Walter J Savitch's Abstract machines and grammars PDF

By Walter J Savitch

ISBN-10: 0316771619

ISBN-13: 9780316771610

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In order to get a more readable grammar, start with this last G' and replace by X, replace S by Y, and then replace S 2 by S. 1. This example worked out particularly well. In general, the Chomsky normal form grammar produced by this algorithm need not be so clean. It can turn out to have many more productions than are needed and can even have productions that are useless. The grammar produced will always be in Chomsky normal form though and will always be equivalent to the grammar we started out with.

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.

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Abstract machines and grammars by Walter J Savitch


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