Prerequisite – Turing Machine, 1’s and 2’s complement of a Binary Number

### Problem-1:

Draw a Turing machine to find 1’s complement of a binary number.

*1’s complement* of a binary number is another binary number obtained by toggling all bits in it, i.e., transforming the 0 bit to 1 and the 1 bit to 0.

**Example:**

**Approach:**

- Scanning input string from left to right
- Converting 1’s into 0’s
- Converting 0’s into 1’s
- Move the head to the start when BLANK is reached.

**Steps:**

**Step-1.**Convert all 0’s into 1’s and all 1’s into 0’s and go right if B found go to left.**Step-2.**Then ignore 0’s and 1’s and go left & if B found go to right**Step-3. Stop the machine.**

Here, **q0** shows the initial state and **q1** shows the transition state and **q2** shows the final state.

And 0, 1 are the variables used and R, L shows right and left.**Explanation:**

- State q0 replace ‘1’ with ‘0’ and ‘0’ with ‘1’ and move to right.
- When BLANK is reached move towards left.
- Using state ‘q2’ we reach start of the string.
- When BLANK is reached move towards right and reaches the final state q2.

#### Problem-2:

Draw a Turing machine to find 2’s complement of a binary number.

*2’s complement* of a binary number is 1 added to the 1’s complement of the binary number.

**Example:**

**Approach:**

- Scanning input string from right to left
- Pass all consecutive ‘0’s
- For first ‘1’ comes, do nothing
- After that, Converting 1’s into 0’s and Converting 0’s into 1’s
- Stop when BLANK is reached.

**Steps:**

**Step-1.**First ignore all 0’s and 1’s and go to right & then if B found go to left.**Step-2.**Then ignore all 0’s and go left, if 1 found go to left.**Step-3.**Convert all 0’s into 1’s and all 1’s into 0’s and go to left & if B found go to right and**stop the machine.**

Here, **q0** shows the initial state and **q1 and q2** shows the transition state and **q3** shows the final state.

And 0, 1 are the variables used and R, L shows right and left.

**Explanation:**

- Using state ‘q0’ we reach end of the string.
- When BLANK is reached move towards left.
- Using state ‘q1’ we passes all 0’s and move left first 1 is found.
- Pass single ‘1’ and move left.
- Using state ‘q2’ we compliment the each digit and move left.
- When BLANK is reached move towards right and reaches the final state q2.