On Weakly Confluent Monadic String-Rewriting Systems

Madlener, K., P. Narendran, F. Otto and L. Zhang, On weakly confluent monadic string-rewriting systems, Theoretical Computer Science 113 (1993) 119-165. 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 z is called weakly confluent if it is confluent on all the congruence classes [a]s, with ao,Su {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 Correspondence to: F. Otto, Fachbereich Mathematik/Informatik, Gesamthochschule Kassel, Postfach 101380, 3500 Kassel, Germany. Email: otto($theory.informatik.uni-kassel.de. *This paper combines and extends the results of three papers that have been presented at the 8th Annual Symposium on Theoretical Aspects of Computer Science (STACS ‘91) at Hamburg. February 1991, at the 18th International Colloquium on Automata, Languages, and Programming (ICALP ‘91) at Madrid, July 1991, and at the 9th International Symposium on Applied Algebra, Algebraic Algorithms and ErrorCorrecting Codes (AAECC-9) at New Orleans, October 1991, and that can be found in the corresponding proceedings. This paper has been prepared while the third author was visiting at the Fachbereich Informatik, Universitat Kaiserslautern, during the winter semester 1991/92. 0304-3975/93/$06.00 c 1993-Elsevier Science Publishers B.V. All rights reserved 120 K. Madlener et al. 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.