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The definition of logspace uniformity is not exactly correct; the definition in both Sipser and Arora & Barak use log-space transducers and not logarithmic space complexity which was implied |
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{{Short description|Model of computational complexity}}
{{Use dmy dates|date=May 2019|cs1-dates=y}}
[[File:Three input boolean circuit.svg|thumb|right|300px|Example Boolean circuit. The <math>\wedge</math> nodes are [[AND gate]]s, the <math>\vee</math> nodes are [[OR gate]]s, and the <math>\neg</math> nodes are [[NOT gate]]s.]]
In [[theoretical computer science]], '''circuit complexity''' is a branch of [[computational complexity theory]] in which [[Boolean function]]s are classified according to the size or depth of the [[Boolean circuit]]s that compute them. A related notion is the circuit complexity of a [[recursive language]] that is [[Machine that always halts|decided]] by a '''uniform''' family of circuits <math>C_{1},C_{2},\ldots</math> (see below).
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==Size and depth==
A Boolean circuit with <math>n</math> input [[bit]]s is a [[directed acyclic graph]] in which every node (usually called ''gates'' in this context) is either an input node of [[in-degree]] 0 labelled by one of the <math>n</math> input bits, an [[AND gate]], an [[OR gate]], or a [[NOT gate]]. One of these gates is designated as the output gate. Such a circuit naturally computes a function of its <math>n</math> inputs. The '''size''' of a circuit is the number of gates it contains and its '''depth''' is the maximal length of a path from an input gate to the output gate.
There are two major notions of circuit complexity.<ref name="Sipser_1997"/> The '''circuit-size complexity''' of a Boolean function <math>f</math> is the minimal size of any circuit computing <math>f</math>. The '''circuit-depth complexity''' of a Boolean function <math>f</math> is the minimal depth of any circuit computing <math>f</math>.
These notions generalize when one considers the circuit complexity of any language that contains strings with different bit lengths, especially infinite [[formal language]]s. Boolean circuits, however, only allow a fixed number of input bits. Thus, no single Boolean circuit is capable of deciding such a language. To account for this possibility, one considers families of circuits <math>C_{1},C_{2},\ldots</math> where each <math>C_{n}</math> accepts inputs of size <math>n</math>. Each circuit family will naturally generate the language by circuit <math>C_{n}</math> outputting <math>1</math> when a length <math>n</math> string is a member of the family, and <math>0</math> otherwise. We say that a family of circuits is '''size minimal''' if there is no other family that decides on inputs of any size, <math>n</math>, with a circuit of smaller size than <math>C_n</math> (respectively for '''depth minimal''' families). Thus, circuit complexity is meaningful even for [[recursive language|non-recursive languages]]. The notion of a '''uniform family''' enables variants of circuit complexity to be related to algorithm based complexity measures of recursive languages. However, the non-uniform variant is helpful to find lower bounds on how complex any circuit family must be in order to decide given languages.
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===Logspace uniform===
A family of Boolean circuits <math>\{C_n:n \in \mathbb{N}\}</math> is ''[[Log-space reduction|logspace uniform]]'' if there exists a [[deterministic Turing machine]] ''M'', such that
* ''M'' runs in logarithmic work space (i.e. ''M'' is a [[log-space transducer]])
* For all <math>n \in \mathbb{N}</math>, ''M'' outputs a description of <math>C_n</math> on input <math>1^n</math>
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==Complexity classes==
{{Unreferenced section|date=October 2024}}
Many circuit complexity classes are defined in terms of class hierarchies. For each non-negative integer ''i'', there is a class [[NC (complexity)|NC<sup>i</sup>]], consisting of polynomial-size circuits of depth <math>O(\log^i(n))</math>, using bounded [[fan-in]] AND, OR, and NOT gates. The union NC of all of these classes is a subject to discussion. By considering unbounded fan-in gates, the classes [[AC (complexity)|AC<sup>i</sup>]] and AC (which is equal to NC) can be constructed. Many other circuit complexity classes with the same size and depth restrictions can be constructed by allowing different sets of gates.
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| title = Foundations of Software Technology and Theoretical Computer Science, 16th Conference, Hyderabad, India, December 18–20, 1996, Proceedings
| volume = 1180
| year = 1996| isbn = 978-3-540-62034-1 }}</ref><ref name="Shannon_1949">{{cite journal |author-last=Shannon |author-first=Claude Elwood |author-link=Claude Elwood Shannon |title=The synthesis of two-terminal switching circuits |journal=[[Bell System Technical Journal]] |date=1949 |volume=28| number=1 |pages=59–98 |doi=10.1002/j.1538-7305.1949.tb03624.x}}</ref>
<ref name="Håstad_1987">{{cite book |author-first=Johan Torkel |author-last=Håstad |author-link=Johan Torkel Håstad |title=Computational limitations of small depth circuits |date=1987 |type=Ph.D. thesis |publisher=Massachusetts Institute of Technology |url=http://www.nada.kth.se/~johanh/thesis.pdf}}</ref>
<ref name="Razborov_1985">{{cite journal |author-first=Aleksandr Aleksandrovich |author-last=Razborov |author-link=Aleksandr Aleksandrovich Razborov |title=Lower bounds on the monotone complexity of some Boolean functions |date=1985 |journal=[[Soviet Mathematics - Doklady]] |issn=0197-6788 |volume=31 |pages=354–357}}</ref>
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