By Walter J Savitch
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Additional info for Abstract machines and grammars
In order to get a more readable grammar, start with this last G' and replace
A. Let A - C 1C2••• C 1 be the first production used. We can set w = w1w2••• w 1 where C; ~ w; in G2 in fewer than k steps. (Some of the C; may be terminals. ) Now P3 contains the production A - Y1 Y2••• Y 1 where Y; = C; if w, -;e. A and where Y; =A if w; =A. Also, by induction hypothesis, if w; -;e. A, then C; ~ w; in G3. So A ~ Y1 Y2••• Y 1 ~ w1 w2••• w 1 = w in G 3• Thus A ~ w in G 3 as desired. 13 Theorem There is an algorithm which, given an arbitrary cfg, will produce an equivalent grammar that is nonerasing and has no productions of the form A - B, where A and Bare nonterminals.
Recall that s is the designated start state and A denotes the empty string. Notice that this definition requires that M be in an accepting state after the entire input is read. If M enters an accepting state before all the input is read, that does not indicate acceptance of the input. The language accepted by M, denoted A (M), is defined to be the set of all strings a accepted by M. finite-state language if L = A (M) for some nondeterministic finite-state acceptor M. 1 We will describe a finite-state acceptor M in two different but equivalent ways: one in terms of an ordered five-tuple and one in our more intuitive notation.
Abstract machines and grammars by Walter J Savitch