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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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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