Decidability of Term Algebras Extending Partial Algebras

Abstract

Let ${\cal A}$ be a partial algebra on a finite signature. We say that ${\cal A}$ has decidable query evaluation problem if there exists an algorithm that given a first order formula $\phi(\bar{x})$ and a tuple $\bar{a}$ from the domain of ${\cal A}$ decides whether or not $\phi(\bar{a})$ holds in ${\cal A}$. Denote by $E({\cal A})$ the total algebra freely generated by ${\cal A}$. We prove that if ${\cal A}$ has a decidable query evaluation problem then so does $E({\cal A})$. In particular, the first order theory of $E({\cal A})$ is decidable. In addition, if ${\cal A}$ has elimination of quantifiers then so does $E({\cal A})$ extended by finitely many definable selector functions and tester predicates. Our proof is a refinement of the quantifier elimination procedure for free term algebras. As an application we show that any finitely presented term algebra has a decidable query evaluation problem. This extends the known result that the word problem for finitely presented term algebras is decidable.