Bounded, Strongly Sequential and Forward-Branching Term Rewriting Systems

Abstract

The class of Strongly Sequential Term Rewriting Systems (SS) was defined in [Huet & Lévy (1979)]. [Strandh (1989)] defined the class of bounded TRSs (B). As a subset of B, Strandh defined the class of forward-branching TRSs (FB); FB strictly contains the class of Strongly-Left Sequential TRSs (SLS) defined by [Hoffmann & O'Donnell (1982)] for their Equational Programming System. For SLS, Hoffmann and O'Donnell found efficient algorithms to compute normal forms. Strandh showed that as efficient algorithms exist for FB.B is defined in terms of the existence of a deterministic pattern matching automaton called an index tree and FB in terms of the existence of a forward-branching index tree. Two open problems set by Strandh were to characterise FB in a simpler way and to find an algorithm to build a forward-branching index tree.This article contains three main parts. In the first part, we introduce the Strongly Sequential class and the Bounded class and we check that B = SS. This insures that FB ⊂ SS and relates Strandh's work to all the works initiated by [Huet & Lévy (1979)]. The second part contains the main result of this article: we give a very simple characterisation of FB. Our proof of the characterisation is constructive; it's then straightforward to extend it to an algorithm that builds a forward-branching index tree. In the third part we give the algorithm and show that it runs in quadratic time.