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.
- Comparing 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.
- 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.
- 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
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