Termination and Confluence of Higher-Order Rewrite Systems

TL;DR

This paper extends Inductive Data Type Systems with higher-order pattern-matching, develops a termination criterion, and compares this unified approach with existing higher-order rewriting frameworks, enhancing tools for proving termination and confluence.

Contribution

It introduces simply-typed CRSs, extends the General Schema for termination proofs, and compares the approach with HRSs for confluence analysis.

Findings

Extended the General Schema to higher-order pattern-matching.

Proved the applicability of the schema to HRSs.

Demonstrated how to analyze confluence using Nipkow's critical pair technique.

Abstract

In the last twenty years, several approaches to higher-order rewriting have been proposed, among which Klop's Combinatory Rewrite Systems (CRSs), Nipkow's Higher-order Rewrite Systems (HRSs) and Jouannaud and Okada's higher-order algebraic specification languages, of which only the last one considers typed terms. The later approach has been extended by Jouannaud, Okada and the present author into Inductive Data Type Systems (IDTSs). In this paper, we extend IDTSs with the CRS higher-order pattern-matching mechanism, resulting in simply-typed CRSs. Then, we show how the termination criterion developed for IDTSs with first-order pattern-matching, called the General Schema, can be extended so as to prove the strong normalization of IDTSs with higher-order pattern-matching. Next, we compare the unified approach with HRSs. We first prove that the extended General Schema can also be applied to HRSs. Second, we show how Nipkow's higher-order critical pair analysis technique for proving local confluence can be applied to IDTSs.