Termination and derivational complexity of confluent one-rule string-rewriting systems

It is not known whether the termination problem is decidable for one-rule string-rewriting systems, though the confluence of such systems is decidable by Wrathall (in: Word Equations and Related Topics, Lecture Notes in Computer Science, vol. 572, 1992, pp. 237-246). In this paper we develop techniques to attack the termination and complexity problems of confluent one-rule string-rewriting systems. With given such a system we associate another rewriting system over another alphabet. The behaviour of the two systems is closely related and the termination problem for the new system is sometimes easier than for the original system. We apply our method to systems of the special type {a(p)b(q) --> t}, where t is an arbitrary word over {a,b}, and give a complete characterization for termination. We also give a complete analysis of the derivational complexity for the system {a(p)b(q) --> b(n)a(m)). (C) 2001 Elsevier Science B.V. All rights reserved.