On Term Rewriting Systems Having a Rational Derivation

TL;DR

This paper investigates the derivation relations of various classes of term rewriting systems, showing that bottom-up, top-down, and suffix systems have rational derivations, whereas prefix systems may not.

Contribution

It introduces a classification of term rewriting systems based on rule overlap and characterizes their derivation relations using rational relations and finite graph grammars.

Findings

Derivation of bottom-up, top-down, and suffix systems is rational.

Prefix systems can have non-recursive derivations.

Provides a finite mechanism to characterize derivations.

Abstract

Several types of term rewriting systems can be distinguished by the way their rules overlap. In particular, we define the classes of prefix, suffix, bottom-up and top-down systems, which generalize similar classes on words. Our aim is to study the derivation relation of such systems (i.e. the reflexive and transitive closure of their rewriting relation) and, if possible, to provide a finite mechanism characterizing it. Using a notion of rational relations based on finite graph grammars, we show that the derivation of any bottom-up, top-down or suffix systems is rational, while it can be non recursive for prefix systems.