BOUNDS ON THE COMPLEXITY OF GRAMMARS††This paper is based on part of J.W. Snively's Ph.D. dissertation. The support of the Information Systems Branch, Office of Naval Research, under Contract Nonr-5144(00), is gratefully acknowledged.

Publisher Summary This chapter discusses bounds on the complexity of grammars. One can study various measures of the complexity of a grammar. The chapter discusses three such measures-rule length, vocabulary size, and number of rules. A difficult problem is that of finding a grammar that has minimum complexity in regard to two or more of the measures. Any language has grammars for which each of these measures individually has a low value. The chapter discusses terms of three measures of the complexity of a grammar: λ, the length of the longest member in any rewriting rule; v, the size of the vocabulary; and ρ, the number of rules. Any language has grammars in which any one of these measures has a low value, but this can in general be achieved only at the expense of increasing one or both of the other two measures. Bounds on v as a function of λ are obtained in a number of cases; in most of these cases, the grammars are required to be context free.