Termination and Confluence of Higher-Order Rewrite Systems

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.

Semantic Labelling for Proving Termination of Combinatory Reduction Systems

“…We need separately a way to prove termination of the labelled system. For this purpose, we use Blanqui's version of the General Schema for CRSs [Bla00] to prove termination of labelled CRSs because in our experience, this is the most powerful decidable method to prove termination of CRSs. The General Schema uses a precedence which is a partial order on function symbols occurring in a CRS.…”

“…Higher-order extensions of term rewriting systems [Ter03] are known as several formats: major representatives are CRSs, Higher-order Rewrite Systems [Nip91], and Inductive Data Type Systems [BJO02]. There exist several termination criteria: higher-order recursive path order (HORPO) [JR07], the General Schema [BJO02,Bla00], hereditary monotone functional interpretation [Pol94], binding algebra interpretation [Ham05]. Recently improvements of HORPO/General Schema are actively investigated [BR01,Raa01,JR06].…”

“…For CRSs, the General Schema is a decidable syntactic criteria of termination [Bla00]. The idea of the General Schema is to control the arguments of the right-hand side recursive calls in a rewrite rule by checking that they are smaller than the left-hand sides ones in the strict subterm order extended in a multiset or lexicographic manner.…”

Static Dependency Pair Method for Simply-Typed Term Rewriting and Related Techniques

“…We define σ as σ(x) = u if We suppose that C is this static recursion component as in Example 5.3. Suppose that π(sum n ) = [3] and π(drop) = [2]. Then the set U(C, π) consists of only three rules for drop described in Example 5.3.…”

Definitions by rewriting in the calculus of constructions

“…This schema has been reformulated and enhanced to deal with definitions on strictly-positive types [6], to higher-order patternmatching [3] and to richer type systems with objectlevel rewriting [1,5].…”