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

TL;DR

This paper develops automated methods to prove or disprove the property of unique normal forms w.r.t. conversion (UNC) in term rewriting systems, extending existing tools and introducing new criteria, with implementation and experimental results.

Contribution

It introduces new automated criteria and methods for UNC, and implements them in the confluence prover ACP, advancing the automation of UNC verification.

Findings

Automated methods for UNC proof and disproof are developed.

New criteria for UNC are proposed and implemented.

Experimental evaluation demonstrates effectiveness of the methods.

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.