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
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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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 \( w \).

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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