Design a mealy machine for 2’s complement

Mealy machine is a finite-state machine, its current state and the current inputs determines the output of this machine. 

There is no final state in Mealy Machine .

Here we are going to design  a Mealy Machine for 2’s Complement

Σ = { 0 , 1 }

2’s complement :
It is the mathematical operation on binary numbers. It is used for computation as a method of signed number representation. Its complement with respect to 2^N defines the two’s complement an N-bit number. 

Logic:-
First calculate 1’s complement of binary number, convert 1 to 0 and 0 to 1 and then add 1 to it. For example, if binary number is 1011 then its 1’s complement is 0100 and its 2’s complement is 0101 

Design mealy machine :

1. Take initial state A. 
2. If there are n number of zeros at initial state, it will remain at initial state. 
3. Whenever first input 1 is found then it gives output 1 and go to state B. 
4. After changing a condition we reverse the output .
5. In state B, if input is zero, output will be 1. And if input is 1 then output will be 0. 
   
   

The approach goes as follows: 

1. Start from right to left. 
2. Ignore all 0’s. 
3. When 1 comes ignore it and then take 1’s complement of every digit. 
   

Mealy Machine for 2’s Complement

Figure – Mealy machine of 2’s complement 

Example-1:

1. Lets take 001 and we know that its 2’s complement is (110+1 = 111). 
2. So scan from right to left. 
3. On state A ‘1’ came first to go to stage B and in output write 1. 
4. On state B replace ‘0’ with ‘1’ and vice-versa. 
5. So finally we got 111 as output. 
6. Be aware that the output is also printed in right to left order. 
   

Example-2:

1. Lets take 01 and we know that its 2’s complement is (10+1 = 11). 
2. So scan from right to left. 
3. On state A ‘1’ came first to go to stage B and in output write 1. 
4. On state B replace ‘0’ with ‘1’ and vice-versa. 
5. So finally we got 11 as output. 
6. Be aware that the output is also printed in right to left order.