An Insertion into the Chomsky Hierarchy?

This review paper will report on some recent discoveries in the area of Formal Languages, chie y by F. Otto, G. Buntrock and G. Niemann. These discoveries have pointed out certain break-throughs connected with the concept of growing contextsensitive languages, which originated in the 1980's with a paper by E. Dahlhaus and M.K. Warmuth. One important result is that the deterministic growing context-sensitive languages turn out to be identical to an interesting family of formal languages de nable in a certain way by con uent reduction systems. 1. Growing context-sensitive languages. There are several reasons for proposing that the family of GrCSL's (growing contextsensitive languages) be considered as a new level in the Chomsky hierarchy of languages. The insertion would be between the CFL's (context-free languages) and the CSL's (context-sensitive languages), in e ect making a new family designated as the \type one-and-a-half languages". This family began to receive attention in 1986, when Dahlhaus and Warmuth published their result [5] that the complexity of its membership problem had a polynomial-time algorithm, in contrast to the P-space-complete membership problem for the larger family of context-sensitive languages. A GrCSG (growing context-sensitive grammar) is one in which j j < j j for every rule ! ; a GrCSL is the language of a GrCSG. This class of grammars is a proper subclass of the class of CSG's (context-sensitive grammars) where rules are also permitted in which j j = j j. The GrCSL's are just one of many studied families properly between the CFL's and the CSL's. Another was the family of CSL's with linear-bounded derivations (see, e.g.,