Skip to main content
SpringerLink
Log in

Opening the AC-unification race

  • Problem Corner
  • Published:
Journal of Automated Reasoning Aims and scope Submit manuscript

Abstract

This note reports about the implementation of AC-unification algorithms, based on the variable-abstraction method of Stickel and on the constant-abstraction method of Livesey, Siekmann, and Herold. We give a set of 105 benchmark examples and compare execution times for implementations of the two approaches. This documents for other researchers what we consider to be the state-of-the-art performance for elementary AC-unification problems.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Boudet, A., Jouannaud, J.-P., and Schmidt-Schauß, M., ‘Unification in Boolean rings and Abelian groups’, Proc. of Conf. on Logic in Computer Science, Edinburgh, to appear (1988).

  2. Boy de la Tour, T., Caferra, R., and Chaminade, G., ‘Some tools for an inference laboratory (ATINF)’, Proc. of 9th Int. Conf. on Automated Deduction, Springer LNCS 310, 744–745 (1988).

    Google Scholar 

  3. Bückert, H.-J., ‘Lazy theory unification in PROLOG: an extension of the Warren abstract machine’, Proc. of German Workshop on Artificial Intelligence, Springer Fachberichte 124, 277–288 (1986).

    Google Scholar 

  4. Bürckert, H.-J., ‘Solving disequations in equational theories’, Proc. of 9th Int. Conf. on Automated Deduction, Springer LNCS 310, 517–526 (1988). See also: SEKI-Report SR-87-15, Universität Kaiserslautern (1987).

    Google Scholar 

  5. Bürckert, H.-J., Herold, A., and Schmidt-Schauß, M., ‘On equational theories, unification, and decidability’, Proc. of Int. Conf. on Rewriting Techniques and Applications, Springer LNCS 256, 204–215 (1987). See also J. of Symb. Comp., Kirchner, C. (ed.), Special Issue on Unification, to appear (1988).

    Google Scholar 

  6. Büttner, W., ‘Unification in the datastructure multisets’, J. of Automated Reasoning 2, 75–88 (1986).

    Google Scholar 

  7. Clausen, M. and Fortenbacher, A., ‘Efficient solution of linear Diophantine equations’, Internal Report 32/87, Universität Karlsruhe (1987).

  8. Eisinger, N. and Ohlbach, H. J., ‘The Markgraf Karl refutation procedure’, in Proc. of 8th Int. Conf. on Automated Deduction, Springer LNCS 230, 682–683 (1986).

  9. Fages, F., ‘Formes Canoniques dans les algèbres booléennes, et application à la démonstration automatique en logique de premier ordre’, Thèse de 3éme Cycle (in French), Université Paris VI (1983).

  10. Fages, F., ‘Associative-commutative unification’, Proc. of 7th Int. Conf. on Automated Deduction, Springer LNCS 170, 194–208 (1984). See also J. Symb. Comb. 3 (1987).

    Google Scholar 

  11. Fages, F. and Huet, G., ‘Complete sets of unifiers and matchers in equational theories’, Proc. of CAAP'83, Springer LNCS 159, 205–220 (1983). See also: J. Theoret. Comp. Sci. 43, 189–200 (1986).

    Google Scholar 

  12. Fortenbacher, A., ‘Algebraische Unifikation’, Diplomarbeit (in German), Universität Karlsruhe (1983).

  13. Fortenbacher, A., ‘An Algebraic Approach to Unification under Associativity and Commutativity’, Proc. of Int. Conf. on Rewriting Techniques and Applications, Springer LNCS 202, 381–397 (1985). See also J. Symb. Comp. 3, 217–229 (1987).

    Google Scholar 

  14. Franzen, M. and Henschen, L. J., ‘A new approach to universal unification and its application to AC-unification’, Proc. of 9th Int. Conf. on Automated Deduction, Springer LNCS 310, 643–657 (1988).

    Google Scholar 

  15. Fribourg, L., ‘A superposition oriented theorem prover’, J. Theoret. Comp. Sci. 35, 124–164 (1985).

    Google Scholar 

  16. Gallier, J. and Raatz, S., ‘SLD resolution methods for Horn clauses with equality based on E-unification’, Proc. of Symp. on Logic Programming, p. 168–179 (1986).

  17. Goguen, J. A. and Meseguer, J., ‘EQLOG — Equality, types and generic modules for logic programming’, in D. DeGroot and G. Lindstrom (eds.), Logic Programming: Functions, Relations, and Equations, Prentice Hall, p. 295–363 (1986).

  18. Guckenbiehl, Th. and Herold, A., ‘Solving linear Diophantine equations’, MEMO-SEKI 85-IV-KL, Universität Kaiserslautern (1985).

  19. Herold, A., ‘Combination of unification algorithms’, Proc. of 8th Int. Conf. on Automated Deduction, Springer LNCS 230, 450–469 (1986).

    Google Scholar 

  20. Herold, A., ‘Combination of unification algorithms in equational theories’, Dissertation, Universität Kaiserslautern (1987).

  21. Herold, A. and Siekmann, J. H., ‘Unification in Abelian semigroups’, J. Automated Reasoning 3, 247–284 (1987).

    Google Scholar 

  22. Hsiang, J., ‘Topics in automated theorem proving and program generation’, Ph.D. Thesis, University of Illinois at Urbana-Champaign (1982).

  23. Hsiang, J., ‘Two results in term rewriting theorem proving’, Proc. of Int. Conf. on Rewrite Techniques and Applications, Springer LNCS 202, 301–324 (1985).

    Google Scholar 

  24. Hsiang, J. and Dershowitz, N., ‘Rewrite methods for clausal and non-clausal theorem proving’, Proc. of 10th ETACS Int. Coll. on Automata, Languages, and Programming (ICALP) (1983).

  25. Huet, G., ‘An algorithm to generate the basis of solutions to homogeneous linear Diophantine equations’, Information Processing Lett. 7, 144–147 (1978).

    Google Scholar 

  26. Huet, G. and Oppen, D. C., ‘Equations and rewrite rules. A survey’, in R. Book (ed.), Formal Languages: Perspectives and Open Problems, Academic Press (1980).

  27. Hullot, J. M., ‘Compilation des formes canoniques dans des théories equationelles’, Thèse du 3ème Cycle (in French), Université de Paris-Sud (1980).

  28. Jaffar, J., Lassez, J.-L. and Maher, M., ‘Logic programming language scheme’, in: D. DeGroot and G. Lindstrom (eds.), Logic Programming: Functions, Relations, Equations, Prentice-Hall (1986).

  29. Jouannaud, J.-P. and Kirchner, H., ‘Completion of a set of rules modulo a set of equations’, Proc. of 11th ACM Conf. on Principles of Programming Languages (1984).

  30. Kapur, D. and Narendran, P., ‘An equational approach to theorem proving in first-order predicate calculus’, Proc. of 7th Int. Joint Conf. on Artificial Intelligence, Los Angeles, pp. 1146–1153 (1985).

  31. Kapur, D. and Zhang, H., ‘RRL: A Rewrite Rule Laboratory — A user's manual’, General Electric (1987).

  32. Kapur, D. and Zhang, H., ‘RRL: A Rewrite Rule Laboratory’, Proc. of 9th Int. Conf. on Automated Deduction, Springer LNCS 310, 768–769 (1988).

    Google Scholar 

  33. Kirchner, C., ‘Methodes et outils de conception systematique d'algorithmes d'unification dans les théories equationelle’, Thèse de Doctorat d'Etat (in French), Université de Nancy (1985).

  34. Kirchner, C. (ed.), ‘Special issue on unification’, J. Symb. Comp., to appear (1988).

  35. Kirchner, C. and Kirchner, H., ‘Implementation of a general completion procedure parametrized by built-in theories and strategies’, Proc. of EUROCAL Conf. (1985).

  36. Lankford, D., ‘A new non-negative integer basis algorithm for linear homogeneous equations with integer coefficients’, unpublished (1985).

  37. Lankford, D. and Ballantyne, R. M., Decision procedures for simple equational theories with commutative-associative axioms: Complete sets of commutative-associative reductions, Internal Report ATP-39, University of Texas, Austin (1977).

    Google Scholar 

  38. Lincoln, P. and Christian, J., ‘Adventures in associative-commutative unification (a summary)’, Proc. of 9th Int. Conf. on Automated Deduction, Springer LNCS 310, 358–367 (1988).

    Google Scholar 

  39. Livesey, M. and Siekmann, J. H., Unification of AC-terms (bags) and ACI-terms (sets)’, Internal Report, University of Essex (1975) and Universität Karlsruhe (1976).

  40. MKRP, ‘The Markgraph Karl Refutation Procedure’, Internal Report, Universität Kaiserslautern (1984).

  41. Müller, J., ‘THEOPOGLES — A theorem prover based on first-order polynomials and a special Knuth-Bendix procedure’, Proc. of German Workshop on Artificial Intelligence, Springer Fachberichte 152, 241–250 (1987).

    Google Scholar 

  42. Ohlbach, H. J., ‘Link inheritance in abstract clause graphs’, J. Automated Reasoning 3, 1–34 (1987).

    Google Scholar 

  43. Peterson, G. E. and Stickel, M. E., ‘Complete sets of reductions for equational theories with complete unification algorithms’, JACM 28, 322–364 (1981).

    Google Scholar 

  44. Plotkin, G., ‘Building in equational theories’, Machine Intelligence 7, 73–90 (1972).

    Google Scholar 

  45. Robinson, J. A., ‘A machine oriented logic based on the resolution principle’, JACM 12, 23–41 (1965).

    Google Scholar 

  46. Schmidt-Schauß, M., ‘Combination of arbitrary disjoint equational theories’, Proc. of 9th Int. Conf. on Automated Deduction, Springer LNCS 310, 378–396 (1988). See also: J. Symb. Comp., Kirchner, C. (ed.), Special Issue on Unification, to appear (1988).

    Google Scholar 

  47. Siekmann, J. H., ‘Unification and matching problems’, Ph.D. Thesis, Essex University (1978).

  48. Siekmann, J. H., ‘Unification theory. A survey’, J. Symb. Comp., C. Kirchner (ed.), Special Issue on Unification, to appear (1988).

  49. Smolka, G., Nutt, W., Goguen, J. A., and Meseguer, J., ‘Order-sorted equational computation’, Proc. of CREAS Workshop, Austin, Texas, to appear (1987).

  50. Stickel, M. E., ‘A complete unification algorithm for associative-commutative functions’, Proc. of 4th Int. Joint Conf. on Artificial Intelligence, Tblisi, p. 71–82 (1975).

  51. Stickel, M. E., ‘Mechanical theorem proving and artificial intelligence languages’, Ph.D. Thesis, Carnegie-Mellon University (1977).

  52. Stickel, M. E., ‘A unification algorithm for associative-commutative functions’, JACM 28, 423–434 (1981).

    Google Scholar 

  53. Stickel, M. E., ‘A case study of theorem Proving by the Knuth-Bendix method discovering that X 3=X implies ring commutativity’, Proc. of 7th Int. Conf. on Automated Deduction, Springer LNCS 170, 248–258 (1984).

    Google Scholar 

  54. Stickel, M. E., ‘Automated deduction by theory resolution’, J. Automated Reasoning 1, 333–357 (1985).

    Google Scholar 

  55. Stickel, M. E., ‘A comparison of the variable-abstraction and constant-abstraction methods for associative-commutative unification’, J. Automated Reasoning 3, 285–289 (1987).

    Google Scholar 

  56. Tiden, E., ‘Unification in combinations of equational theories’, Ph.D. Thesis, Stockholm (1986).

  57. Tiden, E., ‘Unification in combinations of collapse-free theories with disjoint sets of function symbols’, Proc. of 8th Int. Conf. on Automated Deduction, Springer LNCS 230, 431–450 (1986).

    Google Scholar 

  58. Yellick, K., ‘Combining unification algorithms for confined regular equational theories’, Proc. of Int. Conf. on Rewriting Techniques and Applications, Springer LNCS 202, 365–380 (1985).

    Google Scholar 

  59. Zhang, H., ‘An efficient algorithm for simple Diophantine equations’, Technical Report 87-26, Dept. of Computer Sciences, RPI (1987).

Download references

Author information

Authors and Affiliations

  1. FB Informatik, Universität Kaiserslautern, Postfach 3049, D-6750, Kaiserlautern, F.R. Germany

    Hans-Jürgen Bürckert

  2. European Computer-Industry Research Centre, Arabellastr. 17, D-8000, München, F.R. Germany

    Alexander Herold

  3. Dept. of Computer Science, State University of New York at Albany, 12222, Albany, NY, U.S.A.

    Deepak Kapur

  4. FB Informatik, Universität Kaiserlautern, Postfach 3049, D-6750, Kaiserslautern, F.R. Germany

    Jörg H. Siekmann

  5. Artificial Intelligence Center, SRI International, 94025, Menlo Park, CA, U.S.A.

    Mark E. Stickel

  6. FB Informatik (AG Siekmann), Universität Kaiserlautern, Postfach 3049, D-6750, Kaiserslautern, F.R. Germany

    Michael Tepp

  7. Department of Computer Science, Rensselaer Polytechnic Institute, 12180, Troy, NY, U.S.A.

    Hantao Zhang

Authors
  1. Hans-Jürgen Bürckert
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Alexander Herold
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Deepak Kapur
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Jörg H. Siekmann
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Mark E. Stickel
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. Michael Tepp
    View author publications

    You can also search for this author in PubMed Google Scholar

  7. Hantao Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bürckert, HJ., Herold, A., Kapur, D. et al. Opening the AC-unification race. J Autom Reasoning 4, 465–474 (1988). https://doi.org/10.1007/BF00297251

Download citation

  • Received:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF00297251

Key words

Search

Navigation