Turing Machine for Recognizing Palindromes: Design and Implementation

Turing Machine for Recognizing Palindromes

This document describes a Turing Machine (TM) designed to recognize palindromes, strings that read the same forwards and backwards. The TM will receive an input string composed of the characters 'a' and 'b', and determine if it is a palindrome.

Design Principles

1. Marking the Last Character: The TM begins by moving to the last character of the input string and marking it with a special symbol 'X'. It then moves the head back to the beginning of the string.
2. Character Comparison: The TM sequentially compares each character in the string with the corresponding character in its reverse position. If the characters match, the TM moves to the next character in the string. If a mismatch occurs, the TM rejects the string.
3. Acceptance: If the TM reaches the end of the string without encountering any mismatches, it accepts the string as a palindrome.

TM Definition

Q: State set {q0, q1, q2, q3, q4} Σ: Input alphabet {a, b, #}, where '#' represents the blank symbol Γ: Work alphabet {a, b, #, X}, where 'X' represents the marker δ: Transition function:

δ(q0, a) = (q1, X, R)    // Mark the first character as 'a' and move right
δ(q0, b) = (q1, X, R)    // Mark the first character as 'a' and move right
δ(q0, #) = (q4, #, R)    // Accept when the string is empty
δ(q1, a) = (q2, a, R)    // Compare characters and move right if they match
δ(q1, b) = (q3, b, R)    // Compare characters and move right if they match
δ(q2, a) = (q2, a, R)    // Compare characters and move right if they match
δ(q2, b) = (q3, b, R)    // Compare characters and move right if they match
δ(q2, #) = (q0, #, L)    // Move to the last character when the beginning is reached
δ(q3, a) = (q2, a, R)    // Compare characters and move right if they match
δ(q3, b) = (q3, b, R)    // Compare characters and move right if they match
δ(q3, #) = (q0, #, L)    // Move to the last character when the beginning is reached
δ(q4, a) = (q4, a, R)    // Accept when the string is empty
δ(q4, b) = (q4, b, R)    // Accept when the string is empty
δ(q4, #) = (q4, #, R)    // Accept when the string is empty

q0: Initial state q4: Accepting state

Example: Recognizing 'abbbba'

Input: abbbba#

1. q0 a # -> q1 X #
2. q1 b # -> q3 b #
3. q3 b # -> q3 b #
4. q3 b # -> q3 b #
5. q3 b # -> q3 b #
6. q3 a # -> q2 a #
7. q2 b # -> q3 b #
8. q3 b # -> q3 b #
9. q3 b # -> q3 b #
10. q3 b # -> q3 b #
11. q3 a # -> q2 a #
12. q2 a # -> q2 a #
13. q2 b # -> q3 b #
14. q3 b # -> q3 b #
15. q3 b # -> q3 b #
16. q3 b # -> q3 b #
17. q3 a # -> q2 a #
18. q2 # # -> q0 # #
19. q0 # # -> q4 # #

Accept.

This TM successfully recognizes 'abbbba' as a palindrome.