Turing Machine for language { www | w ∈ {a, b} }

Prerequisite – Turing Machine
Design a Turing Machine for a string which contains exactly 3 repetitions of w consecutively.

Approach Used :
First, we will find the position of separation of the first w from the second w.
Now, we will match the first and second w. If both get, matched then the second w string will be converted to a string of \$.

Then, we will match the first and third w. If both get, matched then the third w string will be converted to a string of \$. If the string reaches the Halt state H, then it gets accepted.

Example –

```Input: ababab
Output: Accepted
Input: abbabbabb
Output: Accepted
Input: ^ (Empty string)
Output: Accepted
Input: aba
Output: Not accepted ```

Step-1:
If the symbol is \$, replace it by \$ and move right.

Go to state Q1 and step 2.

Step-2:
If the symbol is a, replace it by A and move right, or

If the symbol is b, replace it by B and move right.

Go to state Q2 and step 3.

————————————————-

If the symbol is A, replace it by A and move left, or

If the symbol is B, replace it by B and move left.

Go to state Q6 and step 7.

————————————————-

If the symbol is \$, replace it by \$, and the string is accepted.

Go to final state H.

Step-3:
If the symbol is a, replace it by a and move right, remain on the same state, or

If the symbol is b, replace it by b and move right, remain on the same state.

————————————————

If the symbol is A replace it by A and move left, or

If the symbol is B replace it by B and move left, or

If the symbol is \$ replace it by \$ and move left.

Go to state Q3 and step 4.

Step-4:
If the symbol is a replace it by A and move left, or

If the symbol is b replace it by B and move left,

Go to state Q4 and step 5

Step-5:
If the symbol is a replace it by A and move left, or

If the symbol is b replace it by B and move left,

Go to state Q5 and step 6

Step-6:
If the symbol is a, replace it by a and move left, remain on the same state, or

If the symbol is b, replace it by b and move left, remain on the same state.

————————————————

If the symbol is A, replace it by A and move right, or

If the symbol is B, replace it by B and move right, or

Go to state Q1 and step 2.

Step-7:
If the symbol is A, replace it by a and move left, remain on the same state, or

If the symbol is B, replace it by b and move left, remain on the same state

————————————————-

If the symbol if \$, replace it by \$ and move right.

Go to state Q7 and step 8.

Step-8:
If the symbol is a, replace it by A and move right.

Go to state Q9 and step 10.

————————————————–

If the symbol is b, replace it by B and move right.

Go to state Q8 and step 9.

————————————————–

If the symbol is \$, replace it by \$ and move left.

Go to state Q11 and step 12.

Step-9:
If the symbol is a, replace it by a and move right, remain on the same state, or

If the symbol is b, replace it by b and move right, remain on the same state, or

If the symbol is \$, replace it by \$ and move right, remain on the same state

————————————————–

If the symbol is B, replace it by \$ and move left.

Go to state Q10 and step 11.

Step-10:
If the symbol is a, replace it by a and move right, remain on the same state, or

If the symbol is b, replace it by b and move right, remain on the same state, or

If the symbol is \$, replace it by \$ and move right, remain on the same state.

————————————————–

If the symbol is A, replace it by \$ and move left.

Go to state Q10 and step 11.

Step-11:
If the symbol is a, replace it by a and move left, remain on the same state, or

If the symbol is b, replace it by b and move left, remain on the same state, or

If the symbol is \$, replace it by \$ and move left.

————————————————–

If the symbol is A, replace it by A and move right, or

If the symbol is B, replace it by B and move right.

Go to state Q7 and step 8.

Step-12:
If the symbol is A, replace it by a and move left, remain on the same state, or

If the symbol is B, replace it by b and move left, remain on the same state.

————————————————–

If the symbol is \$, replace it by \$ and move right.

Go to state Q12 and step 13.

Step-13:
If the symbol is a, replace it by A and move right.

Go to state Q14 and step 15.

————————————————–

If the symbol is b, replace it by B and move right.

Go to state Q13 and step 14.

————————————————–

If the symbol is \$, replace it by \$, and the string is accepted.

Go to final state H.

Step-14:
If the symbol is a, replace it by a and move right, remain on the same state, or

If the symbol is b, replace it by b and move right, remain on the same state, or

If the symbol is \$, replace it by \$ and move right, remain on the same state.

————————————————–

If the symbol is B, replace it by \$ and move left.

Go to state Q15 and step 16.

Step-15:
If the symbol is a, replace it by a and move right, remain on the same state, or

If the symbol is b, replace it by b and move right, remain on the same state, or

If the symbol is \$, replace it by \$ and move right, remain on the same state.

————————————————–

If the symbol is A, replace it by \$ and move left.

Go to state Q15 and step 16.

Step-16:
If the symbol is a, replace it by a and move left, remain on the same state, or

If the symbol is b, replace it by b and move left, remain on the same state, or

If the symbol is \$, replace it by \$ and move left.

————————————————–

If the symbol is A, replace it by A and move right, or

If the symbol is B, replace it by B and move right.

Go to state Q12 and step 13.

Meaning of symbols-

```R- move right

L- move left

a- character a

b- character b

A- character A

B- character B ```

Example:
We have to test the Turing machine for a string “ababab”.

```-> \$ababab\$
-> \$Ababab\$
-> \$AbabaB\$
->\$AbabAB\$
-> \$ABabAB\$
-> \$ABaBAB\$
-> \$ABABAB\$
-> \$AbABAB\$
-> \$abABAB\$
-> \$AbABAB\$
-> \$Ab\$BAB\$
-> \$AB\$BAB\$
-> \$AB\$\$AB\$
-> \$Ab\$\$AB\$
-> \$ab\$\$AB\$
-> \$Ab\$\$AB\$
-> \$Ab\$\$\$B\$
-> \$AB\$\$\$B\$
-> \$AB\$\$\$\$\$ (Accepted) ```

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