Turing Machine for Copying Data in Automata Theory

In this chapter, we will see how we can copy a block using Turing machine. The task is very simple, and we know the Turing machines can perform complex tasks that can be solved using algorithms. Let us see the block copying concept in detail with machine description and state diagrams for a better understanding the concept.

Understanding Turing Machines

In Turing Machine, the Tapes are divided into cells. Each cell can hold a single symbol, representing data. The machine has a head that can read and write symbols on the tape, moving left or right along the tape. The machine operates based on a set of rules, called a transition function. This function dictates what the machine should do depending on the current state and the symbol read from the tape.

The Block Copy Problem

One specific problem we can tackle with Turing machines is block copying. This involves copying a block of data, represented as a sequence of symbols, to a different location on the tape.

Suppose our data is a string of zeros, like "0000". We want to copy this block to the next four blank spaces on the tape.

Planning the Solution

To achieve this, we can use a clever trick. Instead of directly copying the zeros, we will mark the original zeros with a new symbol, say "X", and the blank spaces with another symbol, "Y". This way, we can keep track of which zeros have been copied and which blanks are available.

Here's how it works −

• Mark the Source − We start by marking the first zero as "X", moving to the right and marking the first blank space as "Y".
• Repeat Marking − We repeat this process, marking each zero with "X" and each blank space with "Y".
• Return and Check − Once we have marked all the zeros, we return to the last "X" we marked. If the next symbol is "Y", we know all the zeros have been copied.
• Reset and Copy − We then move to the end of the copied block, reset all "X" and "Y" symbols back to zeros, and finally move to the next blank space to mark the end of the copied data.

This approach allows us to clearly identify the source and target locations for the copy operation.

Designing the Turing Machine

Now, let's translate this plan into a Turing machine. We will define the machine's states, symbols, and transitions.

States

• Q0 − Initial state
• Q1 − Moving right after marking a zero with "X"
• Q2 − Retracing back to the last marked zero
• Q3 − Moving right after finding the last zero
• Q4 − Returning to the start of the copied block
• Qf − Final state, where the machine halts

Symbols

• 0 − The symbol representing the data to be copied
• X − Symbol used to mark the original zeros
• Y − Symbol used to mark the blank spaces
• Blank − Represents an empty cell on the tape

Transition Function

The transition function defines how the machine changes state and manipulates the tape based on the current state and the symbol read. Here's a sample transition table for our block copy Turing machine −

Explanation

• The machine starts in state Q0. If it encounters a "0", it changes it to "X", moves right, and transitions to state Q1.
• In Q1, the machine continues to move right, passing over any "0" or "Y" symbols.
• Upon reaching a blank space, the machine writes "Y", moves left, and enters state Q2.
• State Q2 retraces back to the left, identifying the last "X" and moving right to Q0 to continue the process.
• If Q0 encounters a "Y", it means all zeros have been marked, and it transitions to state Q3 to move to the end of the copied block.
• In Q4, the machine resets the "X" and "Y" symbols back to zeros and moves to the next blank space to signal the end of the copied data.
• The machine finally reaches the final state Qf, where it halts.

Conclusion

Block copy is one of the easy example that we can solve using Turing Machines. We take the input on the tape. Here we are taking set of 0s as input and copy them to another location. To make a copy, we are converting a symbol to X and pasting Y at the right, then after completing all, replacing Y with 0s and converted X with 0s for cleaning up.