Final answer:
Designing a PDA for recognizing palindromes of odd length involves pushing half of the string onto the stack, ignoring the middle character, then matching the remaining characters as the stack unwinds. There is no standard Grafstate code, so a pseudo-description was provided to aid in conceptual understanding of the PDA's operation.
Step-by-step explanation:
The pushdown automaton (PDA) can be used to recognize the language of palindromes of odd length over the alphabet Σ = {a, b}. Since the task requires specific Grafstate code, which is not a standard or widely recognized scripting language for defining automata, the detailed explanation will follow a pseudo-description of a PDA that may be implemented in an actual programming language.
Designing a PDA for Palindromes of Odd Length
For a palindrome of odd length, a PDA needs to push half of the string onto the stack, ignore the middle character, and then match the second half of the string with the stacked characters. The PDA will use a state to transition from processing the first half to processing the second half of the palindrome.
The following steps outline the PDA structure:
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- Initially, the PDA is in a start state and the stack is empty.
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- For each input character, the PDA pushes the character into the stack until it reaches the middle of the string.
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- To handle the odd length, the PDA then transitions to a new state in which it reads one character (the middle one) and does not alter the stack.
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- It moves to another state which is responsible for popping characters from the stack and comparing them with the incoming characters of the second half of the string.
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- If all characters match and the stack is empty when the input is consumed, the PDA accepts the string; otherwise, it rejects it.
Each transition in the PDA should account for both 'a' and 'b' inputs, processing them according to the rules defined above.