# Combinational logic

Classes of automata
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In digital circuit theory, combinational logic (sometimes also referred to as time-independent logic[1] ) is a type of digital logic which is implemented by Boolean circuits, where the output is a pure function of the present input only. This is in contrast to sequential logic, in which the output depends not only on the present input but also on the history of the input. In other words, sequential logic has memory while combinational logic does not.

Combinational logic is used in computer circuits to perform Boolean algebra on input signals and on stored data. Practical computer circuits normally contain a mixture of combinational and sequential logic. For example, the part of an arithmetic logic unit, or ALU, that does mathematical calculations is constructed using combinational logic. Other circuits used in computers, such as half adders, full adders, half subtractors, full subtractors, multiplexers, demultiplexers, encoders and decoders are also made by using combinational logic.

An alternate term is combinatorial logic,[2] though this usage may be considered controversial.[3]

## Representation

Combinational logic is used to build circuits that produce specified outputs from certain inputs. The construction of combinational logic is generally done using one of two methods: a sum of products, or a product of sums. Consider the following truth table:

${displaystyle A}$ ${displaystyle B}$ ${displaystyle C}$ Result Logical equivalent
F F F F ${displaystyle neg Acdot neg Bcdot neg C}$
F F T F ${displaystyle neg Acdot neg Bcdot C}$
F T F F ${displaystyle neg Acdot Bcdot neg C}$
F T T F ${displaystyle neg Acdot Bcdot C}$
T F F T ${displaystyle Acdot neg Bcdot neg C}$
T F T F ${displaystyle Acdot neg Bcdot C}$
T T F F ${displaystyle Acdot Bcdot neg C}$
T T T T ${displaystyle Acdot Bcdot C}$

Using sum of products, all logical statements which yield true results are summed, giving the result:

${displaystyle Acdot neg Bcdot neg C+Acdot Bcdot C,}$

Using Boolean algebra, the result simplifies to the following equivalent of the truth table:

${displaystyle Acdot (neg Bcdot neg C+Bcdot C),}$

## Logic formula minimization

Minimization (simplification) of combinational logic formulas is done using the following rules based on the laws of Boolean algebra:

{displaystyle {begin{aligned}(A+B)cdot (A+C)&=A+(Bcdot C)\(Acdot B)+(Acdot C)&=Acdot (B+C)end{aligned}}}
{displaystyle {begin{aligned}A+(Acdot B)&=A\Acdot (A+B)&=Aend{aligned}}}
{displaystyle {begin{aligned}A+(lnot Acdot B)&=A+B\Acdot (lnot A+B)&=Acdot Bend{aligned}}}
{displaystyle {begin{aligned}(A+B)cdot (lnot A+B)&=B\(Acdot B)+(lnot Acdot B)&=Bend{aligned}}}
{displaystyle {begin{aligned}(Acdot B)+(lnot Acdot C)+(Bcdot C)&=(Acdot B)+(lnot Acdot C)\(A+B)cdot (lnot A+C)cdot (B+C)&=(A+B)cdot (lnot A+C)end{aligned}}}

With the use of minimization (sometimes called logic optimization), a simplified logical function or circuit may be arrived upon, and the logic combinational circuit becomes smaller, and easier to analyse, use, or build.

## References

1. ^ C.J. Savant, Jr.; Martin Roden; Gordon Carpenter. "Electronic Design: Circuits and Systems". 1991. ISBN 0-8053-0285-9 p. 682
2. ^ Clive Maxfield. "FPGAs: World Class Designs". p. 70. 2009. ISBN 1856176215
3. ^ Cliff Cummings. "Common Mistakes In Technical Texts". 2009.
• Michael Predko and Myke Predko, Digital electronics demystified, McGraw-Hill, 2004. ISBN 0-07-144141-7

# Combinational Logic

## English

English Wikipedia has an article on:
Wikipedia

### Noun

1. (electrical engineering) A form of digital circuit logic where the current outputs are entirely a function of the current inputs, without having a memory of previous inputs or states.

#### Related terms

Definitions by https://www.wiktionary.org/

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