Degrees of Undecidability in Rewriting

TL;DR

This paper classifies the undecidability levels of various properties of first order term rewriting systems within the arithmetical and analytic hierarchies, revealing some properties are more complex than previously known.

Contribution

It provides a detailed hierarchy classification of undecidable properties of rewriting systems, including some properties shown to be beyond the arithmetical hierarchy.

Findings

Weak and strong normalization are Sigma-0-1-complete.

Dependency pair problems with minimality flag are Pi-0-2-complete.

Dependency pair problems without minimality flag are Pi-1-1-complete.

Abstract

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 Sigma-0-1-complete, while their uniform versions as well as dependency pair problems with minimality flag are Pi-0-2-complete. We find that confluence is Pi-0-2-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 Pi-0-2-complete while the latter is Sigma-0-1-complete (and thereby recursively enumerable). The most surprising result is on dependency pair problems without minimality flag: we prove this to be Pi-1-1-complete, which means that this property exceeds the arithmetical hierarchy and is essentially analytic.