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In [[theoretical computer science]], and specifically [[computational complexity theory]] and [[circuit complexity]], '''TC''' is a [[complexity class]], and '''TC<sup>i</sup>''' is a hierarchy of complexity classes. Each class '''TC<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]], [[OR gate]]s and [[Majority gate]]s. The class '''TC''' is defined as | |||
:<math>\mbox{TC} = \bigcup_{i \geq 0} \mbox{TC}^i.</math> | |||
==Relation to NC and AC== | |||
The relationship between the TC, [[NC (complexity)|NC]] and the [[AC (complexity)|AC]] hierarchy can be summarized as | |||
:<math>\mbox{NC}^i \subseteq \mbox{AC}^i \subseteq \mbox{TC}^i \subseteq \mbox{NC}^{i+1}.</math> | |||
In particular, we know that | |||
:<math>\mbox{NC}^0 \subsetneq \mbox{AC}^0 \subsetneq \mbox{TC}^0 \subseteq \mbox{NC}^{1}.</math> | |||
The first strict containment follows from the fact that '''NC<sup>0</sup>''' cannot compute any function that depends on all the input bits. Thus choosing a problem that is trivially in '''AC<sup>0</sup>''' and depends on all bits separates the two classes. (For example, consider the OR function.) The strict containment '''AC<sup>0</sup>''' ⊊ '''TC<sup>0</sup>''' follows because parity and majority (which are both in '''TC<sup>0</sup>''') were shown to be not in '''AC<sup>0</sup>'''.<ref>M. Furst, J. B. Saxe, and M. Sipser. Parity, circuits, and the polynomial-time hierarchy. ''Math. Systems Theory'', 17:13–27, 1984.</ref><ref>{{Citation | last1=Håstad | first1=Johan | series=Advances in Computing Research | volume=5 | title=Randomness and Computation | editor-first=Silvio | editor-last=Micali | url=http://reference.kfupm.edu.sa/content/a/l/almost_optimal_lower_bounds_for_small_de_134215.pdf | publisher=JAI Press | year=1989 | chapter=Almost Optimal Lower Bounds for Small Depth Circuits | pages=6–20 | isbn=0-89232-896-7}}</ref> | |||
As an immediate consequence of the above containments, we have that NC = AC = TC. | |||
==References== | |||
<references/> | |||
*{{cite book | last = Vollmer | first = Heribert | title = Introduction to Circuit Complexity | year = 1999 | publisher = Springer | location = Berlin | isbn = 3-540-64310-9}} | |||
{{ComplexityClasses}} | |||
[[Category:Circuit complexity]] | |||
[[Category:Complexity classes]] | |||
Latest revision as of 12:01, 17 September 2013
In theoretical computer science, and specifically computational complexity theory and circuit complexity, TC is a complexity class, and TCi is a hierarchy of complexity classes. Each class TCi consists of the languages recognized by Boolean circuits with depth and a polynomial number of unlimited-fanin AND, OR gates and Majority gates. The class TC is defined as
Relation to NC and AC
The relationship between the TC, NC and the AC hierarchy can be summarized as
In particular, we know that
The first strict containment follows from the fact that NC0 cannot compute any function that depends on all the input bits. Thus choosing a problem that is trivially in AC0 and depends on all bits separates the two classes. (For example, consider the OR function.) The strict containment AC0 ⊊ TC0 follows because parity and majority (which are both in TC0) were shown to be not in AC0.[1][2]
As an immediate consequence of the above containments, we have that NC = AC = TC.
References
- ↑ M. Furst, J. B. Saxe, and M. Sipser. Parity, circuits, and the polynomial-time hierarchy. Math. Systems Theory, 17:13–27, 1984.
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