Decidability of Termination of Grid String Rewriting Rules

Termination of string rewriting is known undecidable. Termination of string rewriting with only one rule is neither known decidable nor known undecidable. This paper presents a decision procedure for rules $u\rightarrow v$ such that some letter b from u occurs as often or less often in v. We call such rules "grid" rules. By far most rules are grid rules. Grid rules cover all rules which terminate by a total division order. Thus total division orders are shown to be irrelevant for the termination problem of one-rule string rewriting.

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