By John E. Hopcroft, Jeffrey D. Ullman
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Extra info for Formal Languages and Their Relation to Automata
Example text
1 SIMPLIFICATION OF CONTEXT-FREE G R A M M A R S 49 In addition to removing variables from which no terminal string can be derived, we can also remove variables which are useless in the sense that they can never appear in a derivation. 3. Given any context-free grammar generating a nonempty cfl L, it is possible to find a grammar G, generating L, such that for each variable A there is a derivation S L_~ w~Aw3 ~ w~w2w3, where wl, w2, and w3 are in V*. 2. If S ~ . ~lAa2, al and a2 in V*, then there exists a derivation S Z~.
Let G be a context-free grammar. 4, we can find an equivalent grammar, G~ = (VN, Vr, P, S), such that P contains no productions of the form A ~ B, where A and B are variables. Thus, if a production has a single symbol on the right, that symbol is a terminal, and the production is already in an acceptable form. Now consider a production in P, of the form A --~ B I B 2 . . B i n , where m __> 2. Each terminal B~ is replaced by a new variable C~, which appears on the right of no other production.
Let R~. denote the set of all,strings x such that 3(q,, x) = qj, and if 3(q,, y) = q~, for any y which is an initial segment of x other than x or ~, then l < k. 6 SOLVABLE PROBLEMS CONCERNING FINITE AUTOMATA 39 That is, R~. is the set of all strings which take the finite automaton from state q~ to state qj without going through any state qz, 1 > k. Note that by "going through a state," we mean both entering and leaving. Thus i or j may be greater than k. We can define R~. recursively" R~. = R~; ~(R~; ~)*R~; ~ u R~ -~ R~°j = {al3(q~, a) = qj}.
Formal Languages and Their Relation to Automata by John E. Hopcroft, Jeffrey D. Ullman
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