specification, Correctness and Implementation of Abstract Data Types', in ‘Current Trends in Programming Methodology, Vol.4, Data Structuring’ (ed. R. T. Yeh), Prentice Hall, (1978)Google Scholar
  6. [Gr 79]
    Grätzer, G., ‘Universal Algebra', Springer-Verlag, (1979)Google Scholar
  7. [He 85]
    Herold, A., ‘A Combination of Unification Algorithms', MEMO SEKI-85-VIII-KL, Universität Kaiserslautern, (1985)Google Scholar
  8. [HO 80]
    Huet, G. and Oppen, D. C., ‘Equations and Rewrite Rules: A Survey', in ‘Formal Languages: Perspectives and Open Problems (ed R. Book), Academic Press, (1980)Google Scholar
  9. [HS85]
    Herold, A. and Siekmann, J., ‘Unification in Abelian Semigroups', MEMO SEKI-85-III-KL, Universität Kaiserslautern, (1985)Google Scholar
  10. [Hu 76]
    Huet, G., ‘Résolution d'équations dans des langages d'ordre 1, 2... ω'. Thèse de doctorat d'état. Université Paris VII, (1976)Google Scholar
  11. [Ki 85]
    Kirchner, C., ‘Methodes et outils de conception systematique d'algorithmes d'unification dans les théories équationelles', Thèse de doctorat d'état, Université de Nancy 1, (1985)Google Scholar
  12. [Lo 78]
    Loveland, D., ‘Automated Theorem Proving', North-Holland, (1978)Google Scholar
  13. [LS 76]
    Livesey, M. and Siekmann, J., ‘Unification of Sets and Multisets', Universität Karisruhe, Techn. Report, (1976)Google Scholar
  14. [Mc 76]
    McNulty, G., ‘The Decision Problem for Equational Bases of Algebras', Annals of Mathematical Logic 10, 193–259, (1976)CrossRefGoogle Scholar
  15. [Ro 65]
    Robinson, J. A., ‘A Machine-Oriented Logic Based on the Resolution Principle', JACM, 12, No. 1, 23–41, (1965)CrossRefGoogle Scholar
  16. [Sc 86]
    Schmidt-Schauß, M., ‘Unification under Associativity and Idempotence is of Type Nullary', MEMO SEKI, Universität Kaiserslautern, (1986), (submitted to JAR)Google Scholar
  17. [Si 84]
    Siekmann, J., ‘Universal Unification', in Proc. of 7th CADE (ed R.E. Shostak), Springer-Verlag, LNCS 170, 1–42, (1984)Google Scholar
  18. [St 81]
    Stickel, M.E., ‘A Unification Algorithm for Associative-Commutative Functions', JACM 28, No. 3, 423–434, (1981)CrossRefGoogle Scholar
  19. [Ta 79]
    Taylor, W., ‘Equational Logic', Houston Journal of Mathematics 5, (1979)Google Scholar
  20. [Ti 85]
    Tidén, E., ‘Unification in Combinations of Theories with Disjoint Sets of Function Symbols', Royal Institute of Technology, Department of Computing Science, S-100 44 Stockholm, Sweden, (1985)Google Scholar
  21. [Ye 85]
    Yelick, K., ‘Combining Unification Algorithms for Confined Regular Equational Theories', in Proc. of ‘Rewriting Techniques and Applications’ (ed J.-P. Jouannaud), Springer-Verlag, LNCS 202, 365–380, (1985)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1986

Authors and Affiliations

  • Alexander Herold
    • 1
  1. 1.Fachbereich InformatikUniversität KaiserslauternKaiserslauternF.R.Germany

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