E-unification with Constants vs. General E-unification

Abstract

We present a solution to Problem #66 from the RTA open problem list. The question is whether there exists an equational theory E such that Eunification with constants is decidable but general E-unification is undecidable. The answer is positive and we show such a theory. The problem has several equivalent formulations, therefore the solution has many consequences. Our result also shows, that there exist two theories E 1 and E 2 over disjoint signatures, such that E 1 -unification with constants and E 2 -unification with constants are decidable, but (E 1 ∪ E 2 )-unification with constants is undecidable.

Building and Combining Matching Algorithms

“…In the same vein, a similar question arises when comparing unification with free constants and general unification: is there an equational theory for which unification with free constants is decidable while general unification is not? A positive answer to this question is given in [44].…”