Abstract. The growing context-sensitive languages have been classified through the shrinking two-pushdown automaton, the deterministic version of which characterizes the class of generalized Church-Rosser languages (Buntrock and Otto 1995). Exploiting this characterization we prove that this latter class coincides with the class of Church-Rosser languages that was introduced by McNaughton, Narendran, and Otto (1988). Based on this result several open problems of McNaughton et al can be answered.
IntroductionIf R is a finite and length-reducing string-rewriting system on some finite alphabet ~, then there exists a linear-time algorithm that, given a string w E ~U* as input, computes an irreducible descendant w0 of w with respect to the reduction relation -+~ that is induced by R [2,3]. If, in addition, the system R is confluent, then the irreducible descendant wo is uniquely determined by w. Hence, in this situation two strings u and v are congruent modulo the Thue congruence +-~ induced by R if and only if their respective irreducible descendants u0 and v0 coincide. Thus, the word problem for a finite, length-reducing, and confluent string-rewriting system is decidable in linear time.Motivated by this result McNaughton, Narendran, and Otto [11] introduced the notion of a Church-Rosser language. A Church-Rosser language L C 2Y* is given through a finite, length-reducing, and confluent string-rewriting system R on some alphabet F properly containing XT, two irreducible strings tl,t2 E (F \ ~)*, and an irreducible letter Y E F \ Z satisfying the following condition for all strings w E ,U*: w E L if and only if tlWt2 -~ Y. Hence, the membership problem for a Church-Rosser language is decidable in linear time, and so the class CRL of Church-Rosser languages is contained in the class CSL of contextsensitive languages.On the other hand, the class CRL contains the class OCFL of deterministic context-free languages, and it contains some languages that are not even contextfree [11]. Hence, the class CRL can be seen as an extension of the class DCFL that preserves the linear-time decidability of the membership problem. As such it is certainly an interesting language class.
