Answer
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Work Step by Step
The design of a Turing machine that recognizes the set of palindromes in \( \{a, b\}^* \) such that the number of 'a's is a multiple of three:
1. **Initialization**: Start in state \( q_0 \) at the leftmost symbol of the input tape.
2. **Scan and Count Phase**: Move right scanning the symbols on the input tape. Count the number of 'a's encountered by transitioning between states \( q_0, q_1, q_2, q_3 \). For every 'a' encountered, move to state \( q_1 \), \( q_2 \), and then back to \( q_0 \) successively. Upon encountering 'b', stay in state \( q_0 \).
3. **Mark the Middle**: When reaching the right end of the input tape, move the tape head back to the leftmost non-blank symbol. This is done to mark the middle of the palindrome.
4. **Palindrome Check**: Move the tape head back to the right, comparing symbols on the left and right sides of the middle. If they match, move to state \( q_4 \) and repeat until all symbols have been compared. If a mismatch is found, move to state \( q_{reject} \).
5. **Acceptance**: If all symbols match, and the input tape is fully scanned, move to state \( q_{accept} \) indicating that the input is a palindrome with the number of 'a's being a multiple of three.
6. **Rejection**: If at any point during the process the input is not a palindrome or the number of 'a's is not a multiple of three, move to state \( q_{reject} \).
This Turing machine design ensures that it recognizes palindromes in \( \{a, b\}^* \) with the condition that the number of 'a's is a multiple of three.
