Context-Free Grammars and Nonnegative Matrices

To apply this fact we first define a context-free grammar and describe how to attach probabilities to the rules of such a grammar, in constructing a behavioral model of sentence production. A context-free grammar consists of two sets of integers, T = (1,. . , m> and C = {HZ + 1,. . ., m + n}, the terminal integers and the nontevminal integers respectively, and a set G = (Y], . . . , yh} of rewrote rules. Each rule is of the form i + c+, where i E C and tc, is a finite sequence of elements from T U C. Each i E C must be on the left of the arrow for at least one rule in G. The sentences generated by the grammar (T, C, G) are all those and only those sequences produced as follows. Start with the sequence consisting of the single term m + 1. Choose from among the rules in G any one which rewrites m + 1, i.e., of form m + 1 + IX. Rewrite (replace) m + 1 with the sequence a. If there are any terms which are elements