Termination Proofs for String Rewriting Systems via Inverse Match-Bounds

Annotating a letter by a number, one can record information about its history during a rewrite derivation. In each rewrite step, numbers in the reduct are updated depending on the redex numbering. A string rewriting system is called match-bounded if there is a global upper bound to these numbers. Match-boundedness is known to be a strong sufficient criterion for both termination and preservation of regular languages. We show that the string rewriting systems whose inverse (left and right hand sides exchanged) is match-bounded, also have exceptional properties, but slightly different ones. Inverse match-bounded systems need not terminate; they effectively preserve context-free languages; their sets of normalizable strings and their sets of immortal strings are effectively regular. These languages can be used to decide the normalization, the uniform normalization, the termination and the uniform termination problem for inverse match-bounded systems. We also prove that the termination problem is decidable in linear time, and that a certain strong reachability problem is decidable, thereby solving two open problems of McNaughton’s. Like match-bounds, inverse match-bounds entail linear derivational complexity on the set of terminating strings.