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Download Formal Languages and Their Relation to Automata by John E. Hopcroft, Jeffrey D. Ullman PDF

By John E. Hopcroft, Jeffrey D. Ullman

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Thus if there is a derivation of c w in G~, then there is a derivation of w in G. N o w suppose that w is in L(G) and consider a leftmost derivation of w in G, say S = S o ~ C q = = ~ . - c ==~c~" = w. If, for 0_< i < n, ~ i ~ c ~ + l by a type y production, then ~ c ~ c~ + 1. Suppose that ~ ~ ~ + ~ by a type x production, but that ~ _ 1 ~-~ ~ by a type y production unless i - 0. Also, suppose that ~+~ ~ ~j ~ ~+2 ==~"" ==~ ~; all by type x productions and t3 ~j+ 1 by a type y production. t3 Then a~, ~ + ~ , .

Define a deterministic fa, M ' = (K', Z, 3', q;, F') as follows. The states of M ' are all the subsets of the set of states of M. That is, K ' = 2 K. M ' will keep track of all the states M could be in at any given time. F ' is the set of all states in K' containing a state of F. An element of K' will be denoted by q~, q 2 , . . , q~, where q~, q 2 , . . , q~ are in K. Note that q; = qo. We define Proof 8'(q~, q2, . , q~, a ) = p~, p 2 , . , P j if and only if 3({q~, q 2 , . . , q,), a) = {p~, P 2 , .

3(p,, a) = (q, L). Here, M next moves left t o cell j - 1. We consult A1 to see what happens next. If ,51(q) = R, then M will never again scan cell j, so it cannot scan j + 1 either. We therefore let A2(pl) = R. If zXl(q) = q', then p~ ÷ 1 = q'. The above process will produce a value for zX2(pl) in all cases, provided that the sequence p~, P 2 , . . never repeats a state. If a state is repeated, M is in a loop and will never reach cell j + 1. In this case, A2(p~) = R. We have now covered all contingencies.

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