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Q43E

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Discrete Mathematics and its Applications
Found in: Page 877
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

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

The result is\({\bf{L(M) = \{ 0,01,11\} }}\).

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Step by Step Solution

TABLE OF CONTENTS :
TABLE OF CONTENTS

Step 1: According to the figure.

Here the given figure contains three states\({{\bf{s}}_{\bf{o}}}{\bf{,}}{{\bf{s}}_{\bf{1}}}{\bf{,}}{{\bf{s}}_{\bf{2}}}\).

If there is an arrow from \({{\bf{s}}_{\bf{i}}}\) to \({{\bf{s}}_{\bf{j}}}\) with label x, then we write down in row \({{\bf{s}}_{\bf{j}}}\)and in the row \({{\bf{s}}_{\bf{i}}}\)and in column x of the following table.

State

0

1

\({{\bf{s}}_{\bf{o}}}\)

\({{\bf{s}}_{\bf{1}}}\),\({{\bf{s}}_{\bf{2}}}\)

\({{\bf{s}}_{\bf{1}}}\)

\({{\bf{s}}_{\bf{1}}}\)

\({{\bf{s}}_{\bf{2}}}\)

\({{\bf{s}}_{\bf{2}}}\)

\({{\bf{s}}_{\bf{o}}}\) is marked as the start state.

Step 2: Find the final result.

To Move from \({{\bf{s}}_{\bf{o}}}\) the final state \({{\bf{s}}_{\bf{2}}}\)(directly), I require that the input is 0 and thus the bit string 0 is in recognized language.

\(0 \in {\bf{L(M)}}\)

To move from\({{\bf{s}}_{\bf{o}}}\)to \({{\bf{s}}_{\bf{1}}}\), the input can be either 0 or 1. To then move from \({{\bf{s}}_{\bf{1}}}\)to \({{\bf{s}}_{\bf{2}}}\), the second input needs to be 1.Combined , I thus require an input of 01 or 11 to arrive at the final state \({{\bf{s}}_{\bf{2}}}\),which then implies that 01 and 11 are both in the recognized language.

\(01,11 \in {\bf{L(M)}}\)

Therefore, the language generated by the machine is

\({\bf{L(M) = \{ 0,01,11\} }}\)

Most popular questions for Math Textbooks

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

Find a phrase-structure grammar for each of these languages.

a) the set consisting of the bit strings 0, 1, and 11

b) the set of bit strings containing only 1s

c) the set of bit strings that start with 0 and end with 1

d) the set of bit strings that consist of a 0 followed by an even number of 1s.

Let \({\bf{G = }}\left( {{\bf{V, T, S, P}}} \right)\) be the context-free grammar with \({\bf{V = }}\left\{ {\left( {\bf{,}} \right){\bf{S,A,B}}} \right\}{\bf{, T = }}\left\{ {\left( {\bf{,}} \right)} \right\}\) starting symbol \({\bf{S}}\), and productions \({\bf{S}} \to {\bf{A,A}} \to {\bf{AB,A}} \to {\bf{B,B}} \to {\bf{A,}}\)and \({\bf{B}} \to {\bf{(),S}} \to {\bf{\lambda }}\)

Construct the derivation trees of these strings.

\({\bf{a)}}\) \({\bf{(())}}\)

\({\bf{b)}}\) \({\bf{()(())}}\)

\({\bf{c)}}\) \({\bf{((()()))}}\)

What is an unsolvable decision problem? Give an example of such a problem.

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