It is Undecidable Whether the Knuth-Bendix Completion Procedure Generates a Crossed Pair

We reduce an instance of Turing machine acceptance to the problem of detecting whether the Knuth-Bendix completion procedure generates a crossed pair of rules. This resolves an open question posed in [5]. Our proof technique generalizes; using similar reductions, we can show that a number of other questions related to whether the Knuth-Bendix completion procedure generates certain types of rules are all undecidable. We suggest that the techniques illustrated herein may be useful in answering a number of related questions about the Knuth-Bendix completion procedure, and discuss several examples; in particular, we demonstrate how our construction provides a simple proof that the universal matching problem is undecidable for regular canonical theories, a result first proved in [4], and prove that the universal unification problem is undecidable for permutative canonical theories.