Degrees of undecidability in rewriting

Undecidability of various properties of first order term rewriting systems is well-known. An undecidable property can be classified by the complexity of the formula defining it. This gives rise to a hierarchy of distinct levels of undecidability, starting from the arithmetical hierarchy classifying properties using first order arithmetical formulas and continuing into the analytic hierarchy, where also quantification over function variables is allowed. In this paper we consider properties of first order term rewriting systems and classify them in this hierarchy. Weak and strong normalization for single terms turn out to be S01-complete, while their uniform versions as well as dependency pair problems with minimality flag are ¿02-complete. We find that confluence is ¿02-complete both for single terms and uniform. Unexpectedly weak confluence for ground terms turns out to be harder than weak confluence for open terms. The former property is ¿02-complete while the latter is S01-complete (and thereby recursively enumerable). The most surprising result is on dependency pair problems without minimality flag: we prove this to be ¿11-complete, which means that this property exceeds the arithmetical hierarchy and is essentially analytic.