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| {{DISPLAYTITLE:TC<sup>0</sup>}}
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| '''TC<sup>0</sup>''' is a [[complexity class]] used in [[circuit complexity]]. It is the first class in the hierarchy of [[TC (complexity)|TC]] classes.
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| TC<sup>0</sup> contains all languages which are decided by [[Boolean circuit]]s with constant depth and polynomial size, containing only unbounded-fanin [[AND gate]]s, [[OR gate]]s, and [[majority gate]]s. Equivalently, [[threshold gate]]s can be used instead of majority gates.
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| TC<sup>0</sup> contains several important problems, such as sorting ''n'' ''n''-bit numbers, and multiplying two ''n''-bit numbers, integer division,<ref>{{cite journal|first1=William|last1=Hesse|first2=Eric|last2=Allender|first3=David|last3=Mix Barrington|title=Uniform constant-depth threshold circuits for division and iterated multiplication|journal=Journal of Computer and System Sciences|volume=65|year=2002|pages=695–716|doi=10.1016/S0022-0000(02)00025-9|url=http://ftp.cs.rutgers.edu/pub/allender/division.pdf}}</ref> recognizing the [[Dyck language]] with two types of parentheses.
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| ==Complexity class relations==
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| We can relate TC<sup>0</sup> to other circuit classes, including [[AC0|AC<sup>0</sup>]] and [[NC1 (complexity)|NC<sup>1</sup>]]; Vollmer 1999 p. 126 states:
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| <blockquote>
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| <math>\mbox{AC}^0 \subsetneq \mbox{AC}^0[p] \subsetneq \mbox{TC}^0 \subseteq \mbox{NC}^1. </math>
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| </blockquote>
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| Vollmer states that the question of whether the last inclusion above is strict is "one of the main open problems in circuit complexity" (ibid.).
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| We also have that uniform <math>\mbox{TC}^0 \subsetneq \mbox{PP}</math>. (Allender 1996, as cited in Burtschick 1999).
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| ==References==
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| {{reflist}}
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| *{{cite book | first=E. | last=Allender | chapter= A note on uniform circuit lower bounds for the counting hierarchy | title=Proceedings 2nd International Computing and Combinatorics Conference (COCOON) | volume=1090 | series=Springer Lecture Notes in Computer Science | pages=127–135 | year=1996 }}
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| *{{cite book | 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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| *{{cite book | last = Vollmer | first = Heribert | title = Introduction to Circuit Complexity. A uniform approach | series=Texts in Theoretical Computer Science | year = 1999 | publisher = [[Springer-Verlag]] | location = Berlin | isbn = 3-540-64310-9 | zbl=0931.68055 }}
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| *{{Cite journal | last1 = Burtschick | first1 = Hans-Jörg | first2 = Heribert | last2= Vollmer | title = Lindström Quantifiers and Leaf Language Definability | id={{ECCC|1996|96|005}}| year = 1999}}
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| ==External links==
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| *{{CZoo|TC<sup>0</sup>|T#tc0}}
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| {{ComplexityClasses}}
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| [[Category:Circuit complexity]]
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| [[Category:Complexity classes]]
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