Skip to main content
SpringerLink
Log in
Book cover

International Conference on Automated Deduction

CADE 1996: Automated Deduction — Cade-13 pp 261–274Cite as

Unification and matching modulo nilpotence

  • Session 4A
  • Conference paper
  • First Online:

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 1104))

Abstract

We consider equational unification and matching problems where the equational theory contains a nilpotent function, i.e., a function f satisfying f(x,x)=0 where 0 is a constant. Nilpotent matching and unification are shown to be JVP-complete. In the presence of associativity and commutativity, the problems still remain NP-complete. But when 0 is also assumed to be the unity for the function f, the problems are solvable in polynomial time. We also show that the problem remains in P even when a homomorphism is added. Second-order matching modulo nilpotence is shown to be undecidable.

Some of the results reported here are in partial fulfillment of Qing Guo's Ph.D. requirements, and will form part of his dissertation. Paliath Narendran was partially supported by NSF grant CCR-9404930.

We are currently working on an implementation of the algorithm for ACUNh-unification, as part of a unification workbench we are developing at the University at Albany (SUNY).

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

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. A. Aiken. Set constraints: results, applications and future directions. In: Proceedings of the Second Workshop on Principles and Practice of Constraints Programming (PPCP'94) Rosairo, Orcas Island, Washington, Lecture Notes in Computer Science 874 (Springer, Berlin, 1994) 326–335.

    Google Scholar 

  2. A. Aiken, D. Kozen, M. Vardi and E. Wimmers. The complexity of set constraints. In: Proceedings of the 1993 Conference on Computer Science Logic (CSL'93) Swansea, UK, Lecture Notes in Computer Science 832 (Springer, Berlin, 1993) 1–17.

    Google Scholar 

  3. F. Baader. Unification in commutative theories, Hilbert's basis theorem, and Gröbner bases. Journal of the ACM 40 (3) (1993) 477 503.

    Article  Google Scholar 

  4. F. Baader and K.U. Schultz. Unification in the union of disjoint equational theories: combining decision procedures. In: Proceedings of the Eleventh Conference on Automated Deduction (CADE-11), Saratoga Springs, New York, Lecture Notes in Artificial Intelligence 607 (Springer, Berlin, 1992) 50–65.

    Google Scholar 

  5. F. Baader and J.S. Siekmann. Unification Theory. In: Handbook of Logic in Artificial Intelligence and Logic Programming (D.M. Gabbay, C.J. Hogger, J.A. Robinson, eds.), (Oxford, 1994) 41–125.

    Google Scholar 

  6. H. Comon, M. Haberstrau, and J.-P. Jouannaud. Decidable problems in shallow equational theories. In: Proceedings of the Seventh Annual IEEE Symposium on Logic in Computer Science (LICS), Santa Cruz, California, (IEEE Computer Society, Washington, D.C., 1992) 255–265.

    Google Scholar 

  7. N. Dershowitz and J.-P. Jouannaud. Rewrite Systems. In: Handbook of Theoretical Computer Science, (Elsevier, Amsterdam, 1990) 245–320.

    Google Scholar 

  8. N. Dershowitz, S. Mitra, and G. Sivakumar. Decidable matching for convergent systems. In: Proceedings of the Eleventh Conference on Automated Deduction (CADE-11), Saratoga Springs, New York, Lecture Notes in Artificial Intelligence 607, (Springer, Berlin, 1992) 589–602.

    Google Scholar 

  9. M.R. Garey and D.S. Johnson. Computers and Intractability: A Guide to the Theory of NP-Completeness. (Freeman, San Francisco, 1979)

    Google Scholar 

  10. W.D., Goldfarb. The undecidability of the second-order unification problem, Theoretical Computer Science 13 (1981) 225–230.

    Article  Google Scholar 

  11. N. Heintze. Set-based analysis of ML programs. In: Proceedings of the 1994 ACM Conference on Lisp and Functional Programming, June 1994.

    Google Scholar 

  12. J.-M. Hullot. Canonical forms and unification. In: Proceedings of the Fifth Conference on Automated Deduction (CADE-5), Les Arcs, France, Lecture Notes in Computer Science 87, (Springer, Berlin, 1980) 318–334.

    Google Scholar 

  13. J.-P. Jouannaud and C. Kirchner. Solving Equations in Abstract Algebras: a Rule-Based Survey of Unification. In: Computational Logic: Essays in Honor of Alan Robinson, (J.-L. Lassez and G.D. Plotkin, Eds.), (MIT, Cambridge, Mass., 1991) 360–394.

    Google Scholar 

  14. E. Kaltofen, M.S. Krishnamoorthy and B.D. Saunders. Fast parallel computation of Hermite and Smith forms of polynomial matrices. SIAM Journal of Algebraic and Discrete Methods 8 (4) October 1987, 683–690.

    Google Scholar 

  15. D. Kapur and P. Narendran. Complexity of unification problems with associative-commutative operators. Journal of Automated Reasoning 9 (2) (1992) 261–288.

    Article  Google Scholar 

  16. D. Kapur and P. Narendran. NP-completeness of the set unification and matching problems. In: Proceedings of the Eighth Conference on Automated Deduction (CADE-8), Oxford, Lecture Notes in Computer Science 230 (Springer, Berlin, 1986) 489–495.

    Google Scholar 

  17. C. Kirchner. Computing unification algorithms. In: Proceedings of the First Annual IEEE Symposium on Logic in Computer Science (LICS), (IEEE Computer Society, Washington, D.C., 1986), 206–216.

    Google Scholar 

  18. A. Martelli and U. Montanari. An efficient unification algorithm ACM Transactions on Programming Languages and Systems Vol. 4, No. 2 (ACM. 1982), 258–282.

    Google Scholar 

  19. P. Narendran, F. Pfenning and R. Statman. On the unification problem for Cartesian Closed Categories. In: Proceedings of the Eighth Annual IEEE Symposium on Logic in Computer Science (LICS), Montreal, Canada, (IEEE Computer Society, Washington, D.C., 1986). (To appear in the Journal of Symbolic Logic).

    Google Scholar 

  20. W. Nutt. Unification in monoidal theories. In: Proceedings of the 10th International Conference on Automated Deduction (CADE-10), Kaiserslautern, West Germany, July 1990 (Springer LNCS 449) 618–632.

    Google Scholar 

  21. G.D. Plotkin. Building-in Equational Theories, In: Machine Intelligence 7, (B. Meltzer and D. Michie, Eds.), (Edinburgh, 1972), 73–90.

    Google Scholar 

  22. T.J. Schaefer. The complexity of satisfiability problems. In: Proceedings of the Tenth Annual ACM Symposium on Theory of Computing (STOC), (ACM, New York, 1978), 216–226.

    Google Scholar 

  23. M. Schmidt-Schauss. An algorithm for distributive unification. Interner Bericht 13/94, Fachbereich Informatik, Universität Frankfurt, Frankfurt, Germany, 1995.

    Google Scholar 

  24. D.A. Wolfram. The Clausal Theory of Types. Volume 21 of Cambridge Tracts in Theoretical Computer Science, (Cambridge, 1993).2

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Programming and Logics, Department of Computer Science, State University of New York at Albany, 12222, Albany, NY, USA

    Qing Guo & Paliath Narendran

  2. Department of Computer Science, The Australian National University, 0200, Canberra, ACT, Australia

    D. A. Wolfram

Authors
  1. Qing Guo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Paliath Narendran
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. D. A. Wolfram
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

M. A. McRobbie J. K. Slaney

Rights and permissions

Reprints and permissions

Copyright information

© 1996 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Guo, Q., Narendran, P., Wolfram, D.A. (1996). Unification and matching modulo nilpotence. In: McRobbie, M.A., Slaney, J.K. (eds) Automated Deduction — Cade-13. CADE 1996. Lecture Notes in Computer Science, vol 1104. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-61511-3_90

Download citation

  • DOI: https://doi.org/10.1007/3-540-61511-3_90

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-61511-8

  • Online ISBN: 978-3-540-68687-3

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation