Term Graph Rewriting and Parallel Term Rewriting

TL;DR

This paper explores the parallel interpretation of term graph rewriting, formalizing how cyclic graphs relate to infinite parallel reductions in rational terms, clarifying the semantics of circular redexes.

Contribution

It formalizes the parallel interpretation of term graph rewriting using complete partial orders, providing a new perspective on reducing cyclic graphs and infinite redexes.

Findings

Parallel interpretation explains circular redex reduction in term graphs.

Formalization using complete partial orders clarifies semantics of infinite reductions.

Differentiates between sequential and parallel interpretations of term graph rewriting.

Abstract

The relationship between Term Graph Rewriting and Term Rewriting is well understood: a single term graph reduction may correspond to several term reductions, due to sharing. It is also known that if term graphs are allowed to contain cycles, then one term graph reduction may correspond to infinitely many term reductions. We stress that this fact can be interpreted in two ways. According to the "sequential interpretation", a term graph reduction corresponds to an infinite sequence of term reductions, as formalized by Kennaway et.al. using strongly converging derivations over the complete metric space of infinite terms. Instead according to the "parallel interpretation" a term graph reduction corresponds to the parallel reduction of an infinite set of redexes in a rational term. We formalize the latter notion by exploiting the complete partial order of infinite and possibly partial terms, and we stress that this interpretation allows to explain the result of reducing circular redexes in several approaches to term graph rewriting.