Turing Machine as Comparator

Prerequisite – Turing Machine**Problem :** Draw a turing machine which compare two numbers. Using unary format to represent the number. For example, 4 is represented by

4 = 1 1 1 1 or 0 0 0 0

Lets use one’s for representation.

**Example:**

**Approach:**

- Camparing two numbers by comparing number of ‘1’s.
- Comparing ‘1’s by marking them ‘X’.
- If ‘1’s are remaining in left of ‘0’, then first number is greater.
- If ‘1’s are remaining in right of ‘0’, then second number is greater.
- If both ‘1’ are finished then both numbers are equal.

**Steps:**

**Step-1:**Convert 1 into X and move right and goto step 2. If symbol is 0 ignore it and move right and goto step-6.**Step-2:**Keep ignoring 1 and move towards right. Ignore 0 move right and goto step-3.**Step-3:**Keep ignoring X and move towards right. Ignore 1 move left and goto step-4. If B is found ignore it and move left and goto step-9.**Step-4:**Keep ignoring X and move towards left. Ignore 0 move left and goto step-5.**Step-5:**Keep ignoring 1 and move towards left. Ignore X move right and goto step-1.**Step-6:**Keep ignoring X and move towards right. If B is found ignore it and move left and goto step-8. If 1 is found ignore it and move right and goto step-7.**Step-7:**Stop the Machine (A < B)**Step-8:**Stop the Machine (A > B)**Step-9:**Stop the Machine (A = B)

**State transition diagram :**

**Comparator for A < B****Comparator for A = B****Comparator for A > B****Universal Comparator for A < B, A = B, A > B**

Here, **Q0** shows the initial state and **Q2, Q3, Q4, Q5** shows the transition state and **(A < B), (A = B)and (A > B)** shows the final state. 0, 1 are the variables used and R, L shows right and left.

**Explanation:**

- Using Q0, when 1 is found make it X and go to leftand to state Q1. and if 0 found move to right to Q5 state.
- On the state Q1, ignore all 1 and goto right.If 0 found ignore it and goto right into next state Q2.
- In Q2, ignore all X and move right.If B found stop the execution and you will goto the state showing A > B, If 1 found make it X move left and to Q3.
- In Q3 state, ignore all X and move left.If 0 found ignore it move left to Q4.
- In Q4, ignore all 1 and move left.If X found ignore it move right.
- In Q5 state, ignore all X and move right.If 1 found ignore it move right and stop the machine it will give the result that A < B.And if, B found move left to stop the machine it will give the result that A = B