To prove that a grammar G is context-free, we need to show that there exists an equivalent nondeterministic pushdown automaton (PDA) that accepts the language produced by G.

Step 1: Convert the grammar G into Chomsky normal form (CNF). This involves rewriting the grammar rules so that each rule is either of the form A - BC or A - a, where A, B, and C are nonterminal symbols and a is a terminal symbol.

Step 2: Create a PDA with a stack and states that simulate the behavior of the CNF grammar. The stack will be used to keep track of the nonterminal symbols during the parsing process.

Step 3: For each rule in the CNF grammar, create a corresponding transition in the PDA. If the rule is of the form A - a, where a is a terminal symbol, the PDA will read the input symbol and pop the top symbol from the stack. If the rule is of the form A - BC, the PDA will push B onto the stack and transition to a new state.

Step 4: Add transitions to handle epsilon (ε) productions in the grammar. If there is a rule of the form A - ε, the PDA will transition to a new state without reading any input symbol and without popping any symbol from the stack.

Step 5: Add transitions to handle the start symbol of the grammar. The PDA will start in an initial state and push the start symbol onto the stack.

Step 6: Add transitions to handle the acceptance of the input string. If the PDA reaches the end of the input string and the stack is empty, it will transition to an accepting state.

Step 7: The PDA is now constructed and is equivalent to the original context-free grammar G. It accepts the language produced by G.

Question

To prove that a grammar G is context-free, we need to show that there exists an equivalent nondeterministic pushdown automaton (PDA) that accepts the language produced by G.

Step 1: Convert the grammar G into Chomsky normal form (CNF). This involves rewriting the grammar rules so that each rule is either of the form A -> BC or A -> a, where A, B, and C are nonterminal symbols and a is a terminal symbol.

Step 2: Create a PDA with a stack and states that simulate the behavior of the CNF grammar. The stack will be used to keep track of the nonterminal symbols during the parsing process.

Step 3: For each rule in the CNF grammar, create a corresponding transition in the PDA. If the rule is of the form A -> a, where a is a terminal symbol, the PDA will read the input symbol and pop the top symbol from the stack. If the rule is of the form A -> BC, the PDA will push B onto the stack and transition to a new state.

Step 4: Add transitions to handle epsilon (ε) productions in the grammar. If there is a rule of the form A -> ε, the PDA will transition to a new state without reading any input symbol and without popping any symbol from the stack.

Step 5: Add transitions to handle the start symbol of the grammar. The PDA will start in an initial state and push the start symbol onto the stack.

Step 6: Add transitions to handle the acceptance of the input string. If the PDA reaches the end of the input string and the stack is empty, it will transition to an accepting state.

Step 7: The PDA is now constructed and is equivalent to the original context-free grammar G. It accepts the language produced by G.

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