Q24P Let E={aibj|i鈮爅and2i鈮爅}. Sh... [FREE SOLUTION]

91影视

Introduction to Theory of Computation

Michael Sipser

Computer Science

3rd Edition 路 458 pages

ISBN: 9781133187790

Chapter 2: Q24P (page 163)

Let $E = {a^{i} b^{j} | i 鈮� j a n d 2 i 鈮� j}$ . Show that $E$ is context-free language.

Short Answer

Expert verified

It is shown that $E$ is a context-free language.

Step by step solution

Define context-free language

The context-free language is generated by context-free grammar. These languages are accepted by Pushdown Automata. These are the supersets of regular languages.

Consider context-free languages $L_{1}$ described as $G_{1} = (V_{1}, S, R_{1}, S_{1}) .$

Consider context-free language $L_{2}$ described as $G_{2} = (V_{2}, S, R_{2}, S_{2}) .$

Explain the solution

Split it as unions of three languages.

$\{a^{i} b^{j} | i > j\} 虁 E \{a^{i} b^{j} | i < j a n d 2 i > j\} 鈭� \{a^{i} b^{j} | 2 i < j\} G^{1} = \{a^{i} b^{j} | i > j\} G^{2} = \{a^{i} b^{j} | i < j a n d 2 i > j\} G^{3} = \{a^{i} b^{j} | 2 i < j\}$

$G^{1} a n d G^{3} a r e q u i t e e a s y t o f i g u r e o u t . f o r G^{1} S 鈫� a S b | a S^{1} S^{1} 鈫� a S^{1} | 蔚$

for G³

$S 鈫� a S b b | S^{1} b S^{1} 鈫� S^{1} b | 蔚 N o w t r i c k y p a r t i s G 2 .$

S 鈫� a S b | a S^{1} b S^{1} 鈫� a S^{1} b b | a S^{2} b b S^{2} 鈫� 蔚

It is proved very clearly, that $S w i l l g e n e r a t e a^{n} a^{m} b^{2 m} b^{n} t h a t i s e q u i v a l e n t t o a^{n + m} b^{n + 2 m}$ ,

where $m, n m u s t b e g r e a t e r t h a n 0 .$

Hence, $E$ is a context-free language.

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91影视

Let $E = {a^{i} b^{j} | i 鈮� j a n d 2 i 鈮� j}$ . Show that $E$ is context-free language.

Short Answer

Step by step solution

Define context-free language

Explain the solution

One App. One Place for Learning.

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91影视

Let E={aibj|i鈮�jand2i鈮�j}. Show thatE is context-free language.

Short Answer

Step by step solution

Define context-free language

Explain the solution

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Computer Science Textbooks

Problem Solving Techniques

Issues in Computer Science

Fintech

Theory of Computation

Algorithms in Computer Science

Functional Programming

Study anywhere. Anytime. Across all devices.

Let $E = {a^{i} b^{j} | i 鈮� j a n d 2 i 鈮� j}$ . Show that $E$ is context-free language.