Design a Turing Machine to Generate 'ww' from 'w'

Generate Turing Machine for L = {w w |- w | w∈ {a, b}}. In general, we have to double the data. For example, if the given word is abaa, then we have to generate abaaabaa.

Let's understand it with the help of an example. We have a string abaa; it will be stored in the tape surrounded by the blank symbols.

We aim to generate abaaabaa, which means we want to double the given str.

Assumption: We will replace 'a' with 'X' or 'P' and 'b' with 'Y' or 'Q'.

Approach Used

The first thing is to start from the beginning of the string, convert a or b from the beginning of the string into X or Y respectively and corresponding to X or Y, and add P or Q respectively at the end of the string. After continuously doing it, a point is reached when all a's and b’s of the initially given string have been converted into X and Y, respectively. The corresponding string with P's and Q's is generated at the end. Here, our first objective is fulfilled. Now you have to come at the end of the string and, while returning to the start, convert all X's and P's to 'a', and all Y's and Q's to 'b.'

Example:

Input: ab
Ouput: abab

Input: aaba
Ouput: aabaaaba

Input: babba
Ouput: babbababba 

Step 1

If the symbol is 'a' replace it with 'X' and move right

Go to state Q1 and Step 2.

———————————————

If the symbol is 'b' replace it with 'Y' and move right

Go to state Q3 and Step-3.

———————————————

If the symbol is 'P' or 'Q' move right

Go to state Q5 and Step 6.

Step-2

Move to the end of the string till we reach a Blank Character.

Replace the 'B' with 'P' and move left

Go to state Q2 and Step-4.

Step-3

Move to the end of the string till we reach a Blank Character.

Replace the 'B' with 'Q' and move left

Go to state Q4 and Step-5.

Step-4

Move left until we reach 'X'. Then move right and get to state Q0

Repeat Step-1.

Step-5

Move left until we reach 'Y'. Then move right and get to state Q0

Repeat Step-1.

Step-6

Move right until we reach the end of the string (till Blank character)

When the Blank character is reached, move left

Go to state Q6 and Step-7.

Step-7

Replace all 'X' and 'P' with 'a', and move left

Replace all 'Y' and 'Q' with 'b', and move left

When a Blank character is reached, move right and go to state Q7, the Final State.

A Turing Machine is expressed as a 7-tuple (Q, T, B, ∑, δ, q0, F) where

Q = {q0, q1, q2, q3, q4, q5, q6, q7} where q0 is initial state.
T = {a, b, X, Y, P, Q, B} where B represents blank.
∑ = {a, b}
F = {q7}

Conclusion

We have successfully designed a Turing Machine to generate the string ww from a given input string w, where w consists of 'a' and 'b' characters. This solution demonstrates the practical application of computational theory in language processing, showcasing the Turing Machine's ability to efficiently double a given string while highlighting the essential principles of the Theory of Computation.