Turing Machine for Palindromes Using Two Tapes

⭐ Turing Machine for Palindromes Using Two Tapes

When we say a string is a palindrome, we simply mean that it looks identical from both directions.
For example:

• “abba”
• “10101”
• “level”

If you flip them, nothing changes — they read the same.

A Turing Machine can check whether a string is a palindrome, and using two tapes makes the whole process much easier and cleaner than using a single tape.

Let’s walk through the idea as if we are sitting together and building the machine step by step.

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🌟 Why Two Tapes Are Helpful

Imagine Tape-1 as your original notebook where the input is written.
Tape-2 acts like a second notebook where you rewrite the same input, but in reverse.

Once you have the reversed version, checking for a palindrome becomes as simple as:

👉 Compare the first symbol of Tape-1 with the first symbol of Tape-2
👉 Move to the next symbol on both tapes
👉 Keep checking until you reach the end

If every pair matches, the string is a palindrome.

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How the Two-Tape Palindrome TM Works

Here’s the entire process broken into small, easy steps.

1. Read the input on Tape-1

The machine is starts the leftmost symbol of input.

2. Copy the input onto Tape-2 in reverse order

While Tape-1 moves right, Tape-2 moves left.
So Tape-2 ends up holding the input backward.

3. Move both heads back to the start

Both tapes are rewound to their left ends to prepare for comparison.

4. Compare characters one by one

At every step:

• If the characters are the same → continue
• If you find even a single mismatch → reject instantly

5. Finish the check

If both tapes reach the blank symbol without any mismatch, the TM accepts the string.

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📊 Simple Diagram (Visual Idea Only)

       Tape-1 : Original Input
   ┌──────────────────────────┐
   │ a │ b │ b │ a │ _ │ _ ...│
   └──────────────────────────┘
           ↑
        Head-1


       Tape-2 : Reversed Copy
   ┌──────────────────────────┐
   │ a │ b │ b │ a │ _ │ _ ...│
   └──────────────────────────┘
           ↑
        Head-2

When both tapes show the same sequence from left to right, the machine accepts.

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🧱 High-Level States (Friendly Overview)

• Start state: Prepare both tapes
• Copy state: Build the reversed version on Tape-2
• Rewind state: Move both heads to the left
• Compare state: Check characters one by one
• Accept state: All symbols match
• Reject state: Found a mismatch

This is not a full formal TM description, but enough to clearly understand the idea.

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