# Construction of the machines to produce residue modulo ‘2’ of binary numbers

Prerequisite: Mealy and Moore Machines, Difference between Mealy machine and Moore machine

In this article, we will see some designing of Finite Automata with Output i.e, Moore and Mealy machines.

**Problem:** Construction of the machines that take the binary number {0, 1} as input and produce residue modulo ‘2’ as output i.e, when the equivalent decimal number of binary input over {0, 1} is divided by 2 then it gives output as it’s remainder.

Assume,

Ε = {0, 1} and Δ = {0, 1}

where Ε and Δ are the input and output alphabet respectively.

**The required Moore machine is constructed below:**

In the above diagram, the initial state ‘X’ on getting ‘0’ as the input it remains in the state of itself and prints ‘0’ as the output and on getting ‘1’ as the input it transmits to a state ‘Y’ and prints ‘1’ as the output so on for the remaining states.

For **example**, when the input string is ’10’ then above Moore machine produce 0 as the output because the decimal equivalent of binary input ’10’ is 2 and 2 divided by 2 is 0 i.e, the remainder is 0. Thus finally above Moore machine can easily produce residue modulo ‘2’ as output i.e, when the equivalent decimal number of binary input over {0, 1} is divided by 2 then it gives output as it’s remainder.

**The required Mealy machine is constructed below:**

In the above diagram, the initial state ‘X’ on getting ‘0’ as the input it remains in the state of itself and prints ‘0’ as the output and on getting ‘1’ as the input it transmits to a state ‘Y’ and prints ‘1’ as the output. The state ‘Y’ on getting ‘1’ as the input it remains in the state of itself and prints ‘1’ as the output and on getting ‘0’ as the input it goes back to the initial state ‘X’ and prints ‘0’ as the output.

For **example**, when the input string is ’10’ then above Mealy machine produce 0 as the output because the decimal equivalent of binary input ’10’ is 2 and 2 divided by 2 is 0 i.e, the remainder is 0. Thus finally above Mealy machine can easily produce residue modulo ‘2’ as output i.e, when the equivalent decimal number of binary input over {0, 1} is divided by 2 then it gives output as it’s remainder.

## Recommended Posts:

- Construction of the machines that produce 'A', 'B', or 'C' if input ends with '1', '0', or nothing
- Moore and Mealy machines to produce 'A', 'B', 'C' depends on inputs that end with ’10’ or with ’11’ else other
- Construction of finite machines to prints ‘1’ as the output for every occurrence of ‘a’ as a substring
- Mealy and Moore Machines in TOC
- Moore and Mealy machines to count number of substring ‘ab’
- Construction of LL(1) Parsing Table
- Construction of Combinational Circuits
- Basics of Signed Binary numbers of ranges of different Datatypes
- NFA machines accepting all strings that ends or not ends with substring 'ab'
- DFA machines accepting odd number of 0’s or/and even number of 1’s
- Complexity of different operations in Binary tree, Binary Search Tree and AVL tree
- 4-bit binary Adder-Subtractor
- Binary Decoder in Digital Logic
- Binary Representations in Digital Logic
- Program for Decimal to Binary Conversion

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