A complete characterization of termination of 1q→ 1r 0s

We characterize termination of one-rule string rewriting systems of the form 0 p 1 q ¿ 1 r 0 s for every choice of positive integers p, q, r, and s. In doing so we introduce a termination proof method that applies to some hard examples. For the simply terminating cases, i.e. string rewriting systems that can be ordered by a division order, we give the precise complexity of derivation lengths. Keywords: String rewriting, Term rewriting, Termination, Simple termination, Transformation ordering, Dummy elimination, Derivation length