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| {{Distinguish2|[[combinatory logic]], a topic in mathematical logic}}
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| In [[digital circuit]] theory, '''combinational logic''' (sometimes also referred to as '''time-independent logic'''<ref>
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| C.J. Savant, Jr.; Martin Roden; Gordon Carpenter.
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| "Electronic Design: Circuits and Systems".
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| 1991.
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| ISBN 0-8053-0285-9
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| p. 682
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| </ref>
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| ) is a type of [[digital logic]] which is implemented by [[Boolean circuit]]s, 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 ''[[computer storage|memory]]'' while combinational logic does not.
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| Combinational logic is used in [[computer]] circuits to perform [[Boolean algebra (logic)|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 adder]]s, [[full adder]]s, [[half subtractor]]s, [[Half subtractor|full subtractor]]s, [[multiplexer]]s, [[Multiplexer|demultiplexer]]s, [[encoder]]s and [[decoder]]s are also made by using combinational logic.
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| ==Representation==
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| 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. A sum of products can be visualized in a truth table, as in this example:
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| {| class="wikitable" style="margin: 1em auto 1em auto; text-align:center;"
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| |-
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| ! <math>A</math> || <math>B</math> || <math>C</math> || Result || [[logical equivalence|Logical equivalent]]
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| |-
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| | F || F || F || F || <math>\neg A \cdot \neg B \cdot \neg C</math>
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| |-
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| | F || F || T || F || <math>\neg A \cdot \neg B \cdot C</math>
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| |-
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| | F || T || F || F || <math>\neg A \cdot B \cdot \neg C</math>
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| |-
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| | F || T || T || F || <math>\neg A \cdot B \cdot C</math>
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| |-
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| | T || F || F || T || <math>A \cdot \neg B \cdot \neg C</math>
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| |-
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| | T || F || T || F || <math>A \cdot \neg B \cdot C</math>
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| |-
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| | T || T || F || F || <math>A \cdot B \cdot \neg C</math>
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| |-
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| | T || T || T || T || <math>A \cdot B \cdot C</math>
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| |}
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| Using sum of products, all logical statements which yield true results are summed, giving the result:
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| : <math>A \cdot \neg B \cdot \neg C + A \cdot B \cdot C \,</math> | |
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| Using [[Boolean algebra (logic)|Boolean algebra]], the result simplifies to the following equivalent of the truth table:
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| : <math>A \cdot (\neg B \cdot \neg C + B \cdot C) \,</math>
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| ==Logic formula minimization==
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| Minimization (simplification) of combinational logic formulas is done using the following rules:
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| : <math>(A + B) \cdot (A + C) = A + (B \cdot C)</math>
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| : <math>\quad (A \cdot B) + (A \cdot C) = A \cdot (B + C)</math>
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| : <math>A + (A \cdot B) = A</math>
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| : <math>\quad A \cdot (A + B) = A</math>
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| : <math>A + (\lnot A \cdot B) = A + B</math>
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| : <math>\quad A \cdot(\lnot A + B) = A \cdot B</math>
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| : <math>(A + B)\cdot(\lnot A + B)=B</math>
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| : <math>\quad (A \cdot B) + (\lnot A \cdot B)=B</math>
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| : <math>(A \cdot B) + (\lnot A \cdot C) + (B \cdot C) = (A \cdot B) + (\lnot A \cdot C)</math>
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| : <math>(A + B) \cdot (\lnot A + C) \cdot (B + C) = (A + B) \cdot (\lnot A + C)</math>
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| 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.
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| == Terminology ==
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| Some people claim that the term "combinatorial logic" is preferred over "combinational circuit",
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| although others make the opposite recommendation.
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| <ref> | |
| Clive Maxfield.
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| [http://books.google.com/books?id=kQuOKBSOz5QC&pg=PA70&dq=%22combinatorial+logic%22&hl=en&sa=X&ei=vBSdULe_LMPHqQGploDoCg&ved=0CEQQ6AEwBw#v=onepage&q=%22combinatorial%20logic%22&f=false "FPGAs: World Class Designs"].
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| p. 70.
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| 2009.
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| ISBN 1856176215
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| </ref><ref> | |
| Cliff Cummings.
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| [http://www.sunburst-design.com/papers/Technical_Text_Mistakes.pdf "Common Mistakes In Technical Texts"]. | |
| 2009.
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| </ref>
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| ==See also==
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| * [[Sequential logic]]
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| * [[Asynchronous logic]]
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| * [[FPGA]]
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| * [[Formal verification]]
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| ==References==
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| {{reflist}}
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| * Michael Predko and Myke Predko, ''Digital electronics demystified'', McGraw-Hill, 2004. ISBN 0-07-144141-7
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| == External links ==
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| * [http://www.ee.surrey.ac.uk/Projects/Labview/combindex.html Combinational Logic & Systems Tutorial Guide] by D. Belton, R. Bigwood.
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| {{DEFAULTSORT:Combinational Logic}}
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| [[Category:Logic in computer science]]
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| [[Category:Digital electronics]]
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