# Construct a Turing machine for L = {aibjck | i>j>k; k ≥ 1}

Prerequisite â€“ Turing Machine

In given language L = {aibjck | i>j>k; k â‰¥ 1}, every string of â€˜aâ€™, â€˜bâ€™ and â€˜câ€™ have certain number of aâ€™s, then certain number of bâ€™s and then certain number of câ€™s. The condition is that count of 3rd symbols should be atleast 1. â€˜aâ€™ and â€˜bâ€™ can have thereafter be as many but count of c is less than count of â€˜bâ€™ and count of â€˜bâ€™ is less than count of â€˜aâ€™. Assume that string is ending with â€˜\$â€™.
Examples:

```Input: a a a b b c
Here a = 3, b = 2, c = 1
Output: ACCEPTED

Input: a a b b c c c
Here a = 2, b = 2, c = 3 but |a|>|b|>|c|
Output: NOT ACCEPTED ```

Tape Representation:

Approach:

1. Comparing two elements by making A & D as a single element.
2. After that Comparing A & D.
3. If |C| is greater than |(A, D)|, then it is not accepted.
4. If |D| is greater than |A|, then it is not accepted.
5. Else it is accepted.

Steps:

• Step-1: Convert A into X and move right and goto step 2.

• Step-2: Keep ignoring A and Y and move towards right. Convert D into Y and move right and goto step-3.
• Step-3: Keep ignoring D and Z and move towards right.If C is found make it Z and move left to step 4.If B is found ignore it and move left and goto step-5.
• Step-4: Keep ignoring Z, A, Y and D and move towards left.If X is found ignore it and move right and goto step-1.
• Step-5: Keep ignoring D, Y and A and move towards left. Ignore X move right and goto step-6.
• Step-6: Convert A into X and move right and goto step-7.
• Step-7: Keep ignoring Y and A and move towards right.If B is found ignore it and move left and goto step-8.If D make it Y and move right and goto step-5.
• Step-8: Stop the Machine (String is accepted)

State transition diagram :

Here, Q0 shows the initial state and Q1, Q2, Q3, Q4, Q5, Q6 shows the transition state and Q7 shows the final state. A, C, D are the variables used and R, L shows right and left.

Explanation:

• Using Q0, when A is found make it X and go to right and to state Q1.
• On the state Q1, ignore all A and Y and goto right.If D found make it Y and goto right into next state Q2.
• In Q2, ignore all D, Z and move right.If B found ignore it, move left and goto the state Q4, If C found make it Z move left and to Q3.
• In Q3 state, ignore all Z, D, Y, A and move left.If X found ignore it move right to Q0.
• In Q4, ignore all A, Y, D and move left.If X found ignore it move right to state Q6.
• In Q6 state, if A found make it X move right to state Q5
• In Q5, ignore all A, Y and move right.If D found make it Y and move right to state Q4.If B is found ignore it and move left to Q7
• If Q7 state is reached it will produced the result of acceptance of string.

Note: For comparison of |A|, |D|, |C|, the concept of Turing Machine as Comparator is used.