Termination of Cycle Rewriting

Abstract

String rewriting can not only be applied on strings, but also on cycles and even on general graphs. In this paper we investigate termination of string rewriting applied on cycles, shortly denoted as cycle rewriting, which is a strictly stronger requirement than termination on strings. Most techniques for proving termination of string rewriting fail for proving termination of cycle rewriting, but match bounds, arctic matrices and tropical matrices can be applied. Further we show how any terminating string rewriting system can be transformed to a terminating cycle rewriting system, preserving derivational complexity.

Efficient Completion of Weighted Automata

“…The matchbound method [4] proves termination of string rewriting (and recently, of cycle rewriting [12]). The method can be presented via automata with weights in the fuzzy semiring F [11].…”

Termination Analysis for Graph Transformation Systems

“…Example 5. Consider the following graph transformation system, adapted from cycle rewriting [12], which consists of the following graph transformation rules:…”

“…In this section we consider graph transformation systems of a specific string-like form, and show that termination of such systems reduces to termination of cycle rewriting [12], which is a variant of string rewriting.…”

“…From a theoretical point of view, it shows that graph transformation shares some characteristics (with respect to termination) with a string-based rewriting formalism, which motivates considering cycle rewriting as a step between graph transformation and term rewriting. In fact, [12] uses an approach similar to the type graph method for proving termination of cycle rewrite systems. For cycle rewrite systems finding termination arguments is easier because of the more restricted format.…”

Proving Termination of Graph Transformation Systems Using Weighted Type Graphs over Semirings

“…Still, adapting these techniques to graph transformation is often non-trivial. A helpful first step is often to modify these techniques to work with cycle rewriting [19,15], which imagines the two ends of a string to be glued together, so that rewriting is indeed performed on a cycle. In this paper we focus exclusively on uniform termination, i.e., there is only a set of graph transformation rules, but no fixed initial graph, and the question is whether the rules terminate on all graphs.…”