Kruskal's Tree Theorem for Acyclic Term Graphs

TL;DR

This paper extends Kruskal's Tree Theorem to acyclic term graphs by defining a morphism-inspired embedding relation, establishing termination criteria for term graph rewriting.

Contribution

It introduces a novel embedding notion for acyclic term graphs and proves a version of Kruskal's Tree Theorem tailored for these structures.

Findings

Established a variant of Kruskal's Tree Theorem for acyclic term graphs

Proposed a new morphism-inspired embedding relation

Developed a lexicographic path order for acyclic term graphs

Abstract

In this paper we study termination of term graph rewriting, where we restrict our attention to acyclic term graphs. Motivated by earlier work by Plump we aim at a definition of the notion of simplification order for acyclic term graphs. For this we adapt the homeomorphic embedding relation to term graphs. In contrast to earlier extensions, our notion is inspired by morphisms. Based on this, we establish a variant of Kruskal's Tree Theorem formulated for acyclic term graphs. In proof, we rely on the new notion of embedding and follow Nash-Williams' minimal bad sequence argument. Finally, we propose a variant of the lexicographic path order for acyclic term graphs.