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International Conference on Automated Deduction

CADE 1996: Automated Deduction — Cade-13 pp 246–260Cite as

Unification algorithms cannot be combined in polynomial time

  • Session 4A
  • Conference paper
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Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 1104))

Abstract

We establish that there is no polynomial-time general combination algorithm for unification in finitary equational theories, unless the complexity class #P of counting problems is contained in the class FP of function problems solvable in polynomial-time. The prevalent view in complexity theory is that such a collapse is extremely unlikely for a number of reasons, including the fact that the containment of #P in FP implies that P=NP. Our main result is obtained by establishing the intractrability of the counting problem for general AG-unification, where AG is the equational theory of Abelian groups. Specifically, we show that computing the cardinality of a minimal complete set of unifiers for general AG-unification is a #P-hard problem. In contrast, AG-unification with constants is solvable in polynomial time. Since an algorithm for general AG-unification can be obtained as a combination of a polynomialtime algorithm for AG-unification with constants and a polynomial-time algorithm for syntactic unification, it follows that no polynomial-time general combination algorithm exists, unless #P is contained in FP.

Research of this author was partially supported by NSF Grants No. CCR-9307758

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Author information

Authors and Affiliations

  1. CNRS, CRIN, BP 239, 54506, Vandœuvre-lès-Nancy, France

    Miki Hermann

  2. INRIA-Lorraine, BP 239, 54506, Vandœuvre-lès-Nancy, France

    Miki Hermann

  3. Computer and Information Sciences, University of California, Santa Cruz, 95064, Santa Cruz, CA, USA

    Phokion G. Kolaitis

Authors
  1. Miki Hermann
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  2. Phokion G. Kolaitis
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Editor information

M. A. McRobbie J. K. Slaney

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© 1996 Springer-Verlag Berlin Heidelberg

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Hermann, M., Kolaitis, P.G. (1996). Unification algorithms cannot be combined in polynomial time. 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_89

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  • DOI: https://doi.org/10.1007/3-540-61511-3_89

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-61511-8

  • Online ISBN: 978-3-540-68687-3

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