Decidability of Term Algebras Extending Partial Algebras

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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.