Turing Machine for Recognizing Strings of the Form ωωR

Turing Machine Design for Recognizing ωωR

This document outlines the design and implementation of a Turing Machine (TM) capable of recognizing strings in the form ωωR, where ω is a string composed of characters 'a' and 'b', and ωR represents the reverse of ω.

Design Philosophy

The TM is designed to function as follows:

1. Input Splitting: The input string is divided into two equal halves. The first half represents ω, and the second half represents ωR.
2. Comparison: The TM compares the characters of the two halves sequentially, starting from the leftmost positions.
3. Acceptance/Rejection: If all corresponding characters match, and the TM reaches the middle of the string without finding any mismatches, the string is accepted. Otherwise, the string is rejected.

Formal Definition

M = 'On input w:

1. Divide the input string w into two equal halves: ω (the first half) and ωR (the second half).
2. Check if the two halves are equal: a. Compare characters at corresponding positions in ω and ωR from left to right. If any mismatch is found, reject the input. b. If the middle position of the string is reached without finding any mismatches, accept the input.
3. If a mismatch is found during comparison, or the middle position is reached without finding any mismatches, reject the input.'

Example: Recognizing 'abbbba'

Let's consider the input string 'abbbba' to illustrate the recognition process:

1. Splitting: The input string is divided into two halves: 'abbbb' and 'abbba'.
2. Comparison: The TM compares the characters of the two halves: 'a' matches 'a', 'b' matches 'b', 'b' matches 'b', 'b' matches 'b', and 'b' matches 'b'.
3. Acceptance: The TM reaches the middle position without finding any mismatches, hence the input string 'abbbba' is accepted.

Therefore, the designed TM successfully recognizes and accepts the input string 'abbbba' because it satisfies the criteria of being composed of two identical halves, where the second half is the reverse of the first (ωωR).