Termination of Cycle Rewriting by Transformation and Matrix Interpretation

TL;DR

This paper introduces transformation techniques and a matrix interpretation framework to automatically prove termination and non-termination of cycle rewriting systems, enhancing existing methods with soundness, completeness, and experimental validation.

Contribution

It presents three new transformations from cycle to string rewriting with proven soundness and completeness, and a unified matrix interpretation framework for termination analysis.

Findings

Transformations are sound and complete for cycle rewriting termination.

The matrix interpretation framework covers most existing approaches.

Experimental results demonstrate the effectiveness of the techniques.

Abstract

We present techniques to prove termination of cycle rewriting, that is, string rewriting on cycles, which are strings in which the start and end are connected. Our main technique is to transform cycle rewriting into string rewriting and then apply state of the art techniques to prove termination of the string rewrite system. We present three such transformations, and prove for all of them that they are sound and complete. In this way not only termination of string rewriting of the transformed system implies termination of the original cycle rewrite system, a similar conclusion can be drawn for non-termination. Apart from this transformational approach, we present a uniform framework of matrix interpretations, covering most of the earlier approaches to automatically proving termination of cycle rewriting. All our techniques serve both for proving termination and relative termination. We present several experiments showing the power of our techniques.