E-unification with Constants vs. General E-unification

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 E-unification 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 E1 and E2 over disjoint signatures, such that E1-unification with constants and E2-unification with constants are decidable, but (E1 ∪ E2)-unification with constants is undecidable.

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