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Some Notes on Basic Syntactic Mutation

Unification modulo convergent term rewrite systems is an important research area with many applications. In their seminal paper Lynch and Morawska gave three conditions on rewrite systems that guarantee that unifiability can be checked in polynomial time (P). We show that these conditions are tight, in the sense that relaxing any one of them will “upset the applecart,” giving rise to unification problems that are not in P (unless P = NP), and in doing so address an open problem posed by Lynch and Morawska. We also investigate a related decision problem: we show the undecidability of subterm-collapse for the restricted term rewriting systems that we are considering.