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 


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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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