Undecidable properties of syntactic theories

Abstract

Since we are looking for unification algorithms for a large enough class of equational theories, we are interested in syntactic theories because they have a nice decomposition property which provides a very simple unification procedure. A presentation is said resolvent if any equational theorem can be proved using at most one equality step at the top position. A theory which has a finite and resolvent presentation is called syntactic. In this paper we give decidability results about open problems in syntactic theories: unifiability in syntactic theories is not decidable, resolventness of a presentation and syntacticness of a theory are even not semidecidable. Therefore we claim that the condition of syntacticness is too weak to get unification algorithms directly.