Some remarks on derivations in general rewriting systems

Abstract

We study a class of general rewriting system derivations called canonical derivations. For context-free rewriting systems, these derivations are in one-to-one correspondence with the usual structural descriptions. Two derivations are similar if one can be obtained from the other by trivial rule rearrangement. We show that every similarity class of derivations includes at most one canonical derivation and that only in general rewriting systems allowing nontrivial derivations from A to A, does there fail to exist a canonical derivation for each similarity class. Whether nontrivial A to A derivations are possible in a general rewriting system is, in general, undecidable, but they are never possible in semi-Thue systems. Transformations preserving canonical derivations are considered, one having the interesting property that, in a certain sense, it modifies a general rewriting system so that only its canonical derivations remain. We also obtain a representation of each recursively enumerable set as the homomorphic image of the intersection of a context-free language and the right-cancelling language. A few remarks about parsing are included.