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| In [[circuit complexity]], '''AC''' is a [[complexity class]] hierarchy. Each class, '''AC<sup>i</sup>''', consists of the [[formal language|languages]] recognized by [[Boolean circuit]]s with depth <math>O(\log^i n)</math> and a [[polynomial|polynomial number]] of [[fanin|unlimited-fanin]] [[AND gate|AND]] and [[OR gate]]s.
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| The name "AC" was chosen by analogy to [[NC (complexity)|NC]], with the "A" in the name standing for "alternating" and referring both to the alternation between the AND and OR gates in the circuits and to [[alternating Turing machine]]s.<ref>{{harvtxt|Regan|1999}}, page 27-18.</ref>
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| The smallest AC class is [[AC0|AC<sup>0</sup>]], consisting of constant-depth unlimited-fanin circuits.
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| The total hierarchy of AC classes is defined as
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| <blockquote>
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| <math>\mbox{AC} = \bigcup_{i \geq 0} \mbox{AC}^i</math> | |
| </blockquote>
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| ==Relation to NC==
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| The AC classes are related to the [[NC (complexity)|NC]] classes, which are defined similarly, but with gates having only constant fanin. For each ''i'', we have<ref name=CK437>{{harvtxt|Clote|Kranakis|2002|p=437}}</ref><ref name=AB118>{{harvtxt|Arora|Barak|2009|p=118}}</ref>
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| :<math>\mbox{NC}^i \subseteq \mbox{AC}^i \subseteq \mbox{NC}^{i+1}.</math>
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| As an immediate consequence of this, we have that NC = AC.<ref name=CK12>{{harvtxt|Clote|Kranakis|2002|p=12}}</ref>
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| It is known that inclusion is strict for ''i'' = 0.<ref name=AB118/>
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| ==Variations==
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| The power of the AC classes can be affected by adding additional gates. If we add gates which calculate the [[modulo operation]] for some modulus ''m'', we have the classes [[ACC (complexity)|ACC<sup>i</sup>[m]]].<ref name=CK12/>
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| ==Notes==
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| {{reflist}}
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| ==References==
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| *{{citation | zbl=1193.68112 | last1=Arora | first1=Sanjeev | authorlink1=Sanjeev Arora | last2=Barak | first2=Boaz | title=Computational complexity. A modern approach | publisher=[[Cambridge University Press]] | year=2009 | isbn=978-0-521-42426-4 }}
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| *{{citation | last1=Clote | first1=Peter | last2=Kranakis |first2=Evangelos | title=Boolean functions and computation models | series=Texts in Theoretical Computer Science. An EATCS Series | location=Berlin | publisher=[[Springer-Verlag]] | year=2002 | isbn=3-540-59436-1 | zbl=1016.94046 }}
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| *{{citation | last = Regan | first = Kenneth W. | contribution = Complexity classes | title = Algorithms and Theory of Computation Handbook | publisher = CRC Press | year = 1999}}.
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| *{{citation | last=Vollmer | first=Heribert | title=Introduction to circuit complexity. A uniform approach | series=Texts in Theoretical Computer Science | location=Berlin | publisher=[[Springer-Verlag]] | year=1998 | isbn=3-540-64310-9 | zbl=0931.68055 }}
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| {{ComplexityClasses}}
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| [[Category:Circuit complexity]]
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| [[Category:Complexity classes]]
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