Improved Descriptional Complexity Results on Generalized Forbidding Grammars

Abstract

In formal language theory, if a grammar consists of rules of the type \((r,F_r)\), where r is a context-free rule and \(F_r\) is an associated finite set of strings called the forbidding set, such that r can be applied to a string only if none of the strings in \(F_r\) is present in the string, then the grammar is said to be a generalized forbidding (GF) grammar. There are four main parameters that describe the size of a GF grammar, namely, (i) d, the maximum length of strings in the forbidding sets, (ii) i, the maximum cardinality of the forbidding sets, (iii) n, the number of nonterminals used in the grammar, and (iv) c, the number of rules with nonempty forbidding set. The family of languages described by a GF grammar of size (at most) (d, i, n, c) is denoted by GF(d, i, n, c). We study for what sizes of generalized forbidding grammars one can obtain the computational power of Turing machines. We specifically show this result for sizes (2, 6, 8, 6), (2, 5, 8, 8), (2, 4, 9, 9) and (2, 3, 20, 18). These results improve on previous results on the descriptional complexity of generalized forbidding grammars.