Unification algorithms cannot be combined in polynomial time

  • Miki Hermann
  • Phokion G. Kolaitis
Session 4A
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1104)


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.


Equational Theory Unification Algorithm Combination Algorithm Counting Problem Boolean Ring 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer-Verlag Berlin Heidelberg 1996

Authors and Affiliations

  • Miki Hermann
    • 1
    • 2
  • Phokion G. Kolaitis
    • 3
  1. 1.CNRSCRINVandœuvre-lès-NancyFrance
  2. 2.INRIA-LorraineVandœuvre-lès-NancyFrance
  3. 3.Computer and Information SciencesUniversity of California, Santa CruzSanta CruzUSA

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