Automated Proofs of Unique Normal Forms w.r.t. Conversion for Term Rewriting Systems

Abstract

The notion of normal forms is ubiquitous in various equivalent transformations. Confluence (CR), one of the central properties of term rewriting systems (TRSs), concerns uniqueness of normal forms. Yet another such property, which is weaker than confluence, is the property of unique normal forms w.r.t. conversion (UNC). Famous examples having UNC but not CR include the TRSs consisting of S,K,I-rules for the combinatory logic supplemented with various pairing rules (de Vrijer, 1999). Recently, automated confluence proof of TRSs has caught attentions leading to investigations of automatable methods for (dis)proving CR of TRSs; some powerful confluence tools have been developed as well. In contrast, there have been little efforts on (dis)proving UNC automatically yet. Indeed, there are few tools that are capable of (dis)proving UNC; furthermore, only few UNC criteria have been elaborated in these tools. In this paper, we address automated methods to prove or disprove UNC of given TRSs. We report automation of some criteria known so far, and also present some new criteria and methods for proving or disproving UNC. Presented methods are implemented over the confluence prover ACP (Aoto et al., 2009) and an experimental evaluation is reported.