Is the validity of some instance of an equational problem decidable?
This problem is known as rigid E-unification. This problem is NP-complete. The first proof was given by Gallier, Narendran, Plaisted and Snyder [1]. 

However a related problem, the Simultaneous rigid E-unification that is to find a substitution that simultaneously satisfies several instances of rigid E-unification is undecidable. This result was proved by Degtyarev and Voronkov [2].

References


Jean H. Gallier, Paliath Narendran, David A. Plaisted, Wayne Snyder: Rigid E-Unification: NP-Completeness and Applications to Equational Matings Inf. Comput. 87(1/2): 129-195 (1990)
Anatoli Degtyarev, Andrei Voronkov: The Undecidability of Simultaneous Rigid E-Unification. Theor. Comput. Sci. 166(1&amp;2): 291-300 (1996)