The Joinability and Unification Problems for Confluent Semi-constructor TRSs

The unification problem for term rewriting systems (TRSs) is the problem of deciding, for aTFLS $R$ and two terms $s$ and $t$ , whether $s$ and $t$ are unifiable modulo $R$. Mitsuhashi et al. have shown that the problem is decidable for confluent simple TRSs. Here, a TRS is simple if the right-hand side of every rewrite rule is a ground term or a variable. In this paper, we extend this result and show that the unification problem for confluent semi-constructor TRSs is decidable. Here, a semi-constructor TRS is such aTRS that every subterm of the right-hand side of each rewrite rule is ground if its root is a defined symbol. We first show the decidability of joinability for confluent semi-constructor TRSs. Then, using the decision algorithm for joinabilty, we obtain a unification algorithm for confluent semi-constructor TRSs.