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build a PDA for the given language.

build a PDA for the given language.

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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For each question, build a PDA for the given language.

## Problem Statement (4) \( \{ w \in \{a, b\}^* : \#_a(w) = 2 \times \#_b(w) \} \). ## Questions 1. **The following PDA accepts L.** - Diagram Explanation: - The diagram is a Pushdown Automaton (PDA) with one state labeled "1". - Transitions include: - \( b, aa; \epsilon \) - \( b, \epsilon; bb \) - \( b, b; b \) - \( a, b; \epsilon \) - \( a, \epsilon; a \) - \( a, c; \epsilon \) - The PDA starts at the state marked with an incoming arrow and has a looping transition. 2. **The following PDA accepts L.** - Diagram Explanation: - Similar to the first diagram, this is also a PDA with one state labeled "1". - Transitions include: - \( b, a; b \) - \( b, \epsilon; bb \) - \( a, b; \epsilon \) - \( a, \epsilon; a \) - \( a, c; a \) - The PDA again starts at the single state with an incoming arrow and has a looping transition. 3. **The provided PDAs above are both deterministic.** - This statement is a checkbox option without a diagram, implying a determination of the determinism of the PDAs described in the above explanations. ### Key Concepts - **PDA (Pushdown Automaton):** A computational model similar to finite automata but with an additional stack-based memory. - **Deterministic PDA:** A PDA where for each state and input symbol, there is at most one possible action and state to transition to. - **Stack Operations:** - **Push:** Add an element to the stack. - **Pop:** Remove the top element from the stack. - **No Operation (ε):** The stack remains unchanged.
Transcribed Image Text:## Problem Statement (4) \( \{ w \in \{a, b\}^* : \#_a(w) = 2 \times \#_b(w) \} \). ## Questions 1. **The following PDA accepts L.** - Diagram Explanation: - The diagram is a Pushdown Automaton (PDA) with one state labeled "1". - Transitions include: - \( b, aa; \epsilon \) - \( b, \epsilon; bb \) - \( b, b; b \) - \( a, b; \epsilon \) - \( a, \epsilon; a \) - \( a, c; \epsilon \) - The PDA starts at the state marked with an incoming arrow and has a looping transition. 2. **The following PDA accepts L.** - Diagram Explanation: - Similar to the first diagram, this is also a PDA with one state labeled "1". - Transitions include: - \( b, a; b \) - \( b, \epsilon; bb \) - \( a, b; \epsilon \) - \( a, \epsilon; a \) - \( a, c; a \) - The PDA again starts at the single state with an incoming arrow and has a looping transition. 3. **The provided PDAs above are both deterministic.** - This statement is a checkbox option without a diagram, implying a determination of the determinism of the PDAs described in the above explanations. ### Key Concepts - **PDA (Pushdown Automaton):** A computational model similar to finite automata but with an additional stack-based memory. - **Deterministic PDA:** A PDA where for each state and input symbol, there is at most one possible action and state to transition to. - **Stack Operations:** - **Push:** Add an element to the stack. - **Pop:** Remove the top element from the stack. - **No Operation (ε):** The stack remains unchanged.
Expert Solution
Step 1: Given


L={w  ∈  {a,b}* : #a(w)=2x#b(w) } 

i.e number of a's in w is equals to twice the number of b's in w.

Hence the strings of the language are : 


L={ε,aab,aba,baa,aaaabb,aaabab,aabaab,abaaab,baaaab,baaaba,baabaa,babaaa,bbaaaa,aaabba,aabbaa,
abbaaa,....}

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