Membership for Growing Context Sensitive Grammars is Polynomial

2. Introduction. Context-sensitive grammars (csgs) are one of the classical grammar families of formal language theory. They were introduced in [Ch59] and have been studied extensively since then (see []3073, Ha78] for an overview). Context-sensitive grammars are defined as rewriting systems, where the length of the right hand side of every production is at least as large as the length of the left hand side. This restriction on the productions is responsible for the fact that memberships for context-sensitive languages (csls) is equivalent to the problem of acceptance for nondeterministic linear bounded automaton [Ku64]. Therefore membership for csls is PSPACE complete [Ka72] and this is true even for certain fixed grammars. In this paper we show that if we restrict ourselves to "growing" productions, i.e. the right hand side of every production is strictly longer than the left hand side, then membership for fixed csls is polynomial. This may appear surprising in view of the results obtained in []3o78]. The growing csls are a subclass of LINEARcs as defined in [G164, Bo71]. Languages of LINEARcs are given by an arbitrary csg which has the property that every word w in the language has a derivation of length at most c I w 1 tt, for some overall constant c which only depends on the grammar. In [13078] it was shown that there are NP-complete languages in LINEARcs. Thus our result that the family of growing csls is in P deserves an explanation. Observe that in LINEARcs "complex derivations" are allowed using non-growing productions; then the final word may be padded such that the length of the overall derivation is