Let's solve **Question 1** step by step as per your instructions. --- ### Step 1: Introduction of the Concept A **Pushdown Automaton (PDA)** is an automaton that employs a stack for extra memory. A **Context-Free Grammar (CFG)** is a set of production rules that describe all possible strings in a context-free language. There is an algorithm to convert a PDA to an equivalent CFG, which generates the same language. The main idea is to construct variables in the CFG that represent transitions in the PDA, specifically how the PDA can move from one stack configuration to another. --- ### Step 2: Correct Answer and Explanation #### **Correct Answer:** Given PDA: - States: {q0} - Stack Symbols: {A, B} - Input Symbols: {a, b} - Transitions: 1. (q0, a, ε) → (q0, A) 2. (q0, b, A) → (q0, ε) 3. (q0, b, ε) → (q0, B) 4. (q0, a, B) → (q0, ε) **CFG Construction:** Let’s denote variable S as the start symbol, which represents the process of emptying the stack from the initial configuration. CFG productions based on transitions: 1. For (q0, a, ε) → (q0, A): \( S \rightarrow aX \) where X is the variable representing stack top A. 2. For (q0, b, A) → (q0, ε): \( X \rightarrow b \) 3. For (q0, b, ε) → (q0, B): \( S \rightarrow bY \) where Y is the variable for stack top B. 4. For (q0, a, B) → (q0, ε): \( Y \rightarrow a \) So, the CFG is: - \( S \rightarrow aX \ | \ bY \) - \( X \rightarrow b \) - \( Y \rightarrow a \) #### **Explanation:** - \( S \) represents the start variable (empty stack). - Each production mimics the action of the PDA on the input and the stack. - \( X \) and \( Y \) correspond to having A and B on the stack, respectively. --- ### Step 3: Why Other Options Are Incorrect & Final Answer - If you were to use rules like \( S \rightarrow aS \) or \( S \rightarrow bS \), you would not be matching the stack operations, and you would generate strings not accepted by the PDA. - Writing productions without variables \( X \) and \( Y \) would ignore the stack content, which is crucial. - Any grammar not reflecting stack pushes/pops as per the PDA transitions would be incorrect. **Final Answer:** The correct CFG is: - \( S \rightarrow aX \ | \ bY \) - \( X \rightarrow b \) - \( Y \rightarrow a \) --- ### Summary of Steps - **Step 1:** Introduced the PDA-to-CFG conversion concept. - **Step 2:** Constructed the correct CFG by examining each PDA transition and mapping it to a CFG production, with explanation. - **Step 3:** Explained why other CFG structures would not properly represent the PDA’s behavior, and finalized the correct CFG. Let me know if you want the answer for Question 2!

Certainly! Here's a structured, step-by-step solution following your instructions for **Question 1**: --- ### **Step 1: Introduction of the Concept** **Concept:** A **Pushdown Automaton (PDA)** is a type of automaton that uses a stack to recognize context-free languages. It transitions between states based on the input symbol and the top of the stack, pushing or popping symbols accordingly. A **Context-Free Grammar (CFG)** is a set of production rules that generate all strings in a language. There is an established algorithm to convert a PDA into an equivalent CFG, which captures the same language by creating variables representing the PDA's stack configurations and transitions. --- ### **Step 2: Convert the given PDA to CFG** **Given PDA:** Transitions: 1. (q0, a, ε) → (q0, A) 2. (q0, b, A) → (q0, ε) 3. (q0, b, ε) → (q0, B) 4. (q0, a, B) → (q0, ε) **Conversion Approach:** - Variables in the CFG represent possible transitions between states with specific stack contents. - The start symbol \( S \) generally represents the process of going from the initial configuration to acceptance (empty stack). **Constructing the CFG:** - For transition 1: \( (q0, a, ε) \to (q0, A) \) The CFG production: \( S \rightarrow aX \) where \(X\) accounts for A on the stack. - For transition 2: \( (q0, b, A) \to (q0, ε) \) The CFG production: \( X \rightarrow b \) - For transition 3: \( (q0, b, ε) \to (q0, B) \) The CFG production: \( S \rightarrow bY \) - For transition 4: \( (q0, a, B) \to (q0, ε) \) The CFG production: \( Y \rightarrow a \) **Final CFG:** \[ \boxed{ \begin{aligned} S &\rightarrow aX \ |\ bY \\ X &\rightarrow b \\ Y &\rightarrow a \end{aligned} } \] --- ### **Step 3: Why Other Options Are Not Correct & Final Answer** - **Incorrect options** might have productions like \( S \rightarrow aS \) or \( S \rightarrow bS \), which do not model the push/pop operations accurately. - Omitting variables \(X\) or \(Y\) or not reflecting the stack operations would fail to produce the correct language. - The provided CFG precisely models the PDA transitions, ensuring the generated language matches. **Final concise answer:** The CFG corresponding to the PDA is: \( S \rightarrow aX \ |\ bY \), with \( X \rightarrow b \) and \( Y \rightarrow a \). --- ### **Summary of What We Did in Each Step** - **Step 1:** Introduced the concepts of PDA and CFG, explaining the conversion process. - **Step 2:** Constructed the CFG from the PDA by mapping each transition to production rules. - **Step 3:** Clarified why other options are incorrect and summarized the correct CFG in few words. --- Let me know if you'd like me to proceed with Question 2!