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2.26 Show that if G is a CFG in Chomsky normal form, then for any string w E L(G) of length n > 1, exactly 2n – 1 steps are required for any derivation of w.

2.26 Show that if G is a CFG in Chomsky normal form, then for any string w E L(G) of length n > 1, exactly 2n – 1 steps are required for any derivation of w.

Database System Concepts
Database System Concepts
7th Edition
ISBN:9780078022159
Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Chapter1: Introduction
Section: Chapter Questions
Problem 1PE
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**2.26** Show that if \( G \) is a CFG in Chomsky normal form, then for any string \( w \in L(G) \) of length \( n \geq 1 \), exactly \( 2n - 1 \) steps are required for any derivation of \( w \).
Transcribed Image Text:**2.26** Show that if \( G \) is a CFG in Chomsky normal form, then for any string \( w \in L(G) \) of length \( n \geq 1 \), exactly \( 2n - 1 \) steps are required for any derivation of \( w \).
**Exercise on Context-Free Grammar and Derivation** **Context**: This exercise is based on Problem 2.26 from page 157 in Sipser's 3rd Edition. It involves analyzing a context-free grammar \( G \) in Chomsky normal form. **Objective**: Prove that for any string \( w \in L(G) \) of length \( n \geq 1 \), exactly \( 2n - 1 \) steps are required for any derivation of \( w \). **Definitions**: - **Variable Rule**: Any rule of the form \( A \rightarrow B C \), where \( A, B, C \) are variables. - **Terminal Rule**: Any rule of the form \( A \rightarrow a \), where \( A \) is a variable and \( a \) is a terminal. **Problem**: Consider a derivation in \( G \) that starts with the start variable and applies variable rules \( s \) times. Terminal rules are applied \( t \) times. Let \( x \) be the string generated by the derivation (note that \( x \) can contain both variables and terminals). **Tasks**: (a) Prove that the string \( x \) contains \( t \) terminals. (b) Prove that the string \( x \) contains \( (1 + s - t) \) variables. (c) Prove that if \( x \in L(G) \) and \( |x| = n \), then \( s + t = 2n - 1 \). This exercise guides you through demonstrating the precise number of steps required for derivations in a grammar setup. The focus is on understanding the relationship between the number of steps, the terminal, and the variable applications in deriving strings from grammars.
Transcribed Image Text:**Exercise on Context-Free Grammar and Derivation** **Context**: This exercise is based on Problem 2.26 from page 157 in Sipser's 3rd Edition. It involves analyzing a context-free grammar \( G \) in Chomsky normal form. **Objective**: Prove that for any string \( w \in L(G) \) of length \( n \geq 1 \), exactly \( 2n - 1 \) steps are required for any derivation of \( w \). **Definitions**: - **Variable Rule**: Any rule of the form \( A \rightarrow B C \), where \( A, B, C \) are variables. - **Terminal Rule**: Any rule of the form \( A \rightarrow a \), where \( A \) is a variable and \( a \) is a terminal. **Problem**: Consider a derivation in \( G \) that starts with the start variable and applies variable rules \( s \) times. Terminal rules are applied \( t \) times. Let \( x \) be the string generated by the derivation (note that \( x \) can contain both variables and terminals). **Tasks**: (a) Prove that the string \( x \) contains \( t \) terminals. (b) Prove that the string \( x \) contains \( (1 + s - t) \) variables. (c) Prove that if \( x \in L(G) \) and \( |x| = n \), then \( s + t = 2n - 1 \). This exercise guides you through demonstrating the precise number of steps required for derivations in a grammar setup. The focus is on understanding the relationship between the number of steps, the terminal, and the variable applications in deriving strings from grammars.
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