Modularity of Convergence and Strong Convergence in Infinitary Rewriting

TL;DR

This paper proves that both convergence and strong convergence are modular properties in non-collapsing infinitary term rewriting systems under certain metric conditions, extending previous results beyond the metric infinity case.

Contribution

It establishes the modularity of convergence and strong convergence in non-collapsing infinitary term rewriting systems with granular metrics, broadening existing theoretical frameworks.

Findings

Convergence is modular in non-collapsing systems with granular metrics.

Strong convergence is also modular under the same conditions.

Results extend known modularity beyond metric infinity.

Abstract

Properties of Term Rewriting Systems are called modular iff they are preserved under (and reflected by) disjoint union, i.e. when combining two Term Rewriting Systems with disjoint signatures. Convergence is the property of Infinitary Term Rewriting Systems that all reduction sequences converge to a limit. Strong Convergence requires in addition that redex positions in a reduction sequence move arbitrarily deep. In this paper it is shown that both Convergence and Strong Convergence are modular properties of non-collapsing Infinitary Term Rewriting Systems, provided (for convergence) that the term metrics are granular. This generalises known modularity results beyond metric \infty.