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Abstract

Rewriting is traditionally presented as a method to compute normal forms in varieties. Conceptually, however, its essence are commutation properties. We develop rewriting as a general theory of commutation for two possibly non-symmetric transitive relations modulo a congruence and prove a generalization of the standard Church-Rosser theorem. The theorems of equational rewriting, including the existence of normal forms, derive as corollaries to this result. Completion also is purely commutational and we show how to extend it to plain transitive relations. Nevertheless the loss of symmetry introduces some unpleasant consequences: unique normal forms do not exist, rewrite proofs cannot be found by don't-care nondeterministic rewriting and also simplification during completion requires backtracking. On the non-ground level, variable critical pairs have to be considered.