The first-order theory of lexicographic path orderings is undecidable

We show, under some assumption on the signature, that the *This formula not viewable on a Text-Browser* fragment of the theory of any lexicographic path ordering is undecidable. This applies to partial and to total precedences. Our result implies in particular that the simplification rule of ordered completion is undecidable

Weak orthogonality implies confluence: the higher-order case

In this paper we prove confluence for weakly orthogonal Higher-Order Rewriting Systems. This generalises all the known `confluence by orthogonality' results

The undecidability of the first-order theories of one step rewriting in linear canonical systems

By reduction from the halting problem for Minsky's two-register machines we prove that there is no algorithm capable of deciding the EAAA-theory of one step rewriting of an arbitrary finite linear confluent finitely terminating term rewriting system (weak undecidability). We also present a fixed such system with undecidable EA...A-theory of one step rewriting (strong undecidability). This improves over all previously known results of the same kind