The article introduces a novel notion of lazy rewriting. By annotating argument positions as lazy, redundant rewrite steps are avoided, and the termination behaviour of a term rewriting system can be improved. Some transformations of rewrite rules enable an implementation using the same primitives as an implementation of eager rewriting. 1

Abstract. Kamperman and Walters proposed the notion of a simulation of one rewrite system by another one, whereby each term of the simulating rewrite system is related to a term in the original rewrite system. In this paper it is shown that if such a simulation is sound and complete and preserves termination, then the transformation of the original into the simulating rewrite system constitutes a correct step in the compilation of the original rewrite system. That is, the normal forms of a term in the original rewrite system can then be obtained by computing the normal forms of a related term in the simulating rewrite system. 1

We consider transformations of reduction systems in an abstract setting. We study some sets of correctness criteria for such transformations, adapt a notion of simulation proposed by Kamperman and Walters, and show that the resulting !-simulation behaves well w.r.t. the criteria. We apply our results in an investigation of a transformation proposed by Thatte, and prove that this transformation preserves semicompleteness for weakly persistent systems. 1 Introduction With the emergence of the use of Reduction Systems as semantical basis of programming paradigms, the use of transformation methods for Reduction Systems is growing. Some systems are more suitable for execution on a machine than others, for instance because they have a normalizing reduction strategy that is easily decidable, or because they are in a specific format. Transformation methods are then used to translate an arbitrary system into such a suitable system. Evidently, such a transformation should not be applied if it ...

Reduction Systems : : : : : : : : : : : : : : : : : : : : : : : : : : : : 3 2.3 Term Rewriting Systems : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 5 2.3.1 Terms : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 5 2.3.2 Rules : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 6 2.3.3 Subclasses : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 7 3 Correctness Criteria 9 3.1 Mappings : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 9 3.2 Preservation of Properties : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 10 3.3 Preservation of Reduction Graphs : : : : : : : : : : : : : : : : : : : : : : : : 12 4 Simulation 13 4.1 A Preorder : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 13 4.2 Preservation of Normal Forms : : : : : : : : : : : : : : : : : : : : : : : : : : : 14 4.3 Preservation of Confluence : : : : : : : : : : : : : : : : : : : : : : : :...

We define a lifting of the notion of transformation of term rewriting systems to arbitrary relational structures, that ties in with the simulation relations of concurrency theory. We illustrate the resulting concepts in an examination of Thatte's transformation of term rewriting systems. We refute an earlier claim that this transformation preserves confluence for weakly persistent systems. We conclude by proving the preservation of weak normalization, and of confluence in weakly normalizing systems and in nonoverlapping systems with linear subtemplates. 1