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Discrete Mathematics and its Applications
Found in: Page 900
Discrete Mathematics and its Applications

Discrete Mathematics and its Applications

Book edition 7th
Author(s) Kenneth H. Rosen
Pages 808 pages
ISBN 9780073383095

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Short Answer

a) Define a phrase-structure grammar.

b) What does it mean for a string to be derivable from a string w by a phrase-structure grammar G?

  1. A phrase structure grammar (V, T, S, and P) represents a language's description; V is the alphabet, T is a collection of terminal symbols, S is the start symbol, and P is a production set.
  2. If a string w can be produced from the start symbol S using a series of productions, then the string w can be derived from the phase structure grammar.
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Step by Step Solution

TABLE OF CONTENTS :
TABLE OF CONTENTS

Step 1: Definitions

A collection of pieces that can be used to build strings is called an alphabet.

The set of all possible strings across an alphabet is divided into languages.

Step 2: Phrase-structure grammar

(a)

The description of a language is represented in a phrase structure grammar by the letters V, T, S, and P, which stand for the alphabet, a set of terminal symbols, a set of start symbols, and a set of production symbols, respectively.

Step 3: The concept of derivable

(b)

The phase structure grammar can be used to derive a string w if the start symbol S can be used to construct the string through a succession of productions.

Therefore, the require definition of phrase structure grammar.

Most popular questions for Math Textbooks

In Exercises 43–49 find the language recognized by the given nondeterministic finite-state automaton.

let \({{\bf{G}}_{\bf{1}}}\) and \({{\bf{G}}_{\bf{2}}}\) be context-free grammars, generating the language\({\bf{L}}\left( {{{\bf{G}}_{\bf{1}}}} \right)\) and \({\bf{L}}\left( {{{\bf{G}}_{\bf{2}}}} \right)\), respectively. Show that there is a context-free grammar generating each of these sets.

a) \({\bf{L}}\left( {{{\bf{G}}_{\bf{1}}}} \right){\bf{UL}}\left( {{{\bf{G}}_{\bf{2}}}} \right)\)

b) \({\bf{L}}\left( {{{\bf{G}}_{\bf{1}}}} \right){\bf{L}}\left( {{{\bf{G}}_{\bf{2}}}} \right)\)

c) \({\bf{L}}{\left( {{{\bf{G}}_{\bf{1}}}} \right)^{\bf{*}}}\)

Suppose that A is a subset of\({{\bf{V}}^{\bf{*}}}\)where V is an alphabet.Prove or disprove each of these statements.

\(\begin{array}{l}{\bf{a)}}\,\,{\bf{A}} \subseteq {{\bf{A}}^{\bf{2}}}\\{\bf{b)}}\,\,{\bf{if}}\,{\bf{A = }}{{\bf{A}}^{\bf{2}}}{\bf{,then}}\,{\bf{\lambda }} \in {\bf{A}}\\{\bf{c)}}\,\,{\bf{A\{ \lambda \} = A}}\\{\bf{d)}}\,\,{{\bf{(}}{{\bf{A}}^{\bf{*}}}{\bf{)}}^{\bf{*}}}{\bf{ = }}{{\bf{A}}^{\bf{*}}}\\{\bf{e)}}\,\,{{\bf{A}}^{\bf{*}}}{\bf{A = }}{{\bf{A}}^{\bf{*}}}\\{\bf{f)}}\,\,\left| {{{\bf{A}}^{\bf{n}}}} \right|{\bf{ = }}{\left| {\bf{A}} \right|^{\bf{n}}}\end{array}\)

a) Show that the grammar \({{\bf{G}}_{\bf{1}}}\) given in Example 6 generates the set\({\bf{\{ }}{{\bf{0}}^{\bf{m}}}{{\bf{1}}^{\bf{n}}}{\bf{|}}\,{\bf{m,}}\,{\bf{n = 0,}}\,{\bf{1,}}\,{\bf{2,}}\,...{\bf{\} }}\).

b) Show that the grammar \({{\bf{G}}_{\bf{2}}}\) in Example 6 generates the same set.

Find the output generated from the input string 01110 for the finite-state machine with the state table in

a) Exercise 1(a).

b) Exercise 1(b).

c) Exercise 1(c).
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