### Termination modulo equations by abstract commutation with an application to iteration

Wan Fokkink, Hans Zantema
<span title="">1997</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/elaf5sq7lfdxfdejhkqbtz6qoq" style="color: black;">Theoretical Computer Science</a> </i> &nbsp;
We generalize a termination theorem in term rewriting, based on an abstract commutation technique, to rewriting modulo equations. This result is applied in the setting of process algebra with iteration. * Revised version of 'Prefix iteration in basic process algebra: applying termination techniques', which appeared in the proceedings of the ACP'95 workshop in Eindhoven, April 1995. We present an extensive example of an application of abstract commutation modulo equations, which concerns the
more &raquo; ... ry operator x * y from Kleene [26], called Kleene star or iteration. The process term p * q can choose to execute either p, after which it evolves into p * q again, or q, after which it terminates. Milner [28] was the first to study the Kleene star in process algebra, modulo bisimulation equivalence from Park [29] . Recently, a paper by Bergstra, Bethke, and Ponse [10] has caused a resurgence in this line of research, mostly dealing with complete axiomatizations for iteration [20, 19] and with variants of iteration [18, 1, 2] . In process algebra, rewriting is usually applied in order to obtain normal forms for which the syntax and their semantics are closely related. In the case of iteration, such rewriting strategies have the tendency to produce self-embedding rewrite rules, where the right-hand side can be obtained from the left-hand side by the elimination of function symbols. For such a rewrite system, standard techniques to prove termination such as path orderings and weight functions in the natural numbers cannot be applied, see for example [32] . Hence, the only sensible strategy to prove termination of such a rewrite system is to transform it into a rewrite system without self-embedding rules, for which termination can be derived. Transformation techniques which can be applied for this purpose are based either on commutations [8, 34] or on semantic properties [24, 33] . In [20], a rewrite system for iteration in process algebra was applied, which contains self-embedding rules. It was proved to be terminating by the technique of semantic labelling from [33] . Here, we consider another rewrite system for iteration in process algebra, motivated by the aim to find normal forms for which the syntax and their semantics are closely related. Again this rewrite system contains self-embedding rules. We present an elegant proof that it is terminating, based on the technique of abstract commutation from [34] extended to rewriting modulo equations.
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