Leftmost Derivations of Propagating Scattered Context Grammars: A New Proof

Propagating scattered context grammars, introduced in [3], represent an important type of semi-parallel rewriting systems. Since their introduction, however, the exact relationship of the family of languages they generate to the family of context-sensitive languages is unknown. The language family generated by these grammars is included in the family of context-sensitive languages; on the other hand, the question of whether this inclusion is proper represents an open problem in formal language theory. There have been several attempts to modify the definition of propagating scattered context grammars to obtain the family of context-sensitive languages (see [1, 2, 7, 9, 11]). The approach discussed in [11] allows the productions to be applied only in a leftmost way and, thereby, obtain the family of context-sensitive languages generated by these grammars. This result is of some interest as the use of context-free, contextsensitive, and unrestricted productions in a leftmost way in the corresponding grammars of the Chomsky hierarchy does not have any impact on their generative power. The proof in [11] consists of two parts; first, two preliminary lemmas (Lemma 2 and Lemma 3) are given and then the main result, stated in Theorem 2, is presented as a straightforward corollary of these two lemmas. In Lemma 2 it is demonstrated how any sentence of some context-sensitive language can be derived by a propagating scattered context grammar which uses leftmost derivations. Every sentence