### Tuesday, December 4, 2012

# Finite State Machine that adds two integers

**Q.**Construct a ﬁnite-state machine or finite state automaton that adds two integers (using binary representation of integers).

**Sol:**

Finite Automata (FA) or Finite State Machine to add two integers can be constructed using two states:

·

**q**: Start state to represent carry bit is 0_{0}
·

**q**: State to represent carry bit is 1_{1}
The inputs to FA will be pair of bits i.e. 00, 01, 10, and 11

The FA starts in state 1 (since carry is 0) and inputs a pair of bits. If the pair is 11, the FA outputs a 0 and switches to state 2 (since the carry is 1), where the next pair of bits is input and is added to a carry bit of 1.

Example: Consider the addition of 52 and 21

110100 - (binary representation of 52)

010101 - (binary representation of 21)

Since adding numbers is done from right to left, The first input symbol is 01, representing a 0 in the rightmost (binary) digit of 52 and a 1 in the rightmost digit of 21. The machine enters state

**q**(since there is no carry) and outputs a 1. The next input is 00 because both numbers have zero as the second rightmost digit. The machine enters state_{0 }**q**and outputs 0. The next input is 11. The machine enters state_{0}**q**(since the carry is 1) and outputs 0. Being in state_{1}**q**means that there is a carry from this position into the next. And the remaining bits can be worked out to get 1001001 (i.e. 73)._{1}
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AWESOME :)

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