On weakly confluent monadic string-rewriting systems

Abstract

It is investigated as to how far the various decidability results for finite, monadic, and confluent string-rewriting systems can be carried over to the class of finite monadic string-rewriting systems that are only weakly confluent. Here a monadic string-rewriting system R on some alphabet Σ is called weakly confluent if it is confluent on all the congruence classes [ a] R , with a ∈ Σ ⋓ {e}. After establishing that the property of weak confluence is tractable for finite monadic string-rewriting systems, we prove that many decision problems that are tractable for finite, monadic, and confluent systems are, in fact, undecidable for finite monadic systems that are only weakly confluent. An example is the word problem. On the other hand, for finite, monadic, and weakly confluent systems that present groups, the validation problem for linear sentences is decidable. Many decision problems, among them the word problem and the generalized word problem, can be expressed through linear sentences and, hence, they all are decidable in this setting. The paper closes with a specialized completion procedure for finite, monadic string-rewriting systems presenting groups. Given a system of this form, the completion procedure tries to construct an equivalent system of the same form that, in addition, is weakly confluent. The correctness and completeness of this procedure are shown, and some detailed examples are presented. This procedure, together with the decidability results mentioned before, presents an elegant and uniform way to perform computations in context-free groups effectively.