https://complexityzoo.net/api.php?action=feedcontributions&user=LegionMammal978&feedformat=atomComplexity Zoo - User contributions [en]2024-03-28T21:26:09ZUser contributionsMediaWiki 1.35.0https://complexityzoo.net/index.php?title=Complexity_Zoo:A&diff=6531Complexity Zoo:A2017-05-13T22:11:52Z<p>LegionMammal978: /* AC0: Unbounded Fanin Constant-Depth Circuits */ claim appears to be erroneous</p>
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<div>__NOTOC__<br />
{{CZ-Letter-Menu|A}}<br />
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===== <span id="a0pp" style="color:red">A<sub>0</sub>PP</span>: One-Sided Analog of [[#awpp|AWPP]] =====<br />
Same as [[Complexity Zoo:S#sbp|SBP]], except that f is a nonnegative-valued [[Complexity Zoo:G#gapp|GapP]] function rather than a [[Complexity Zoo:Symbols#sharpp|#P]] function.<br />
<br />
Defined in [[zooref#vya03|[Vya03]]], where the following was also shown:<br />
<ul><br />
<li>A<sub>0</sub>PP contains [[Complexity Zoo:Q#qma|QMA]], [[#awpp|AWPP]], and [[Complexity Zoo:C#cocequalsp|coC<sub>=</sub>P]].</li><br />
<li>A<sub>0</sub>PP is contained in [[Complexity Zoo:P#pp|PP]].</li><br />
<li>If A<sub>0</sub>PP = [[Complexity Zoo:P#pp|PP]] then [[Complexity Zoo:P#ph|PH]] is contained in [[Complexity Zoo:P#pp|PP]].</li><br />
</ul><br />
<br />
Kuperberg ([[zooref#kup09|[Kup09]]]) showed that A<sub>0</sub>PP = [[Complexity Zoo:S#sbqp|SBQP]].<br />
----<br />
<br />
===== <span id="ac" style="color:red">AC</span>: Unbounded Fanin Polylogarithmic-Depth Circuits =====<br />
AC<sup>i</sup> is the class of decision problems solvable by a nonuniform family of Boolean circuits, with polynomial size, depth O(log<sup>i</sup>(n)), and unbounded fanin. The gates allowed are AND, OR, and NOT.<br />
<br />
Then AC is the union of AC<sup>i</sup> over all nonnegative i.<br />
<br />
AC<sup>i</sup> is contained in [[Complexity Zoo:N#nc|NC]]<sup>i+1</sup>; thus, AC = [[Complexity Zoo:N#nc|NC]].<br />
<br />
Contains [[Complexity Zoo:N#nl|NL]].<br />
<br />
For a random oracle A, (AC<sup>i</sup>)<sup>A</sup> is strictly contained in (AC<sup>i+1</sup>)<sup>A</sup>, and (uniform) AC<sup>A</sup> is strictly contained in P<sup>A</sup>, with probability 1 [[zooref#mil92|[Mil92]]].<br />
<br />
fo-uniform AC with depth <math>t(n)</math> is equal to [[#Complexity_Zoo:F#fot|FO[<math>t(n)</math>]]]<br />
----<br />
<br />
===== <span id="ac0" style="color:red">AC<sup>0</sup></span>: Unbounded Fanin Constant-Depth Circuits =====<br />
An especially important subclass of [[#ac|AC]], corresponding to constant-depth, unbounded-fanin, polynomial-size circuits with AND, OR, and NOT gates.<br />
<br />
Computing the [[Complexity_Garden#parity|Parity]] or [[Complexity_Garden#majority|Majority]] of n bits is not in AC<sup>0</sup> [[zooref#fss84|[FSS84]]].<br />
<br />
There are functions in AC<sup>0</sup> that are pseudorandom for all statistical tests in AC<sup>0</sup> [[zooref#nw94|[NW94]]]. But there are no functions in AC<sup>0</sup> that are pseudorandom for all statistical tests in [[Complexity Zoo:Q#qp|QP]] (quasipolynomial time) [[zooref#lmn93|[LMN93]]].<br />
<br />
[[zooref#lmn93|[LMN93]]] showed furthermore that functions with AC<sup>0</sup> circuits of depth d are learnable in [[#qp|QP]], given their outputs on O(2<sup>log(n)^O(d)</sup>) randomly chosen inputs. On the other hand, this learning algorithm is essentially optimal, unless there is a 2<sup>n^o(1)</sup> algorithm for [[Complexity_Garden#integer_factorization|factoring]] [[zooref#kha93|[Kha93]]].<br />
<br />
Although there are no good pseudorandom <i>functions</i> in AC<sup>0</sup>, [[zooref#in96|[IN96]]] showed that there are pseudorandom <i>generators</i> that stretch n bits to n+&#920;(log n), assuming the hardness of a problem based on subset sum.<br />
<br />
AC<sup>0</sup> contains [[Complexity Zoo:N#nc0|NC<sup>0</sup>]], and is contained in [[Complexity Zoo:Q#qacf0|QAC<sub>f</sub><sup>0</sup>]] and [[Complexity Zoo:M#mac0|MAC<sup>0</sup>]].<br />
<br />
In descriptive complexity, uniform AC<sup>0</sup> can be characterized as the class of problems expressible by first-order predicates with addition and multiplication operators - or indeed, with ordering and multiplication, or ordering and division (see [[zooref#lee02|[Lee02]]]). So it's equivalent to the class [[Complexity_Zoo:F#fo|FO]].<br />
<br />
[[zooref#blm98|[BLM+98]]] showed the following problem is complete for depth-k AC<sup>0</sup> circuits (with a uniformity condition):<br />
<ul> Given a grid graph of polynomial length and width k, decide whether there is a path between vertices s and t (which can be given as part of the input). </ul><br />
<br />
----<br />
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===== <span id="ac0m" style="color:red">AC<sup>0</sup>[m]</span>: [[#ac0|AC<sup>0</sup>]] With MOD m Gates =====<br />
Same as [[#ac0|AC<sup>0</sup>]], but now "MOD m" gates (for a specific m) are allowed in addition to AND, OR, and NOT gates. (A MOD m gate outputs 0 if the sum of its inputs is congruent to 0 modulo m, and 1 otherwise.)<br />
<br />
If m is a power of a prime p, then for any prime q not equal to p, deciding whether the sum of n bits is congruent to 0 modulo q is not in AC<sup>0</sup>[m] [[zooref#raz87|[Raz87]]] [[zooref#smo87|[Smo87]]]. It follows that, for any such m, AC<sup>0</sup>[m] is strictly contained in [[Complexity Zoo:N#nc1|NC<sup>1</sup>]].<br />
<br />
However, if m is a product of distinct primes (e.g. 6), then it is not even known whether AC<sup>0</sup>[m] = [[Complexity Zoo:N#np|NP]]!<br />
<br />
See also: [[Complexity Zoo:Q#qac0m|QAC<sup>0</sup>[m]]].<br />
<br />
----<br />
<br />
===== <span id="ac1" style="color:red">AC<sup>1</sup></span>: Unbounded Fanin Log-Depth Circuits =====<br />
See [[#ac|AC]].<br />
<br />
----<br />
===== <span id="acc0" style="color:red">ACC<sup>0</sup></span>: [[#ac0|AC<sup>0</sup>]] With Arbitrary MOD Gates =====<br />
Same as [[#ac0m|AC<sup>0</sup>[m]]], but now the constant-depth circuit can contain MOD m gates for <i>any</i> m.<br />
<br />
Contained in [[Complexity Zoo:T#tc0|TC<sup>0</sup>]].<br />
<br />
Indeed, can be simulated by depth-3 threshold circuits of quasipolynomial size [[zooref#yao90|[Yao90]]].<br />
<br />
According to [[zooref#all96|[All96]]], there is no good evidence for the existence of cryptographically secure functions in ACC<sup>0</sup>. <br />
<br />
There is no non-uniform ACC<sup>0</sup> circuits of polynomial size for [[Complexity Zoo:R#N:ntime|NTIMES[2<sup>n</sup>]]] and no ACC<sup>0</sup> circuit of size 2<sup>n<sup>O(1)</sup></sup> for E<sup>NP</sup> (The class [[Complexity Zoo:E#e|E]] with an [[Complexity Zoo:N#np|NP]] oracle). These are the only two known nontrivial lower bounds against ACC<sup>0</sup> and were recently discovered by [[zooref#wil11|[Wil11]]]. <br />
<br />
Contains 4-[[Complexity Zoo:P#kpbp|PBP]] [[zooref#bt88|[BT88]]].<br />
<br />
See also: [[Complexity Zoo:Q#qacc0|QACC<sup>0</sup>]].<br />
<br />
----<br />
<br />
===== <span id="ah" style="color:red">AH</span>: Arithmetic Hierarchy =====<br />
The analog of [[Complexity Zoo:P#ph|PH]] in computability theory.<br />
<br />
Let &#916;<sub>0</sub> = &#931;<sub>0</sub> = &#928;<sub>0</sub> = [[Complexity Zoo:R#r|R]]. Then for i&gt;0, let<br />
<ul><br />
<li>&#916;<sub>i</sub> = [[Complexity Zoo:R#r|R]] with &#931;<sub>i-1</sub> oracle.</li><br />
<li>&#931;<sub>i</sub> = [[Complexity Zoo:R#re|RE]] with &#931;<sub>i-1</sub> oracle.</li><br />
<li>&#928;<sub>i</sub> = [[Complexity Zoo:C#core|coRE]] with &#931;<sub>i-1</sub> oracle.</li><br />
</ul><br />
Then AH is the union of these classes for all nonnegative constant i.<br />
<br />
Each level of AH strictly contains the levels below it.<br />
<br />
An equivalent definition is: <math>\Sigma_0=\Delta_0=\Pi_0</math> is the set of numbers decided by formula with one free variable and bounded quantifier, where the primitives are + and <math>\times</math>. A bounded quantifier is of the form <math> \phi=\forall i<j \psi </math> or <math>\phi=\exists i<j \psi</math> where <math>j</math> is considered to be free in <math>\phi</math>. Then <math>\Sigma_{i+1}</math> is the sets of number validating a formula of the form <math>\exists X_1\dots\exists X_n,\psi</math> with <math>\psi\in\Delta_i</math>. <math>\Pi_i</math> is the set of formula who are negation of <math>\Sigma_i</math> formula. <math>\Delta_i=\Sigma_i\cap\Pi_i</math> <br />
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===== <span id="al" style="color:red">AL</span>: Alternating [[Complexity_Zoo:L#l|L]] =====<br />
Same as [[#ap|AP]], but for logarithmic-space instead of polynomial-time.<br />
<br />
AL = [[Complexity Zoo:P#p|P]] [[zooref#cks81|[CKS81]]].<br />
<br />
----<br />
<br />
===== <span id="all" style="color:red">ALL</span>: The Class of All Languages =====<br />
Literally, the class of ALL languages.<br />
<br />
ALL is a gargantuan beast that's been wreaking havoc in the Zoo of late.<br />
<br />
First [[zooref#aar04b|[Aar04b]]] observed that [[Complexity Zoo:P#pp|PP]]/rpoly ([[Complexity Zoo:P#pp|PP]] with polynomial-size randomized advice) equals ALL, as does [[Complexity Zoo:P#postbqp|PostBQP]]/qpoly ([[Complexity Zoo:P#postbqp|PostBQP]] with polynomial-size quantum advice).<br />
<br />
Then [[zooref#raz05|[Raz05]]] showed that [[Complexity Zoo:Q#qip|QIP]]/qpoly, and even [[Complexity Zoo:I#ip|IP]](2)/rpoly, equal ALL.<br />
<br />
Nor is it hard to show that [[Complexity Zoo:M#maexp|MA<sub>EXP</sub>]]/rpoly = ALL.<br />
<br />
On the other hand, even though [[Complexity Zoo:P#pspace|PSPACE]] contains [[Complexity Zoo:P#pp|PP]], and [[Complexity Zoo:E#expspace|EXPSPACE]] contains [[#maexp|MA<sub>EXP</sub>]], it's easy to see that [[Complexity Zoo:P#pspace|PSPACE]]/rpoly = [[Complexity Zoo:P#pspace|PSPACE]]/poly and [[Complexity Zoo:E#expspace|EXPSPACE]]/rpoly = [[Complexity Zoo:E#expspace|EXPSPACE]]/poly are not ALL.<br />
<br />
So does ALL have no respect for complexity class inclusions at ALL? (Sorry.)<br />
<br />
It is not as contradictory as it first seems. The deterministic base class in all of these examples is modified by computational non-determinism ''after'' it is modified by advice. For example, [[Complexity Zoo:M#maexp|MA<sub>EXP</sub>]]/rpoly means M(A<sub>EXP</sub>/rpoly), while ([[Complexity Zoo:M#maexp|MA<sub>EXP</sub>]])/rpoly equals [[Complexity Zoo:M#maexp|MA<sub>EXP</sub>]]/poly by a standard argument. In other words, it's only the verifier, not the prover or post-selector, who receives the randomized or quantum advice. The prover knows a description of the advice state, but not its measured values. Modification by /rpoly does preserve class inclusions when it is applied after other changes.<br />
<br />
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===== <span id="alogtime" style="color:red">ALOGTIME</span>: Logarithmic time alternating RAM =====<br />
<br />
ALOGTIME is the class of languages decidable in logarithmic time by a random access alternating Turing machine.<br />
<br />
Known to be equal to U<sub>E<sup>*</sup></sub>-uniform [[Complexity Zoo:N#nc1|NC<sup>1</sup>]].<br />
<br />
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<br />
===== <span id="algppoly" style="color:red">AlgP/poly</span>: Polynomial-Size Algebraic Circuits =====<br />
The class of multivariate polynomials over the integers that can be evaluated using a polynomial (in the input size n) number of additions, subtractions, and multiplications, together with the constants -1 and 1. The class is nonuniform, in the sense that the polynomial for each input size n can be completely different.<br />
<br />
Named in [[zooref#imp02|[Imp02]]], though it has been considered since the 1970's.<br />
<br />
If [[Complexity Zoo:P#p|P]] = [[Complexity Zoo:B#bpp|BPP]] (or even [[Complexity Zoo:B#bpp|BPP]] is contained in [[Complexity Zoo:N#ne|NE]]), then either [[Complexity Zoo:N#nexp|NEXP]] is not in [[Complexity Zoo:P#ppoly|P/poly]], or else the permanent polynomial of a matrix is not in AlgP/poly [[zooref#ki02|[KI02]]].<br />
<br />
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===== <span id="almostnp" style="color:red">Almost-[[Complexity Zoo:N#np|NP]]</span>: Languages Almost Surely in [[Complexity Zoo:N#np|NP]]<sup>A</sup> =====<br />
The class of problems that are in [[Complexity Zoo:N#np|NP]]<sup>A</sup> with probability 1, where A is an oracle chosen uniformly at random.<br />
<br />
Equals [[#am|AM]] [[zooref#nw94|[NW94]]].<br />
<br />
----<br />
===== <span id="almostp" style="color:red">Almost-[[Complexity Zoo:P#p|P]]</span>: Languages Almost Surely in [[Complexity Zoo:P#p|P]]<sup>A</sup> =====<br />
The class of problems that are in [[Complexity Zoo:P#p|P]]<sup>A</sup> with probability 1, where A is an oracle chosen uniformly at random.<br />
<br />
Equals [[Complexity Zoo:B#bpp|BPP]] [[zooref#bg81|[BG81]]].<br />
<br />
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===== <span id="almostpspace" style="color:red">Almost-[[Complexity Zoo:P#pspace|PSPACE]]</span>: Languages Almost Surely in [[Complexity Zoo:P#pspace|PSPACE]]<sup>A</sup> =====<br />
The class of problems that are in [[Complexity Zoo:P#pspace|PSPACE]]<sup>A</sup> with probability 1, where A is an oracle chosen uniformly at random.<br />
<br />
Almost-PSPACE is not known to equal [[Complexity Zoo:P#pspace|PSPACE]] -- rather surprisingly, given the fact that [[Complexity Zoo:P#pspace|PSPACE]] equals BPPSPACE and even [[Complexity Zoo:P#ppspace|PPSPACE]].<br />
<br />
What's known is that Almost-PSPACE = BP<sup>exp</sup>&#149;[[Complexity Zoo:P#pspace|PSPACE]], where [[Zoo Operators#bpexp|BP<sup>exp</sup>&#149;]] is like the [[Zoo Operators#bp|BP&#149;]] operator but with exponentially-long strings [[zooref#bvw98|[BVW98]]]. It follows that Almost-PSPACE is contained in [[Complexity Zoo:N#nexp|NEXP]]<sup>[[Complexity Zoo:N#np|NP]]</sup> &#8745; [[Complexity Zoo:A#conexp|coNEXP]]<sup>[[Complexity Zoo:N#np|NP]]</sup>.<br />
<br />
Whereas both BP<sup>exp</sup>&#149;[[Complexity Zoo:P#pspace|PSPACE]] and BPPSPACE machines are allowed exponentially many random bits, the former has a reusable record of all of these bits on a witness tape, while the latter can only preserve a fraction of them on the work tape.<br />
<br />
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===== <span id="am" style="color:red">AM</span>: Arthur-Merlin =====<br />
The class of decision problems for which a "yes" answer can be verified by an <i>Arthur-Merlin protocol</i>, as follows.<br />
<br />
Arthur, a [[Complexity Zoo:B#bpp|BPP]] (i.e. probabilistic polynomial-time) verifier, generates a "challenge" based on the input, and sends it together with his random coins to Merlin. Merlin sends back a response, and then Arthur decides whether to accept. Given an algorithm for Arthur, we require that<br />
<ol><br />
<li>If the answer is "yes," then Merlin can act in such a way that Arthur accepts with probability at least 2/3 (over the choice of Arthur's random bits).</li><br />
<li>If the answer is "no," then however Merlin acts, Arthur will reject with probability at least 2/3.</li><br />
</ol><br />
Surprisingly, it turns out that such a system is just as powerful as a <i>private-coin</i> one, in which Arthur does not need to send his random coins to Merlin [[zooref#gs86|[GS86]]]. So, Arthur never needs to hide information from Merlin.<br />
<br />
Furthermore, define AM[k] similarly to AM, except that Arthur and Merlin have k rounds of interaction. Then for all constant k&gt;2, AM[k] = AM[2] = AM [[zooref#bm88|[BM88]]]. Also, the result of [[zooref#gs86|[GS86]]] can then be stated as follows: [[Complexity Zoo:I#ip|IP]][k] is contained in AM[k+2] for every k (constant or non-constant).<br />
<br />
AM contains [[Complexity_Garden#graph_isomorphism|graph nonisomorphism]].<br />
<br />
Contains [[Complexity Zoo:N#np|NP]], [[Complexity Zoo:B#bpp|BPP]], and [[Complexity Zoo:S#szk|SZK]], and is contained in [[Complexity Zoo:P#pi2p|&#928;<sub>2</sub>P]] and [[Complexity Zoo:N#nppoly|NP/poly]].<br />
<br />
If AM contains [[Complexity Zoo:C#conp|coNP]] then [[Complexity Zoo:P#ph|PH]] collapses to [[Complexity Zoo:S#sigma2p|&#931;<sub>2</sub>P]] &#8745; [[Complexity Zoo:P#pi2p|&#928;<sub>2</sub>P]] [[zooref#bhz87|[BHZ87]]].<br />
<br />
There exists an oracle relative to which AM is not contained in [[Complexity Zoo:P#pp|PP]] [[zooref#ver92|[Ver92]]].<br />
<br />
AM = [[Complexity Zoo:N#np|NP]] under a strong derandomization assumption: namely that some language in [[Complexity Zoo:N#ne|NE]] &#8745; [[Complexity Zoo:C#cone|coNE]] requires nondeterministic circuits of size 2<sup>&#937;(n)</sup> ([[zooref#mv99|[MV99]]], improving [[zooref#km99|[KM99]]]). (A nondeterministic circuit C has two inputs, x and y, and accepts on x if there exists a y such that C(x,y)=1.)<br />
<br />
----<br />
===== <span id="amcc" style="color:red">AM<sup>cc</sup></span>: Communication Complexity [[#am|AM]] =====<br />
<br />
Here, Alice and Bob collectively constitute "Arthur", and Merlin sends a message that depends on the input and all the randomness, and the cost is defined to be the bit length of Merlin's message plus the communication cost of the ensuing verification protocol between Alice and Bob. (Without loss of generality, the verification protocol consists only of checking containment in a rectangle, since Merlin could always include the transcript of the verification in his message.)<br />
<br />
It is open to prove that there exists an explicit two-party function that is not in AM<sup>cc</sup>.<br />
<br />
Contained in [[Complexity Zoo:P#phcc|PH<sup>cc</sup>]].<br />
<br />
AM<sup>cc</sup> &#8745; coAM<sup>cc</sup> is not contained in [[Complexity Zoo:P#ppcc|PP<sup>cc</sup>]] if partial functions are allowed [[zooref#kla11|[Kla11]]].<br />
<br />
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===== <span id="amexp" style="color:red">AM<sub>EXP</sub></span>: Exponential-Time [[#am|AM]] =====<br />
Same as [[#am|AM]], except that Arthur is exponential-time and can exchange exponentially long messages with Merlin.<br />
<br />
Contains [[Complexity Zoo:M#maexp|MA<sub>EXP</sub>]], and is contained in [[Complexity Zoo:E#eh|EH]] and indeed [[Complexity Zoo:S#s2exppnp|S<sub>2</sub>-EXP&#149;P<sup>NP</sup>]].<br />
<br />
If [[Complexity Zoo:C#conp|coNP]] is contained in [[#ampolylog|AM[polylog]]] then [[Complexity Zoo:E#eh|EH]] collapses to AM<sub>EXP</sub> [[zooref#pv04|[PV04]]].<br />
<br />
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===== <span id="amicoam" style="color:red">AM &#8745; coAM</span> =====<br />
The class of decision problems for which both "yes" and "no" answers can be verified by an [[#am|AM]] protocol.<br />
<br />
If [[Complexity Zoo:E#exp|EXP]] requires exponential time even for [[#am|AM]] protocols, then AM &#8745; coAM = [[Complexity Zoo:N#npiconp|NP &#8745; coNP]] [[zooref#gst03|[GST03]]].<br />
<br />
There exists an oracle relative to which AM &#8745; coAM is not contained in [[Complexity Zoo:P#pp|PP]] [[zooref#ver95|[Ver95]]].<br />
<br />
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===== <span id="ampolylog" style="color:red">AM[polylog]</span>: [[#am|AM]] With Polylog Rounds =====<br />
Same as [[#am|AM]], except that we allow polylog(n) rounds of interaction between Arthur and Merlin instead of a constant number.<br />
<br />
Not much is known about AM[polylog] -- for example, whether it sits in [[Complexity Zoo:P#ph|PH]]. However, [[zooref#ss04|[SS04]]] show that if AM[polylog] contains [[Complexity Zoo:C#conp|coNP]], then [[Complexity Zoo:E#eh|EH]] collapses to [[Complexity Zoo:S#s2exppnp|S<sub>2</sub>-EXP&#149;P<sup>NP</sup>]]. ([[zooref#pv04|[PV04]]] improved the collapse to [[#amexp|AM<sub>EXP</sub>]].)<br />
<br />
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===== <span id="ampmp" style="color:red">AmpMP</span>: Amplifiable [[Complexity Zoo:M#mp2|MP]] =====<br />
The class of decision problems such that for some [[Complexity Zoo:Symbols#sharpp|#P]] function f(x,0<sup>m</sup>),<br />
<ol><br />
<li>The answer on input x is 'yes' if and only if the middle bit of f(x) is 1.</li><br />
<li>The m bits of f(x) to the left and right of the middle bit are all 0.</li><br />
</ol><br />
Defined in [[zooref#gkr95|[GKR+95]]].<br />
<br />
Contains [[Complexity Zoo:P#ph|PH]] and Contained in [[Complexity Zoo:M#mp2|MP]].<br />
<br />
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===== <span id="amppbqp" style="color:red">AmpP-BQP</span>: [[Complexity Zoo:B#bqp|BQP]] Restricted To [[Zoo_Exhibit#ampp|AmpP]] States =====<br />
Similar to [[Complexity Zoo:T#treebqp|TreeBQP]] except that the quantum computer's state at each time step is restricted to being exponentially close to a state in [[Zoo_Exhibit#ampp|AmpP]] (that is, a state for which the amplitudes are computable by a classical polynomial-size circuit).<br />
<br />
Defined in [[zooref#aar03b|[Aar03b]]], where it was also observed that AmpP-BQP is contained in the third level of [[Complexity Zoo:P#ph|PH]], just as [[Complexity Zoo:T#treebqp|TreeBQP]] is.<br />
<br />
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===== <span id="ap" style="color:red">AP</span>: Alternating [[Complexity Zoo:P#p|P]] =====<br />
An <i>alternating Turing machine</i> is a nondeterministic machine with two kinds of states, AND states and OR states. It accepts if and only if the tree of all computation paths, considered as an AND-OR tree, evaluates to 1. (Here 'Accept' corresponds to 1 and 'Reject' to 0.)<br />
<br />
Then AP is the class of decision problems solvable in polynomial time by an alternating Turing machine.<br />
<br />
AP = [[Complexity Zoo:P#pspace|PSPACE]] [[zooref#cks81|[CKS81]]].<br />
<br />
The abbreviation AP is also used for Approximable in Polynomial Time, see [[#axp|AxP]].<br />
<br />
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===== <span id="app" style="color:red">APP</span>: Amplified [[Complexity Zoo:P#pp|PP]] =====<br />
Roughly, the class of decision problems for which the following holds. For all polynomials p(n), there exist [[Complexity Zoo:G#gapp|GapP]] functions f and g such that for all inputs x with n=|x|,<br />
<ol><br />
<li>If the answer is "yes" then 1 &gt; f(x)/g(1<sup>n</sup>) &gt; 1-2<sup>-p(n)</sup>.</li><br />
<li>If the answer is "no" then 0 &lt; f(x)/g(1<sup>n</sup>) &lt; 2<sup>-p(n)</sup>.</li><br />
</ol><br />
Defined in [[zooref#li93|[Li93]]], where the following was also shown:<br />
<ul><br />
<li>APP is contained in [[Complexity Zoo:P#pp|PP]], and indeed is low for [[Complexity Zoo:P#pp|PP]].</li><br />
<li>APP is closed under intersection, union, and complement.</li><br />
</ul><br />
APP contains [[#awpp|AWPP]] [[zooref#fen02|[Fen02]]].<br />
<br />
The abbreviation APP is also used for Approximable in Probabilistic Polynomial Time, see [[#axpp|AxPP]].<br />
<br />
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<br />
===== <span id="apx" style="color:red">APX</span>: Approximable =====<br />
The subclass of [[Complexity Zoo:N#npo|NPO]] problems that admit constant-factor approximation algorithms. (I.e., there is a polynomial-time algorithm that is guaranteed to find a solution within a constant factor of the optimum cost.)<br />
<br />
Contains [[Complexity Zoo:P#ptas|PTAS]].<br />
<br />
Equals the closure of [[Complexity Zoo:M#maxsnp|MaxSNP]] and of [[Complexity Zoo:M#maxnp|MaxNP]] under [[Complexity Zoo:P#ptas|PTAS]] reduction [[zooref#kms99|[KMS+99]]], [[zooref#ct94|[CT94]]].<br />
<br />
Defined in [[zooref#acg99|[ACG+99]]].<br />
<br />
----<br />
===== <span id="atime" style="color: red">ATIME</span>: Alternating [[Complexity Zoo:D#dtime|TIME]] =====<br />
'''ATIME'''(f(n)) is the class of problems for which there are alternating Turing machines (see [[#ap|AP]]) which decide the problem in time bounded by f(n).<br />
<br />
In particular, [[#ap|AP]] = ATIME(poly(n)).<br />
<br />
----<br />
<br />
===== <span id="aucspace" style="color:red">AUC-SPACE(f(n))</span>: Randomized Alternating f(n)-Space =====<br />
The class of problems decidable by an O(f(n))-space Turing machine with three kinds of quantifiers: existential, universal, and randomized.<br />
<br />
Contains [[Complexity Zoo:G#ganspace|GAN-SPACE(f(n))]].<br />
<br />
AUC-SPACE(poly(n)) = [[Complexity Zoo:S#saptime|SAPTIME]] = [[Complexity Zoo:P#pspace|PSPACE]] [[zooref#pap83|[Pap83]]].<br />
<br />
[[zooref#con92|[Con92]]] shows that AUC-SPACE(log n) has a natural complete problem, and is contained in [[Complexity Zoo:N#npiconp|NP &#8745; coNP]].<br />
<br />
----<br />
===== <span id="auxpda" style="color:red">AuxPDA</span>: Auxiliary Pushdown Automata =====<br />
Equivalent to [[Complexity Zoo:N#nauxpdap|NAuxPDA<sup>p</sup>]] without the running-time restriction.<br />
<br />
Equals [[Complexity Zoo:P#p|P]] [[zooref#coo71b|[Coo71b]]].<br />
<br />
----<br />
===== <span id="avbpp" style="color:red">AVBPP</span>: Average-Case [[Complexity Zoo:B#bpp|BPP]] =====<br />
Defined in [[zooref#ow93|[OW93]]] to be the class of decision problems that have a good average-case [[Complexity Zoo:B#bpp|BPP]] algorithm, whenever the input is chosen from an efficiently samplable distribution.<br />
<br />
Note that this is <i>not</i> the same as the [[Complexity Zoo:B#bpp|BPP]] version of [[#avgp|AvgP]].<br />
<br />
----<br />
===== <span id="avge" style="color:red">AvgE</span>: Average Exponential-Time With Linear Exponent =====<br />
Has the same relation to [[Complexity Zoo:E#e|E]] as [[#avgp|AvgP]] does to [[Complexity Zoo:P#p|P]].<br />
<br />
----<br />
===== <span id="avgp" style="color:red">AvgP</span>: Average Polynomial-Time =====<br />
A <i>distributional problem</i> consists of a decision problem A, and a probability distribution &#956; over problem instances.<br />
<br />
A function f, from strings to integers, is <i>polynomial on &#956;-average</i> if there exists a constant &#949;&gt;0 such that the expectation of f<sup>&#949;</sup>(x) is finite, when x is drawn from &#956;.<br />
<br />
Then (A,&#956;) is in AvgP if there is an algorithm for A whose running time is polynomial on &#956;-average.<br />
<br />
This convoluted definition is due to Levin [[zooref#lev86|[Lev86]]], who realized that simpler definitions lead to classes that fail to satisfy basic closure properties. Also see [[zooref#gol97|[Gol97]]] for more information.<br />
<br />
If AvgP = [[Complexity Zoo:D#distnp|DistNP]] then [[Complexity Zoo:E#exp|EXP]] = [[Complexity Zoo:N#nexp|NEXP]] [[zooref#bcg92|[BCG+92]]].<br />
<br />
Strictly contained in [[Complexity Zoo:H#heurp|HeurP]] [[zooref#ns05|[NS05]]].<br />
<br />
See also: [[Complexity Zoo:N#nppsamp|(NP,P-samplable)]].<br />
<br />
----<br />
<br />
===== <span id="awp" style="color:red">AW[P]</span>: Alternating [[Complexity Zoo:W#wp|W[P]]] =====<br />
Same as [[#awsat|AW[SAT]]] but with 'circuit' instead of 'formula.'<br />
<br />
Has the same relation to [[#awsat|AW[SAT]]] as [[Complexity Zoo:W#wp|W[P]]] has to [[Complexity Zoo:W#wsat|W[SAT]]].<br />
<br />
Defined in [[zooref#df99|[DF99]]].<br />
<br />
----<br />
===== <span id="awpp" style="color:red">AWPP</span>: Almost [[Complexity Zoo:W#wpp|WPP]] =====<br />
The class of decision problems solvable by an [[Complexity Zoo:N#np|NP]] machine such that for some polynomial-time computable (i.e. [[Complexity Zoo:F#fp|FP]]) function f,<br />
<ol><br />
<li>If the answer is "no," then the difference between the number of accepting and rejecting paths is non-negative and at most 2<sup>-poly(n)</sup>f(x).</li><br />
<li>If the answer is "yes," then the difference is between (1-2<sup>-poly(n)</sup>)f(x) and f(x).</li><br />
</ol><br />
Defined in [[zooref#ffk94|[FFK94]]].<br />
<br />
Contains [[Complexity Zoo:B#bqp|BQP]] [[zooref#fr98|[FR98]]], [[Complexity Zoo:W#wapp|WAPP]] [[zooref#bgm02|[BGM02]]], [[Complexity Zoo:L#lwpp|LWPP]], and [[Complexity Zoo:W#wpp|WPP]].<br />
<br />
Contained in [[#app|APP]] [[zooref#fen02|[Fen02]]].<br />
<br />
----<br />
<br />
===== <span id="awsat" style="color:red">AW[SAT]</span>: Alternating [[Complexity Zoo:W#wsat|W[SAT]]] =====<br />
Basically has the same relation to [[Complexity Zoo:W#wsat|W[SAT]]] as [[Complexity Zoo:P#pspace|PSPACE]] does to [[Complexity Zoo:N#np|NP]].<br />
<br />
The class of decision problems of the form (x,r,k<sub>1</sub>,...,k<sub>r</sub>) (r,k<sub>1</sub>,...,k<sub>r</sub> parameters), that are fixed-parameter reducible to the following problem, for some constant h:<br />
<ul><br />
'''Parameterized QBFSAT:''' Given a Boolean formula F (with no restriction on depth), over disjoint variable sets S<sub>1</sub>,...,S<sub>r</sub>. Does there exist an assignment to S<sub>1</sub> of Hamming weight k<sub>1</sub>, such that for all assignments to S<sub>2</sub> of Hamming weight k<sub>2</sub>, etc. (alternating 'there exists' and 'for all'), F is satisfied?<br />
</ul><br />
See [[Complexity Zoo:W#w1|W[1]]] for the definition of fixed-parameter reducibility.<br />
<br />
Defined in [[zooref#df99|[DF99]]].<br />
<br />
Contains [[#awstar|AW[*]]], and is contained in [[#awp|AW[P]]].<br />
<br />
----<br />
===== <span id="awstar" style="color:red">AW[*]</span>: Alternating [[Complexity Zoo:W#wstar|W[*]]] =====<br />
The union of [[#awt|AW[t]]] over all t.<br />
<br />
----<br />
===== <span id="awt" style="color:red">AW[t]</span>: Alternating [[Complexity Zoo:W#wt|W[t]]] =====<br />
Has the same relation to [[Complexity Zoo:W#wt|W[t]]] as [[Complexity Zoo:P#pspace|PSPACE]] does to [[Complexity Zoo:N#np|NP]].<br />
<br />
Same as [[#awsat|AW[SAT]]], except that the formula F can have depth at most t.<br />
<br />
Defined in [[zooref#df99|[DF99]]].<br />
<br />
Contained in [[#awstar|AW[*]]].<br />
<br />
[[zooref#dft98|[DFT98]]] show that for all t, [[#awt|AW[t]]] = [[#awstar|AW[*]]].<br />
<br />
----<br />
<br />
===== <span id="axp" style="color:red">AxP</span>: Approximable in Polynomial Time =====<br />
Usually called AP in the literature. I've renamed it AxP to distinguish it from the "other" [[#ap|AP]].<br />
<br />
The class of real-valued functions from {0,1}<sup>n</sup> to [0,1] that can be approximated within any &epsilon;>0 by a deterministic Turing machine in time polynomial in n and 1/&epsilon;.<br />
<br />
Defined by [[zooref#krc00|[KRC00]]], who also showed that the set of AxP machines is in [[Complexity Zoo:R#re|RE]].<br />
<br />
----<br />
===== <span id="axpp" style="color:red">AxPP</span>: Approximable in Probabilistic Polynomial Time =====<br />
Usually called APP. I've renamed it AxPP to distinguish it from the "other" [[#app|APP]].<br />
<br />
The class of real-valued functions from {0,1}<sup>n</sup> to [0,1] that can be approximated within any &epsilon;>0 by a probabilistic Turing machine in time polynomial in n and 1/&epsilon;.<br />
<br />
Defined by [[zooref#krc00|[KRC00]]], who also show the following:<br />
<ul><br />
<li>Approximating the acceptance probability of a Boolean circuit is AxPP-complete. The authors argue that this makes AxPP a more natural class than [[Complexity Zoo:B#bpp|BPP]], since the latter is not believed to have complete problems.</li><br />
<li>If AxPP = [[#axp|AxP]], then [[Complexity Zoo:B#bpp|BPP]] = [[Complexity Zoo:P#p|P]].</li><br />
<li>On the other hand, there exists an oracle relative to which [[Complexity Zoo:B#bpp|BPP]] = [[Complexity Zoo:P#p|P]] but AxPP does not equal [[#axp|AxP]].</li><br />
</ul><br />
AxPP is recursively enumerable [[zooref#jer07|[Jeř07]]].</div>LegionMammal978https://complexityzoo.net/index.php?title=Complexity_Zoo:F&diff=6530Complexity Zoo:F2017-05-13T22:11:23Z<p>LegionMammal978: Just removed the claim altogether</p>
<hr />
<div>__NOTOC__<br />
{{CZ-Letter-Menu|F}}<br />
<br />
===== <span id="fbpp" style="color:red">FBPP</span>: Function [[Complexity Zoo:B#bpp|BPP]] =====<br />
Has the same relation to [[Complexity Zoo:B#bpp|BPP]] as [[#fnp|FNP]] does to [[Complexity Zoo:N#np|NP]]. Equivalently, it is the randomised analogue of [[#fp|FP]].<br />
<br />
----<br />
<br />
===== <span id="fbqp" style="color:red">FBQP</span>: Function [[Complexity Zoo:B#bqp|BQP]] =====<br />
Has the same relation to [[Complexity Zoo:B#bqp|BQP]] as [[#fnp|FNP]] does to [[Complexity Zoo:N#np|NP]].<br />
<br />
There exists an oracle relative to which [[Complexity Zoo:P#pls|PLS]] is not contained in [[#fbqp|FBQP]] [[zooref#aar03|[Aar03]]].<br />
<br />
----<br />
===== <span id="fert" style="color:red">FERT</span>: Fixed Error Randomized Time =====<br />
FERT and [[#fpert|FPERT]] are parameterized classes. FERT formally defined as the class of decision problems of the form (x, k), decidable in polynomial time by a probabilistic Turing Machine such that<br />
<ol><br />
<li>If the answer is yes, the probability of acceptance is at least 1/2 + min(f(k),1/|x|<sup>c</sup>)</li><br />
<li>If the answer is no, the probability of acceptance is at most 1/2</li><br />
</ol><br />
<br />
Here, f is an arbitrary function (from the reals to <0,1/2]).<br />
<br />
Defined in [[zooref#kw15|[KW15]]]. Contains [[Complexity Zoo:B#bpp|BPP]] and is contained in para-[[Complexity Zoo:P#pp|PP]] and in [[#fpert|FPERT]].<br />
----<br />
<br />
===== <span id="fpert" style="color:red">FPERT</span>: Fixed Parameter and Error Randomized Time =====<br />
[[#fert|FERT]] and FPERT are parameterized classes. FPERT is formally defined as the class of decision problems of the form (x, k1, k2), decidable in time f1(k1) * p(|x|) by a probabilistic Turing Machine such that<br />
<ol><br />
<li>If the answer is yes, the probability of acceptance is at least 1/2 + min(f2(k2),1/|x|<sup>c</sup>)</li><br />
<li>If the answer is no, the probability of acceptance is at most 1/2</li><br />
</ol><br />
<br />
Here, f1 and f2 are arbitrary functions (f2 from the reals to <0,1/2]) and p is a polynomial.<br />
<br />
Defined in [[zooref#kw15|[KW15]]]. Contains [[#fert|FERT]] and [[#fpt|FPT]] and is contained in para-[[Complexity Zoo:N#np|NP]]<sup>[[Complexity Zoo:P#pp|PP]]</sup>. <br />
----<br />
<br />
===== <span id="few" style="color:red">Few</span>: [[#fewp|FewP]] With Flexible Acceptance Mechanism =====<br />
The class of decision problems solvable by an [[Complexity Zoo:N#np|NP]] machine such that<br />
<ol><br />
<li>The number of accepting paths a is bounded by a polynomial in the size of the input x.</li><br />
<li>For some polynomial-time predicate Q, Q(x,a) is true if and only if the answer is 'yes.'</li><br />
</ol><br />
Also called FewPaths.<br />
<br />
Defined in [[zooref#ch89|[CH89]]].<br />
<br />
Contains [[#fewp|FewP]], and is contained in [[Complexity Zoo:P#p|P]]<sup>[[#fewp|FewP]]</sup> [[zooref#kob89|[Kob89]]] and in [[Complexity Zoo:S#spp|SPP]] [[zooref#ffk94|[FFK94]]].<br />
<br />
See also the survey [[zooref#tor90|[Tor90]]].<br />
<br />
----<br />
===== <span id="fewexp" style="color:red">FewEXP</span>: {{zcls|n|nexp}} With Few Witnesses =====<br />
The class of decision problems solvable by an {{zcls|n|nexp}} machine such that, given a "yes" instance, no more than an exponential number of computation paths accept.<br />
<br />
Contained in {{zcls|m|mip}}<nowiki>[</nowiki>{{zcls|n|np}}<sup>FewEXP</sup><nowiki>]</nowiki> (MIP where provers are not unbounded in computational power, but are limited to NP<sup>FewEXP</sup>) {{zcite|AKS+94}}.<br />
<br />
Alternatively, FewEXP can refer to the sparsity of accepting paths in a given instance. In {{zcite|AKR+03}}, the authors show that the conjectures "FewEXP search instances are {{zcls|e|exp}}-solvable" and "FewEXP decision instances are EXP/poly-solvable" are equivalent. That is, for all NEXP machines <math>N</math>, the following conditions are equivalent:<br />
# There exists an EXP machine <math>M</math> such that if given a string <math>x</math>, <math>N(x)</math> accepts and has exponentially bounded accepting paths, then <math>M(x)</math> produces one such path.<br />
# There exists an EXP/poly machine <math>M</math> which accepts a string <math>x</math> if and only <math>N(x)</math> accepts.<br />
<br />
----<br />
<br />
===== <span id="fewp" style="color:red">FewP</span>: [[Complexity Zoo:N#np|NP]] With Few Witnesses =====<br />
The class of decision problems solvable by an [[Complexity Zoo:N#np|NP]] machine such that<br />
<ol><br />
<li>If the answer is 'no,' then all computation paths reject.</li><br />
<li>If the answer is 'yes,' then at least one path accepts; furthermore, the number of accepting paths is upper-bounded by a polynomial in n, the size of the input.</li><br />
</ol><br />
Defined in [[zooref#ar88|[AR88]]].<br />
<br />
Is contained in [[Complexity Zoo:Symbols#parityp|&#8853;P]] [[zooref#ch89|[CH89]]].<br />
<br />
There exists an oracle relative to which [[Complexity Zoo:P#p|P]], [[Complexity Zoo:U#up|UP]], FewP, and [[Complexity Zoo:N#np|NP]] are all distinct [[zooref#rub88|[Rub88]]].<br />
<br />
Also, there exists an oracle relative to which FewP does not have a Turing-complete set [[zooref#hjv93|[HJV93]]].<br />
<br />
Contained in [[#few|Few]].<br />
<br />
See also the survey [[zooref#tor90|[Tor90]]].<br />
<br />
----<br />
<br />
===== <span id="fh" style="color:red">FH</span>: Fourier Hierarchy =====<br />
FH<sub>k</sub> is the class of problems solvable by a uniform family of polynomial-size quantum circuits, with k levels of Hadamard gates and all other gates preserving the computational basis. (Conditional phase flip gates are fine, for example.) Thus<br />
<ul><br />
<li>FH<sub>0</sub> = [[Complexity Zoo:P#p|P]]</li><br />
<li>FH<sub>1</sub> = [[Complexity Zoo:B#bpp|BPP]]</li><br />
<li>FH<sub>2</sub> contains [[Complexity_Garden#integer_factorization|factoring]] because of Kitaev's phase estimation algorithm</li><br />
</ul><br />
It is an open problem to show that the Fourier hierarchy is infinite relative to an oracle (that is, FH<sub>k</sub> is strictly contained in FH<sub>k+1</sub>).<br />
<br />
Defined in [[zooref#shi03|[Shi03]]].<br />
<br />
----<br />
<br />
===== <span id="fixp" style="color:red">FIXP</span>: Fixed Point =====<br />
The class of fixed point problems. In the framework of fixed point problems, an instance I is associated with a (continuous) function F<sub>I</sub>, and a solution of I is a fixed point of F<sub>I</sub>. <br />
<br />
Properties of FIXP problems:<br />
<ol><br />
<li> the function F<sub>I</sub> is represented by an algebraic circuit over {+, -, *, /, max, min} with rational constants<br />
<li> there is a polynomial time algorithm that computes the circuit from I.<br />
</ol><br />
<br />
Every FIXP problem has Partial Computation, Decision, (Strong) Approximation, and Existence counterparts; these can all be solved in PSPACE.<br />
<br />
The Nash equilibrium problem for 3 or more players is FIXP-complete.<br />
<br />
Linear-FIXP = PPAD.<br />
<br />
Defined in [[zooref#ey07|[EY07]]].<br />
----<br />
<br />
===== <span id="fnl" style="color:red">FNL</span>: Function [[Complexity Zoo:N#nl|NL]] =====<br />
Has the same relation to [[Complexity Zoo:N#nl|NL]] as [[#fnp|FNP]] does to [[Complexity Zoo:N#np|NP]].<br />
<br />
Defined by [[zooref#aj93|[AJ93]]], who also showed that if [[Complexity Zoo:N#nl|NL]] = [[Complexity Zoo:U#ul|UL]], then FNL is contained in [[Complexity Zoo:Symbols#sharpl|#L]].<br />
<br />
----<br />
<br />
===== <span id="fnlpoly" style="color:red">FNL/poly</span>: Nonuniform [[#fnl|FNL]] =====<br />
Has the same relation to [[#fnl|FNL]] as [[Complexity Zoo:P#ppoly|P/poly]] does to [[Complexity Zoo:P#p|P]].<br />
<br />
Contained in [[Complexity Zoo:Symbols#sharplpoly|#L/poly]] [[zooref#ra00|[RA00]]].<br />
<br />
----<br />
===== <span id="fnp" style="color:red">FNP</span>: Function [[Complexity Zoo:N#np|NP]] =====<br />
The class of function problems of the following form:<br />
<ul> Given an input x and a polynomial-time predicate F(x,y), if there exists a y satisfying F(x,y) then output any such y, otherwise output 'no.' </ul><br />
FNP generalizes [[Complexity Zoo:N#np|NP]], which is defined in terms of decision problems only.<br />
<br />
Actually the word "function" is misleading, since there could be many valid outputs y. That's unavoidable, since given a predicate F there's no "syntactic" criterion ensuring that y is unique.<br />
<br />
[[#fp|FP]] = FNP if and only if [[Complexity Zoo:P#p|P]] = [[Complexity Zoo:N#np|NP]].<br />
<br />
Contains [[Complexity Zoo:T#tfnp|TFNP]].<br />
<br />
A basic question about FNP problems is whether they're <i>self-reducible</i>; that is, whether they reduce to the corresponding [[Complexity Zoo:N#np|NP]] decision problems. Although this is true for all [[Complexity Zoo:N#npc|NPC]] problems, [[zooref#bg94|[BG94]]] shows that if [[Complexity Zoo:E#ee|EE]] does not equal [[Complexity Zoo:N#nee|NEE]], then there is a problem in [[Complexity Zoo:N#np|NP]] such that <i>no</i> corresponding FNP problem can be reduced to it. [[zooref#bg94|[BG94]]] cites Impagliazzo and Sudan as giving the same conclusion under the assumption that [[Complexity Zoo:N#ne|NE]] does not equal [[Complexity Zoo:C#cone|coNE]].<br />
<br />
----<br />
===== <span id="fo" style="color:red">FO</span>: First-Order logic =====<br />
First order logic is the smallest logical class of logic language. It is the base of [[zooref#imm98|Descriptive complexity]] and equal to the class [[Complexity_Zoo:A#ac0|AC<sup>0</sup>]].<br />
<br />
When we use logic formalism, the input is a structure, usually it is either strings (of bits or over an alphabet) whose elements are position of the strings, or graphs whose elements are vertices. The mesure of the input will there be the size of the structure.<br />
Whatever the structure is, we can suppose that there are relation that you can test, by example <math>E(x,y)</math> is true iff there is an edge from <math>x</math> to <math>y</math> if the structure is a graph, and <math>P(n)</math> is true iff the nth letter of the string is 1. We also have constant, who are special elements of the structure, by example if we want to check reachability in a graph, we will have to choose two constant s and t.<br />
<br />
In descriptive complexity we almost always suppose that there is a total order over the elements and that we can check equality between elements. This let us consider elements as number, <math>x</math> is the number <math>n</math> iff there is <math>(n-1)</math> elements <math>y</math> such that <math>y<x</math>. Thanks to this we also want the primitive "bit", where <math>bit(x,y)</math> is true if only the <math>y</math>th bith of <math>x</math> is 1. (We can replace <math>bit</math> by plus and time, ternary relation such that <math>plus(x,y,z)</math> is true iff <math>x+y=z</math> and <math>times(x,y,z)</math> is true iff <math>x*y=z</math>).<br />
<br />
Since in a computer elements are only pointers or string of bit, thoses assumptions make sens, and those primitive function can be calculated in most of the small complexity classes. We can also imagine FO without those primitives, which gives us smaller complexity classes.<br />
<br />
The language FO is then defined as the closure by conjunction ( <math>\wedge</math>), negation (<math>\neg</math>) and universal quantification (<math>\forall</math>) over element of the structures. We also often use existantial quantification (<math>\exists</math>) and disjunction (<math>\vee</math>) but those can be defined thanks to the 3 first symbols. <br />
<br />
The semantics of the formulae in FO is straightforward, <math>\neg A</math> is true iff <math>A</math> is false, <math>A\wedge B</math> is true iff <math>A</math> is true and <math>B</math> is true, and (<math>\forall x P</math>) is true iff whatever element we decide that <math>x</math> is we can choose, <math>P</math> is true.<br />
<br />
<br />
A querie in FO will then be to check if a FO formulae is true over a given structure, this structure is the input of the problem. One should not confuse this kind of problem with checking if a quantified boolean formula is true, which is the definition of QBF, which is [[Complexity_Zoo:P#PSPACE|Pspace]]-complete. The difference between those two problem is that in QBF the size of the problem is the size of the formula and elements are just boolean value, whereas in FO the size of the problem is the size of the structure and the formula is fixed.<br />
<br />
Every formulae is equivalent to a formulae in prenexe normal form where we put recursively every quantifier and then a quantifier-free formulae. <br />
----<br />
<br />
===== <span id="fodtc" style="color:red">FO(DTC)</span>: First-order with deterministic transitive closure =====<br />
FO(DTC) is defined as [[#Complexity_Zoo:F#fotc|FO(tc)]] where the transitive closure operator is deterministic, which means that when we apply DTC(<math>\phi_{u,v}</math>), we know that for all <math>u</math>, there exist at most one <math>v</math> such that phi(u,v).<br />
<br />
We can suppose that DTC(<math>\phi_{u,v}</math>) is syntactic sugar for TC(<math>\psi_{u,v}</math>) where <math>\psi(u,v)=\phi(u,v)\wedge \forall x, (x=v \vee \neg \psi(u,x))</math>.<br />
<br />
It was shown in [[zooref#imm99|[Imm99]]] page 144 that this class is equal to [[#Complexity_Zoo:L#L|L]]. <br />
----<br />
<br />
===== <span id="folfp" style="color:red">FO(LFP)</span>: First-order with least fixed point =====<br />
FO(LFP) is the set of boolean queries definable with [[#Complexity_Zoo:S#fopfp|first-order fixed-point]] formulae where the partial fixed point is limited to be monotone, which means that if the second order variable is <math>P</math>, then <math>P_i(x)</math> always implies <math>P_{i+1}(x)</math>.<br />
<br />
We can obtain the monotony by restricting the formula <math>\phi</math> to have only positive occurrences of <math>P</math> (i.e. there is an even number of negations before every occurrence of <math>P</math>). We can also describe LFP(<math>\phi_{P,x}</math>) as syntactic sugar of PFP(<math>\psi_{P,x}</math>) where <math>\psi(P,x)=\phi(P,x)\vee P(x)</math>.<br />
<br />
Thanks to monotonicity we only add and never remove vectors to the truth table of <math>P</math>, and since there is only <math>n^k</math> possible vectors we always find a fixed point before <math>n^k</math> iterations. Hence it was shown in [[zooref#imm82|[Imm82]]] that FO(LFP)=P. This definition is equivalent to [[Complexity_Zoo:F#fot|FO(<math>n^{O(1)}</math>)]].<br />
----<br />
<br />
===== <span id="fopfp" style="color:red">FO(PFP)</span>: First-order with partial fixed point =====<br />
FO(pfp) is the set of boolean queries definable with [[#Complexity_Zoo:S#FO|first-order]] formulae with a partial fixed point operator.<br />
<br />
Let <math>k</math> be an integer, <math>x, y</math> vectors of <math>k</math> variables, <math>P</math> a second-order variable of arity <math>k</math>, and <math>\phi</math> a FO(PFP) function using <math>x</math> and <math>P</math> as variables, then we can define iteratively <math>(P_i)_{i\in N}</math> such that <math>P_0(x)=false</math> and <math>P_i(x)=\phi(P_{i-1},x)</math> which means that the property <math>P_i</math> is true on the input <math>x</math> if <math>\phi</math> is true on input <math>x</math>, and when the variable <math>P</math> is replaced by <math>P_{i-1}</math>. Then, either there is a fixed point, or the list of <math>(P_i)</math> is looping. <br />
<br />
PFP(<math>\phi_{P,x})(y)</math> is defined as the value of the fixed point of <math>(P_i)</math> on y if there is a fixed point, else as false.<br />
<br />
Since <math>P</math>s are property of arity <math>k</math>, there is at most <math>2^{n^k}</math> values for the <math>P_i</math>s, so with a poly-space counter we can check if there is a loop or not.<br />
<br />
It was proved in [[zooref#imm98|[Imm98]]] that FO(pfp) is equal to [[#Complexity_Zoo:P#PSPACE|PSPACE]].<br />
----<br />
<br />
===== <span id="fotc" style="color:red">FO(TC)</span>: First-order with transitive closure =====<br />
FO(TC) is the set of boolean queries definable with [[#Complexity_Zoo:S#FO|first-order]] formulae with a transitive closure (TC) operator. <br />
<br />
TC is defined this way: let <math>k</math> be a positiver integer and <math>u,v,x,y</math> be vectors of <math>k</math> variables, then TC(<math>\phi_{u,v})(x,y)</math> is true if there exist <math>n</math> variables <math>(x_i)</math> such that <math>x_1=x, x_n=y</math> and for all <math>i<n</math> <math>\phi_{u,v}(x_i,x_{i+1})</math>. Here, <math>\phi_{u,v}</math> is a formula over <math>u,v</math> written in FO(TC) and <math>\phi_{u,v}(x,y)</math> means that the variables <math>u</math> and <math>v</math> are replaced by <math>x</math> and <math>y</math>.<br />
<br />
Every formula of TC can be written in a normal form FO(<math>\phi_{u,v})(0,max)</math> where <math>\phi</math> is a FO formula and we suppose that there is an order on the model where variables are quantified, so we can choose the minimum and maximum element.<br />
<br />
It was shown in [[zooref#imm98|[Imm98]]] page 150 that this class is equal to [[#Complexity_Zoo:N#NL|NL]]. <br />
----<br />
<br />
===== <span id="fot" style="color:red">FO[<math>t(n)</math>]</span>: Iterated First-Order logic =====<br />
Let <math>t(n)</math> be a function from integers to integers.<br />
<math>(\forall x P) Q</math> abbreviates <math>(\forall x (P\Rightarrow Q))</math> and <math>(\exists x P) Q</math> abbreviates <math>(\exists x (P \wedge Q))</math>.<br />
<br />
A quantifier block is a list <math>(Q_1 x_1. \phi_1)...(Q_k x_k. \phi_k)</math> where the <math>\phi_i</math>s are quantifier free [[#Complexity_Zoo:F#FO|FO]]-formulae and each <math>Q_i</math>s is either <math>\forall</math> or <math>\exists</math>.<br />
If <math>Q</math> is a quantifier block then <math>[Q]^{t(n)}</math> is the block consisting of <math>t(n)</math> iterated copies of <math>Q</math>. <br />
Note that there are <math>k*t(n)</math> quantifiers in the list, but only k variables; each variable is used <math>t(n)</math> times.<br />
<br />
FO[<math>t(n)</math>] consists of the FO-formulae with quantifier blocks that are iterated <math>\Theta(t(n))</math> times.<br />
<br />
In [[zooref#imm98|Descriptive complexity]] we can see that :<br />
<br />
*FO[<math>(\log n)^i</math>] is equal to fo-uniform [[#Complexity_Zoo:A#AC|AC<sup>i</sup>]], and in fact FO[<math>t(n)</math>] is fo-uniform AC of depth <math>t(n)</math><br />
*FO[<math>(\log n)^{O(1)}</math>] is equal to [[#Complexity_Zoo:N#NC|NC]]<br />
*FO[<math>n^{O(1)}</math>] is equal to [[#Complexity_Zoo:P#P|P]] and [[#Complexity_Zoo:F#folfp|FO(LFP)]]<br />
*FO[<math>2^{n^{O(1)}}</math>] is equal to [[#Complexity_Zoo:P#pspace|PSPACE]] and [[#Complexity_Zoo:F#fopfp|FO(PFP)]]<br />
----<br />
<br />
===== <span id="foll" style="color:red">FOLL</span>: First-Order log log n =====<br />
The class of decision problems solvable by a uniform family of polynomial-size, unbounded-fanin, depth O(log&nbsp;log&nbsp;''n'') circuits with AND, OR, and NOT gates. Equals [[#fo|FO]](log&nbsp;log&nbsp;''n'').<br />
<br />
Defined in [[zooref#bkl00|[BKL+00]]], where it was also shown that many problems on finite groups are in FOLL.<br />
<br />
Contains uniform [[Complexity Zoo:A#ac0|AC<sup>0</sup>]], and is contained in uniform [[Complexity Zoo:A#ac|AC<sup>1</sup>]].<br />
<br />
Is not known to be comparable to [[Complexity Zoo:L#l|L]] or [[Complexity Zoo:N#nl|NL]].<br />
<br />
----<br />
<br />
===== <span id="fp" style="color:red">FP</span>: Function Polynomial-Time =====<br />
Sometimes defined as the class of functions computable in polynomial time by a Turing machine. (Generalizes [[Complexity Zoo:P#p|P]], which is defined in terms of decision problems only.)<br />
<br />
However, if we want to compare FP to [[#fnp|FNP]], we should instead define it as the class of [[#fnp|FNP]] problems (that is, polynomial-time predicates P(x,y)) for which there exists a polynomial-time algorithm that, given x, outputs <i>any</i> y such that P(x,y). That is, there could be more than one valid output, even though any given algorithm only returns one of them.<br />
<br />
FP = [[#fnp|FNP]] if and only if [[Complexity Zoo:P#p|P]] = [[Complexity Zoo:N#np|NP]].<br />
<br />
If FP<sup>[[Complexity Zoo:N#np|NP]]</sup> = FP<sup>[[Complexity Zoo:N#np|NP]][log]</sup> (that is, allowed only a logarithmic number of queries), then [[Complexity Zoo:P#p|P]] = [[Complexity Zoo:N#np|NP]] [[zooref#kre88|[Kre88]]]. The corresponding result for [[Complexity Zoo:P#pnp|P<sup>NP</sup>]] versus [[Complexity Zoo:P#pnplog|P<sup>NP[log]</sup>]] is not known, and indeed fails relative to some oracles (see [[zooref#har87b|[Har87b]]]).<br />
<br />
----<br />
<br />
===== <span id="fpnplog" style="color:red">FP<sup>NP[log]</sup></span>: [[#fp|FP]] With Logarithmically Many Queries To [[Complexity Zoo:N#np|NP]] =====<br />
Given a graph, the problem of outputting the size of its maximum clique is complete for FP<sup>NP[log]</sup>.<br />
<br />
----<br />
===== <span id="fpr" style="color:red">FPR</span>: Fixed-Parameter Randomized =====<br />
Has the same relation to [[#fpt|FPT]] as [[Complexity Zoo:R#rp|RP]] does to [[Complexity Zoo:P#p|P]].<br />
<br />
Defined in [[zooref#ar01|[AR01]]], where it was shown that, if the Resolution proof system is <i>automatizable</i> (that is, if a refutation can always be found in time polynomial in the length of the shortest refutation), then [[Complexity Zoo:W#wp|W[P]]] is contained in FPR.<br />
<br />
----<br />
<br />
===== <span id="fpras" style="color:red">FPRAS</span>: Fully Polynomial Randomized Approximation Scheme =====<br />
The subclass of [[Complexity Zoo:Symbols#sharpp|#P]] counting problems whose answer, y, is approximable in the following sense. There exists a randomized algorithm that, with probability at least 1-&delta;, approximates y to within an &epsilon; multiplicative factor in time polynomial in n (the input size), 1/&epsilon;, and log(1/&delta;).<br />
<br />
The permanent of a nonnegative matrix is in FPRAS [[zooref#jsv01|[JSV01]]].<br />
<br />
----<br />
===== <span id="fpt" style="color:red">FPT</span>: Fixed-Parameter Tractable =====<br />
The class of decision problems of the form (x,k), k a parameter, that are solvable in time f(k)p(|x|), where f is arbitrary and p is a polynomial.<br />
<br />
The basic class of the theory of <i>fixed-parameter tractability</i>, as described by Downey and Fellows [[zooref#df99|[DF99]]].<br />
<br />
To separate FPT and [[Complexity Zoo:W#w2|W[2]]], one could show there is no proof system for CNF formulae that admits proofs of size f(k)n<sup>O(1)</sup>, where f is a computable function and n is the size of the formula.<br />
<br />
Contained in [[#fptnu|FPT<sub>nu</sub>]], [[Complexity Zoo:W#w1|W[1]]], and [[#fpr|FPR]].<br />
<br />
Contains [[#fptas|FPTAS]] [[zooref#cc97|[CC97]]], as well as [[#fptsu|FPT<sub>su</sub>]].<br />
<br />
Contains [[Complexity Zoo:E#eptas|EPTAS]] unless FPT = [[Complexity Zoo:W#w1|W[1]]] [[zooref#baz95|[Baz95]]].<br />
<br />
----<br />
<br />
===== <span id="fptnu" style="color:red">FPT<sub>nu</sub></span>: Fixed-Parameter Tractable (nonuniform) =====<br />
Same as [[#fpt|FPT]] except that the algorithm can vary with the parameter k (though its running time must always be O(p(|x|)), for a fixed polynomial p).<br />
<br />
An alternate view is that a single algorithm can take a polynomial-length advice string, depending on k.<br />
<br />
Defined in [[zooref#df99|[DF99]]] (though they did not use our notation).<br />
<br />
----<br />
===== <span id="fptsu" style="color:red">FPT<sub>su</sub></span>: Fixed-Parameter Tractable (strongly uniform) =====<br />
Same as [[#fpt|FPT]] except that f has to be recursive.<br />
<br />
Defined in [[zooref#df99|[DF99]]] (though they did not use our notation).<br />
<br />
----<br />
===== <span id="fptas" style="color:red">FPTAS</span>: Fully Polynomial-Time Approximation Scheme =====<br />
The subclass of [[Complexity Zoo:N#npo|NPO]] problems that admit an approximation scheme in the following sense. For any &#949;&gt;0, there is an algorithm that is guaranteed to find a solution whose cost is within a 1+&#949; factor of the optimum cost. Furthermore, the running time of the algorithm is polynomial in n (the size of the problem) and in 1/&#949;.<br />
<br />
Contained in [[Complexity Zoo:P#ptas|PTAS]].<br />
<br />
Defined in [[zooref#acg99|[ACG+99]]].<br />
<br />
Contained in [[#fpt|FPT]] [[zooref#cc97|[CC97]]].<br />
<br />
----<br />
===== <span id="fqma" style="color:red">FQMA</span>: Function [[Complexity Zoo:Q#qma|QMA]] =====<br />
The class of problems for which the task is to output a quantum certificate for a [[Complexity Zoo:Q#qma|QMA]] problem, when such a certificate exists. Thus, the desired output is a quantum state.<br />
<br />
Defined in [[zooref#jwb03|[JWB03]]], where it is also shown that state preparation for 3-local Hamiltonians is FQMA-complete. The authors also observe that, in contrast to the case of [[#fnp|FNP]] versus [[Complexity Zoo:N#np|NP]], there is no obvious reduction of FQMA problems to [[Complexity Zoo:Q#qma|QMA]] problems.<br />
<br />
----<br />
===== <span id="frip" style="color:red">frIP</span>: Function-Restricted [[Complexity Zoo:I#ip|IP]] Proof Systems =====<br />
The class of problems L that have a <i>decider</i> in the following sense. There exists a [[Complexity Zoo:B#bpp|BPP]] machine D such that for all inputs x,<br />
<ol><br />
<li>If the answer is "yes" then D<sup>L</sup>(x) (D with oracle for L) accepts with probability at least 2/3.</li><br />
<li>If the answer is "no" then D<sup>A</sup>(x) accepts with probability at most 1/3 for all oracles A.</li><br />
</ol><br />
<br />
Contains [[Complexity Zoo:C#compip|compIP]] [[zooref#bg94|[BG94]]] and [[#check|Check]] [[zooref#bk89|[BK89]]].<br />
<br />
Contained in [[Complexity Zoo:M#mip|MIP]] = [[Complexity Zoo:N#nexp|NEXP]] [[zooref#frs88|[FRS88]]].<br />
<br />
Assuming [[Complexity Zoo:N#nee|NEE]] is not contained in [[Complexity Zoo:B#bpee|BPEE]], [[Complexity Zoo:N#np|NP]] (and indeed [[Complexity Zoo:N#np|NP]] &#8745; [[Complexity Zoo:C#coh|Coh]]) is not contained in [[#frip|frIP]] [[zooref#bg94|[BG94]]].<br />
<br />
----<br />
===== <span id="ftape" style="color:red">F-TAPE(f(n))</span>: Provable [[Complexity Zoo:D#dspace|DSPACE(f(n))]] For Formal System F =====<br />
The class of decision problems that can be <i>proven</i> to be solvable in O(f(n)) space on a deterministic Turing machine, from the axioms of formal system F.<br />
<br />
Defined in [[zooref#har78|[Har78]]].<br />
<br />
See also [[#ftime|F-TIME(f(n))]]. The results about F-TAPE mirror those about [[#ftime|F-TIME]], but in some cases are sharper.<br />
<br />
----<br />
===== <span id="ftime" style="color:red">F-TIME(f(n))</span>: Provable [[Complexity Zoo:D#dtime|DTIME(f(n))]] For Formal System F =====<br />
The class of decision problems that can be <i>proven</i> to be solvable in O(f(n)) time on a deterministic Turing machine, from the axioms of formal system F.<br />
<br />
Defined in [[zooref#har78|[Har78]]], where the following was also shown:<br />
<ul><br />
<li>If F-TIME(f(n)) = [[Complexity Zoo:D#dtime|DTIME(f(n))]], then [[Complexity Zoo:D#dtime|DTIME(f(n))]] is strictly contained in [[Complexity Zoo:D#dtime|DTIME(f(n)g(n))]] for any nondecreasing, unbounded, recursive g(n).</li><br />
<li>There exist recursive, monotonically increasing f(n) such that F-TIME(f(n)) is strictly contained in [[Complexity Zoo:D#dtime|DTIME(f(n))]].</li><br />
</ul><br />
See also [[#ftape|F-TAPE(f(n))]].</div>LegionMammal978https://complexityzoo.net/index.php?title=Complexity_Zoo:F&diff=6526Complexity Zoo:F2017-04-30T23:55:29Z<p>LegionMammal978: /* FO: First-Order logic */ spent hours tracking down this error</p>
<hr />
<div>__NOTOC__<br />
{{CZ-Letter-Menu|F}}<br />
<br />
===== <span id="fbpp" style="color:red">FBPP</span>: Function [[Complexity Zoo:B#bpp|BPP]] =====<br />
Has the same relation to [[Complexity Zoo:B#bpp|BPP]] as [[#fnp|FNP]] does to [[Complexity Zoo:N#np|NP]]. Equivalently, it is the randomised analogue of [[#fp|FP]].<br />
<br />
----<br />
<br />
===== <span id="fbqp" style="color:red">FBQP</span>: Function [[Complexity Zoo:B#bqp|BQP]] =====<br />
Has the same relation to [[Complexity Zoo:B#bqp|BQP]] as [[#fnp|FNP]] does to [[Complexity Zoo:N#np|NP]].<br />
<br />
There exists an oracle relative to which [[Complexity Zoo:P#pls|PLS]] is not contained in [[#fbqp|FBQP]] [[zooref#aar03|[Aar03]]].<br />
<br />
----<br />
===== <span id="fert" style="color:red">FERT</span>: Fixed Error Randomized Time =====<br />
FERT and [[#fpert|FPERT]] are parameterized classes. FERT formally defined as the class of decision problems of the form (x, k), decidable in polynomial time by a probabilistic Turing Machine such that<br />
<ol><br />
<li>If the answer is yes, the probability of acceptance is at least 1/2 + min(f(k),1/|x|<sup>c</sup>)</li><br />
<li>If the answer is no, the probability of acceptance is at most 1/2</li><br />
</ol><br />
<br />
Here, f is an arbitrary function (from the reals to <0,1/2]).<br />
<br />
Defined in [[zooref#kw15|[KW15]]]. Contains [[Complexity Zoo:B#bpp|BPP]] and is contained in para-[[Complexity Zoo:P#pp|PP]] and in [[#fpert|FPERT]].<br />
----<br />
<br />
===== <span id="fpert" style="color:red">FPERT</span>: Fixed Parameter and Error Randomized Time =====<br />
[[#fert|FERT]] and FPERT are parameterized classes. FPERT is formally defined as the class of decision problems of the form (x, k1, k2), decidable in time f1(k1) * p(|x|) by a probabilistic Turing Machine such that<br />
<ol><br />
<li>If the answer is yes, the probability of acceptance is at least 1/2 + min(f2(k2),1/|x|<sup>c</sup>)</li><br />
<li>If the answer is no, the probability of acceptance is at most 1/2</li><br />
</ol><br />
<br />
Here, f1 and f2 are arbitrary functions (f2 from the reals to <0,1/2]) and p is a polynomial.<br />
<br />
Defined in [[zooref#kw15|[KW15]]]. Contains [[#fert|FERT]] and [[#fpt|FPT]] and is contained in para-[[Complexity Zoo:N#np|NP]]<sup>[[Complexity Zoo:P#pp|PP]]</sup>. <br />
----<br />
<br />
===== <span id="few" style="color:red">Few</span>: [[#fewp|FewP]] With Flexible Acceptance Mechanism =====<br />
The class of decision problems solvable by an [[Complexity Zoo:N#np|NP]] machine such that<br />
<ol><br />
<li>The number of accepting paths a is bounded by a polynomial in the size of the input x.</li><br />
<li>For some polynomial-time predicate Q, Q(x,a) is true if and only if the answer is 'yes.'</li><br />
</ol><br />
Also called FewPaths.<br />
<br />
Defined in [[zooref#ch89|[CH89]]].<br />
<br />
Contains [[#fewp|FewP]], and is contained in [[Complexity Zoo:P#p|P]]<sup>[[#fewp|FewP]]</sup> [[zooref#kob89|[Kob89]]] and in [[Complexity Zoo:S#spp|SPP]] [[zooref#ffk94|[FFK94]]].<br />
<br />
See also the survey [[zooref#tor90|[Tor90]]].<br />
<br />
----<br />
===== <span id="fewexp" style="color:red">FewEXP</span>: {{zcls|n|nexp}} With Few Witnesses =====<br />
The class of decision problems solvable by an {{zcls|n|nexp}} machine such that, given a "yes" instance, no more than an exponential number of computation paths accept.<br />
<br />
Contained in {{zcls|m|mip}}<nowiki>[</nowiki>{{zcls|n|np}}<sup>FewEXP</sup><nowiki>]</nowiki> (MIP where provers are not unbounded in computational power, but are limited to NP<sup>FewEXP</sup>) {{zcite|AKS+94}}.<br />
<br />
Alternatively, FewEXP can refer to the sparsity of accepting paths in a given instance. In {{zcite|AKR+03}}, the authors show that the conjectures "FewEXP search instances are {{zcls|e|exp}}-solvable" and "FewEXP decision instances are EXP/poly-solvable" are equivalent. That is, for all NEXP machines <math>N</math>, the following conditions are equivalent:<br />
# There exists an EXP machine <math>M</math> such that if given a string <math>x</math>, <math>N(x)</math> accepts and has exponentially bounded accepting paths, then <math>M(x)</math> produces one such path.<br />
# There exists an EXP/poly machine <math>M</math> which accepts a string <math>x</math> if and only <math>N(x)</math> accepts.<br />
<br />
----<br />
<br />
===== <span id="fewp" style="color:red">FewP</span>: [[Complexity Zoo:N#np|NP]] With Few Witnesses =====<br />
The class of decision problems solvable by an [[Complexity Zoo:N#np|NP]] machine such that<br />
<ol><br />
<li>If the answer is 'no,' then all computation paths reject.</li><br />
<li>If the answer is 'yes,' then at least one path accepts; furthermore, the number of accepting paths is upper-bounded by a polynomial in n, the size of the input.</li><br />
</ol><br />
Defined in [[zooref#ar88|[AR88]]].<br />
<br />
Is contained in [[Complexity Zoo:Symbols#parityp|&#8853;P]] [[zooref#ch89|[CH89]]].<br />
<br />
There exists an oracle relative to which [[Complexity Zoo:P#p|P]], [[Complexity Zoo:U#up|UP]], FewP, and [[Complexity Zoo:N#np|NP]] are all distinct [[zooref#rub88|[Rub88]]].<br />
<br />
Also, there exists an oracle relative to which FewP does not have a Turing-complete set [[zooref#hjv93|[HJV93]]].<br />
<br />
Contained in [[#few|Few]].<br />
<br />
See also the survey [[zooref#tor90|[Tor90]]].<br />
<br />
----<br />
<br />
===== <span id="fh" style="color:red">FH</span>: Fourier Hierarchy =====<br />
FH<sub>k</sub> is the class of problems solvable by a uniform family of polynomial-size quantum circuits, with k levels of Hadamard gates and all other gates preserving the computational basis. (Conditional phase flip gates are fine, for example.) Thus<br />
<ul><br />
<li>FH<sub>0</sub> = [[Complexity Zoo:P#p|P]]</li><br />
<li>FH<sub>1</sub> = [[Complexity Zoo:B#bpp|BPP]]</li><br />
<li>FH<sub>2</sub> contains [[Complexity_Garden#integer_factorization|factoring]] because of Kitaev's phase estimation algorithm</li><br />
</ul><br />
It is an open problem to show that the Fourier hierarchy is infinite relative to an oracle (that is, FH<sub>k</sub> is strictly contained in FH<sub>k+1</sub>).<br />
<br />
Defined in [[zooref#shi03|[Shi03]]].<br />
<br />
----<br />
<br />
===== <span id="fixp" style="color:red">FIXP</span>: Fixed Point =====<br />
The class of fixed point problems. In the framework of fixed point problems, an instance I is associated with a (continuous) function F<sub>I</sub>, and a solution of I is a fixed point of F<sub>I</sub>. <br />
<br />
Properties of FIXP problems:<br />
<ol><br />
<li> the function F<sub>I</sub> is represented by an algebraic circuit over {+, -, *, /, max, min} with rational constants<br />
<li> there is a polynomial time algorithm that computes the circuit from I.<br />
</ol><br />
<br />
Every FIXP problem has Partial Computation, Decision, (Strong) Approximation, and Existence counterparts; these can all be solved in PSPACE.<br />
<br />
The Nash equilibrium problem for 3 or more players is FIXP-complete.<br />
<br />
Linear-FIXP = PPAD.<br />
<br />
Defined in [[zooref#ey07|[EY07]]].<br />
----<br />
<br />
===== <span id="fnl" style="color:red">FNL</span>: Function [[Complexity Zoo:N#nl|NL]] =====<br />
Has the same relation to [[Complexity Zoo:N#nl|NL]] as [[#fnp|FNP]] does to [[Complexity Zoo:N#np|NP]].<br />
<br />
Defined by [[zooref#aj93|[AJ93]]], who also showed that if [[Complexity Zoo:N#nl|NL]] = [[Complexity Zoo:U#ul|UL]], then FNL is contained in [[Complexity Zoo:Symbols#sharpl|#L]].<br />
<br />
----<br />
<br />
===== <span id="fnlpoly" style="color:red">FNL/poly</span>: Nonuniform [[#fnl|FNL]] =====<br />
Has the same relation to [[#fnl|FNL]] as [[Complexity Zoo:P#ppoly|P/poly]] does to [[Complexity Zoo:P#p|P]].<br />
<br />
Contained in [[Complexity Zoo:Symbols#sharplpoly|#L/poly]] [[zooref#ra00|[RA00]]].<br />
<br />
----<br />
===== <span id="fnp" style="color:red">FNP</span>: Function [[Complexity Zoo:N#np|NP]] =====<br />
The class of function problems of the following form:<br />
<ul> Given an input x and a polynomial-time predicate F(x,y), if there exists a y satisfying F(x,y) then output any such y, otherwise output 'no.' </ul><br />
FNP generalizes [[Complexity Zoo:N#np|NP]], which is defined in terms of decision problems only.<br />
<br />
Actually the word "function" is misleading, since there could be many valid outputs y. That's unavoidable, since given a predicate F there's no "syntactic" criterion ensuring that y is unique.<br />
<br />
[[#fp|FP]] = FNP if and only if [[Complexity Zoo:P#p|P]] = [[Complexity Zoo:N#np|NP]].<br />
<br />
Contains [[Complexity Zoo:T#tfnp|TFNP]].<br />
<br />
A basic question about FNP problems is whether they're <i>self-reducible</i>; that is, whether they reduce to the corresponding [[Complexity Zoo:N#np|NP]] decision problems. Although this is true for all [[Complexity Zoo:N#npc|NPC]] problems, [[zooref#bg94|[BG94]]] shows that if [[Complexity Zoo:E#ee|EE]] does not equal [[Complexity Zoo:N#nee|NEE]], then there is a problem in [[Complexity Zoo:N#np|NP]] such that <i>no</i> corresponding FNP problem can be reduced to it. [[zooref#bg94|[BG94]]] cites Impagliazzo and Sudan as giving the same conclusion under the assumption that [[Complexity Zoo:N#ne|NE]] does not equal [[Complexity Zoo:C#cone|coNE]].<br />
<br />
----<br />
===== <span id="fo" style="color:red">FO</span>: First-Order logic =====<br />
First order logic is the smallest logical class of logic language. It is the base of [[zooref#imm98|Descriptive complexity]] and equal to the class [[Complexity_Zoo:A#ac0|AC<sup>0</sup>]] and to [[Complexity_Zoo:A#AL|AL]], the alternating logtime hierarchy.<br />
<br />
When we use logic formalism, the input is a structure, usually it is either strings (of bits or over an alphabet) whose elements are position of the strings, or graphs whose elements are vertices. The mesure of the input will there be the size of the structure.<br />
Whatever the structure is, we can suppose that there are relation that you can test, by example <math>E(x,y)</math> is true iff there is an edge from <math>x</math> to <math>y</math> if the structure is a graph, and <math>P(n)</math> is true iff the nth letter of the string is 1. We also have constant, who are special elements of the structure, by example if we want to check reachability in a graph, we will have to choose two constant s and t.<br />
<br />
In descriptive complexity we almost always suppose that there is a total order over the elements and that we can check equality between elements. This let us consider elements as number, <math>x</math> is the number <math>n</math> iff there is <math>(n-1)</math> elements <math>y</math> such that <math>y<x</math>. Thanks to this we also want the primitive "bit", where <math>bit(x,y)</math> is true if only the <math>y</math>th bith of <math>x</math> is 1. (We can replace <math>bit</math> by plus and time, ternary relation such that <math>plus(x,y,z)</math> is true iff <math>x+y=z</math> and <math>times(x,y,z)</math> is true iff <math>x*y=z</math>).<br />
<br />
Since in a computer elements are only pointers or string of bit, thoses assumptions make sens, and those primitive function can be calculated in most of the small complexity classes. We can also imagine FO without those primitives, which gives us smaller complexity classes.<br />
<br />
The language FO is then defined as the closure by conjunction ( <math>\wedge</math>), negation (<math>\neg</math>) and universal quantification (<math>\forall</math>) over element of the structures. We also often use existantial quantification (<math>\exists</math>) and disjunction (<math>\vee</math>) but those can be defined thanks to the 3 first symbols. <br />
<br />
The semantics of the formulae in FO is straightforward, <math>\neg A</math> is true iff <math>A</math> is false, <math>A\wedge B</math> is true iff <math>A</math> is true and <math>B</math> is true, and (<math>\forall x P</math>) is true iff whatever element we decide that <math>x</math> is we can choose, <math>P</math> is true.<br />
<br />
<br />
A querie in FO will then be to check if a FO formulae is true over a given structure, this structure is the input of the problem. One should not confuse this kind of problem with checking if a quantified boolean formula is true, which is the definition of QBF, which is [[Complexity_Zoo:P#PSPACE|Pspace]]-complete. The difference between those two problem is that in QBF the size of the problem is the size of the formula and elements are just boolean value, whereas in FO the size of the problem is the size of the structure and the formula is fixed.<br />
<br />
Every formulae is equivalent to a formulae in prenexe normal form where we put recursively every quantifier and then a quantifier-free formulae. <br />
----<br />
<br />
===== <span id="fodtc" style="color:red">FO(DTC)</span>: First-order with deterministic transitive closure =====<br />
FO(DTC) is defined as [[#Complexity_Zoo:F#fotc|FO(tc)]] where the transitive closure operator is deterministic, which means that when we apply DTC(<math>\phi_{u,v}</math>), we know that for all <math>u</math>, there exist at most one <math>v</math> such that phi(u,v).<br />
<br />
We can suppose that DTC(<math>\phi_{u,v}</math>) is syntactic sugar for TC(<math>\psi_{u,v}</math>) where <math>\psi(u,v)=\phi(u,v)\wedge \forall x, (x=v \vee \neg \psi(u,x))</math>.<br />
<br />
It was shown in [[zooref#imm99|[Imm99]]] page 144 that this class is equal to [[#Complexity_Zoo:L#L|L]]. <br />
----<br />
<br />
===== <span id="folfp" style="color:red">FO(LFP)</span>: First-order with least fixed point =====<br />
FO(LFP) is the set of boolean queries definable with [[#Complexity_Zoo:S#fopfp|first-order fixed-point]] formulae where the partial fixed point is limited to be monotone, which means that if the second order variable is <math>P</math>, then <math>P_i(x)</math> always implies <math>P_{i+1}(x)</math>.<br />
<br />
We can obtain the monotony by restricting the formula <math>\phi</math> to have only positive occurrences of <math>P</math> (i.e. there is an even number of negations before every occurrence of <math>P</math>). We can also describe LFP(<math>\phi_{P,x}</math>) as syntactic sugar of PFP(<math>\psi_{P,x}</math>) where <math>\psi(P,x)=\phi(P,x)\vee P(x)</math>.<br />
<br />
Thanks to monotonicity we only add and never remove vectors to the truth table of <math>P</math>, and since there is only <math>n^k</math> possible vectors we always find a fixed point before <math>n^k</math> iterations. Hence it was shown in [[zooref#imm82|[Imm82]]] that FO(LFP)=P. This definition is equivalent to [[Complexity_Zoo:F#fot|FO(<math>n^{O(1)}</math>)]].<br />
----<br />
<br />
===== <span id="fopfp" style="color:red">FO(PFP)</span>: First-order with partial fixed point =====<br />
FO(pfp) is the set of boolean queries definable with [[#Complexity_Zoo:S#FO|first-order]] formulae with a partial fixed point operator.<br />
<br />
Let <math>k</math> be an integer, <math>x, y</math> vectors of <math>k</math> variables, <math>P</math> a second-order variable of arity <math>k</math>, and <math>\phi</math> a FO(PFP) function using <math>x</math> and <math>P</math> as variables, then we can define iteratively <math>(P_i)_{i\in N}</math> such that <math>P_0(x)=false</math> and <math>P_i(x)=\phi(P_{i-1},x)</math> which means that the property <math>P_i</math> is true on the input <math>x</math> if <math>\phi</math> is true on input <math>x</math>, and when the variable <math>P</math> is replaced by <math>P_{i-1}</math>. Then, either there is a fixed point, or the list of <math>(P_i)</math> is looping. <br />
<br />
PFP(<math>\phi_{P,x})(y)</math> is defined as the value of the fixed point of <math>(P_i)</math> on y if there is a fixed point, else as false.<br />
<br />
Since <math>P</math>s are property of arity <math>k</math>, there is at most <math>2^{n^k}</math> values for the <math>P_i</math>s, so with a poly-space counter we can check if there is a loop or not.<br />
<br />
It was proved in [[zooref#imm98|[Imm98]]] that FO(pfp) is equal to [[#Complexity_Zoo:P#PSPACE|PSPACE]].<br />
----<br />
<br />
===== <span id="fotc" style="color:red">FO(TC)</span>: First-order with transitive closure =====<br />
FO(TC) is the set of boolean queries definable with [[#Complexity_Zoo:S#FO|first-order]] formulae with a transitive closure (TC) operator. <br />
<br />
TC is defined this way: let <math>k</math> be a positiver integer and <math>u,v,x,y</math> be vectors of <math>k</math> variables, then TC(<math>\phi_{u,v})(x,y)</math> is true if there exist <math>n</math> variables <math>(x_i)</math> such that <math>x_1=x, x_n=y</math> and for all <math>i<n</math> <math>\phi_{u,v}(x_i,x_{i+1})</math>. Here, <math>\phi_{u,v}</math> is a formula over <math>u,v</math> written in FO(TC) and <math>\phi_{u,v}(x,y)</math> means that the variables <math>u</math> and <math>v</math> are replaced by <math>x</math> and <math>y</math>.<br />
<br />
Every formula of TC can be written in a normal form FO(<math>\phi_{u,v})(0,max)</math> where <math>\phi</math> is a FO formula and we suppose that there is an order on the model where variables are quantified, so we can choose the minimum and maximum element.<br />
<br />
It was shown in [[zooref#imm98|[Imm98]]] page 150 that this class is equal to [[#Complexity_Zoo:N#NL|NL]]. <br />
----<br />
<br />
===== <span id="fot" style="color:red">FO[<math>t(n)</math>]</span>: Iterated First-Order logic =====<br />
Let <math>t(n)</math> be a function from integers to integers.<br />
<math>(\forall x P) Q</math> abbreviates <math>(\forall x (P\Rightarrow Q))</math> and <math>(\exists x P) Q</math> abbreviates <math>(\exists x (P \wedge Q))</math>.<br />
<br />
A quantifier block is a list <math>(Q_1 x_1. \phi_1)...(Q_k x_k. \phi_k)</math> where the <math>\phi_i</math>s are quantifier free [[#Complexity_Zoo:F#FO|FO]]-formulae and each <math>Q_i</math>s is either <math>\forall</math> or <math>\exists</math>.<br />
If <math>Q</math> is a quantifier block then <math>[Q]^{t(n)}</math> is the block consisting of <math>t(n)</math> iterated copies of <math>Q</math>. <br />
Note that there are <math>k*t(n)</math> quantifiers in the list, but only k variables; each variable is used <math>t(n)</math> times.<br />
<br />
FO[<math>t(n)</math>] consists of the FO-formulae with quantifier blocks that are iterated <math>\Theta(t(n))</math> times.<br />
<br />
In [[zooref#imm98|Descriptive complexity]] we can see that :<br />
<br />
*FO[<math>(\log n)^i</math>] is equal to fo-uniform [[#Complexity_Zoo:A#AC|AC<sup>i</sup>]], and in fact FO[<math>t(n)</math>] is fo-uniform AC of depth <math>t(n)</math><br />
*FO[<math>(\log n)^{O(1)}</math>] is equal to [[#Complexity_Zoo:N#NC|NC]]<br />
*FO[<math>n^{O(1)}</math>] is equal to [[#Complexity_Zoo:P#P|P]] and [[#Complexity_Zoo:F#folfp|FO(LFP)]]<br />
*FO[<math>2^{n^{O(1)}}</math>] is equal to [[#Complexity_Zoo:P#pspace|PSPACE]] and [[#Complexity_Zoo:F#fopfp|FO(PFP)]]<br />
----<br />
<br />
===== <span id="foll" style="color:red">FOLL</span>: First-Order log log n =====<br />
The class of decision problems solvable by a uniform family of polynomial-size, unbounded-fanin, depth O(log&nbsp;log&nbsp;''n'') circuits with AND, OR, and NOT gates. Equals [[#fo|FO]](log&nbsp;log&nbsp;''n'').<br />
<br />
Defined in [[zooref#bkl00|[BKL+00]]], where it was also shown that many problems on finite groups are in FOLL.<br />
<br />
Contains uniform [[Complexity Zoo:A#ac0|AC<sup>0</sup>]], and is contained in uniform [[Complexity Zoo:A#ac|AC<sup>1</sup>]].<br />
<br />
Is not known to be comparable to [[Complexity Zoo:L#l|L]] or [[Complexity Zoo:N#nl|NL]].<br />
<br />
----<br />
<br />
===== <span id="fp" style="color:red">FP</span>: Function Polynomial-Time =====<br />
Sometimes defined as the class of functions computable in polynomial time by a Turing machine. (Generalizes [[Complexity Zoo:P#p|P]], which is defined in terms of decision problems only.)<br />
<br />
However, if we want to compare FP to [[#fnp|FNP]], we should instead define it as the class of [[#fnp|FNP]] problems (that is, polynomial-time predicates P(x,y)) for which there exists a polynomial-time algorithm that, given x, outputs <i>any</i> y such that P(x,y). That is, there could be more than one valid output, even though any given algorithm only returns one of them.<br />
<br />
FP = [[#fnp|FNP]] if and only if [[Complexity Zoo:P#p|P]] = [[Complexity Zoo:N#np|NP]].<br />
<br />
If FP<sup>[[Complexity Zoo:N#np|NP]]</sup> = FP<sup>[[Complexity Zoo:N#np|NP]][log]</sup> (that is, allowed only a logarithmic number of queries), then [[Complexity Zoo:P#p|P]] = [[Complexity Zoo:N#np|NP]] [[zooref#kre88|[Kre88]]]. The corresponding result for [[Complexity Zoo:P#pnp|P<sup>NP</sup>]] versus [[Complexity Zoo:P#pnplog|P<sup>NP[log]</sup>]] is not known, and indeed fails relative to some oracles (see [[zooref#har87b|[Har87b]]]).<br />
<br />
----<br />
<br />
===== <span id="fpnplog" style="color:red">FP<sup>NP[log]</sup></span>: [[#fp|FP]] With Logarithmically Many Queries To [[Complexity Zoo:N#np|NP]] =====<br />
Given a graph, the problem of outputting the size of its maximum clique is complete for FP<sup>NP[log]</sup>.<br />
<br />
----<br />
===== <span id="fpr" style="color:red">FPR</span>: Fixed-Parameter Randomized =====<br />
Has the same relation to [[#fpt|FPT]] as [[Complexity Zoo:R#rp|RP]] does to [[Complexity Zoo:P#p|P]].<br />
<br />
Defined in [[zooref#ar01|[AR01]]], where it was shown that, if the Resolution proof system is <i>automatizable</i> (that is, if a refutation can always be found in time polynomial in the length of the shortest refutation), then [[Complexity Zoo:W#wp|W[P]]] is contained in FPR.<br />
<br />
----<br />
<br />
===== <span id="fpras" style="color:red">FPRAS</span>: Fully Polynomial Randomized Approximation Scheme =====<br />
The subclass of [[Complexity Zoo:Symbols#sharpp|#P]] counting problems whose answer, y, is approximable in the following sense. There exists a randomized algorithm that, with probability at least 1-&delta;, approximates y to within an &epsilon; multiplicative factor in time polynomial in n (the input size), 1/&epsilon;, and log(1/&delta;).<br />
<br />
The permanent of a nonnegative matrix is in FPRAS [[zooref#jsv01|[JSV01]]].<br />
<br />
----<br />
===== <span id="fpt" style="color:red">FPT</span>: Fixed-Parameter Tractable =====<br />
The class of decision problems of the form (x,k), k a parameter, that are solvable in time f(k)p(|x|), where f is arbitrary and p is a polynomial.<br />
<br />
The basic class of the theory of <i>fixed-parameter tractability</i>, as described by Downey and Fellows [[zooref#df99|[DF99]]].<br />
<br />
To separate FPT and [[Complexity Zoo:W#w2|W[2]]], one could show there is no proof system for CNF formulae that admits proofs of size f(k)n<sup>O(1)</sup>, where f is a computable function and n is the size of the formula.<br />
<br />
Contained in [[#fptnu|FPT<sub>nu</sub>]], [[Complexity Zoo:W#w1|W[1]]], and [[#fpr|FPR]].<br />
<br />
Contains [[#fptas|FPTAS]] [[zooref#cc97|[CC97]]], as well as [[#fptsu|FPT<sub>su</sub>]].<br />
<br />
Contains [[Complexity Zoo:E#eptas|EPTAS]] unless FPT = [[Complexity Zoo:W#w1|W[1]]] [[zooref#baz95|[Baz95]]].<br />
<br />
----<br />
<br />
===== <span id="fptnu" style="color:red">FPT<sub>nu</sub></span>: Fixed-Parameter Tractable (nonuniform) =====<br />
Same as [[#fpt|FPT]] except that the algorithm can vary with the parameter k (though its running time must always be O(p(|x|)), for a fixed polynomial p).<br />
<br />
An alternate view is that a single algorithm can take a polynomial-length advice string, depending on k.<br />
<br />
Defined in [[zooref#df99|[DF99]]] (though they did not use our notation).<br />
<br />
----<br />
===== <span id="fptsu" style="color:red">FPT<sub>su</sub></span>: Fixed-Parameter Tractable (strongly uniform) =====<br />
Same as [[#fpt|FPT]] except that f has to be recursive.<br />
<br />
Defined in [[zooref#df99|[DF99]]] (though they did not use our notation).<br />
<br />
----<br />
===== <span id="fptas" style="color:red">FPTAS</span>: Fully Polynomial-Time Approximation Scheme =====<br />
The subclass of [[Complexity Zoo:N#npo|NPO]] problems that admit an approximation scheme in the following sense. For any &#949;&gt;0, there is an algorithm that is guaranteed to find a solution whose cost is within a 1+&#949; factor of the optimum cost. Furthermore, the running time of the algorithm is polynomial in n (the size of the problem) and in 1/&#949;.<br />
<br />
Contained in [[Complexity Zoo:P#ptas|PTAS]].<br />
<br />
Defined in [[zooref#acg99|[ACG+99]]].<br />
<br />
Contained in [[#fpt|FPT]] [[zooref#cc97|[CC97]]].<br />
<br />
----<br />
===== <span id="fqma" style="color:red">FQMA</span>: Function [[Complexity Zoo:Q#qma|QMA]] =====<br />
The class of problems for which the task is to output a quantum certificate for a [[Complexity Zoo:Q#qma|QMA]] problem, when such a certificate exists. Thus, the desired output is a quantum state.<br />
<br />
Defined in [[zooref#jwb03|[JWB03]]], where it is also shown that state preparation for 3-local Hamiltonians is FQMA-complete. The authors also observe that, in contrast to the case of [[#fnp|FNP]] versus [[Complexity Zoo:N#np|NP]], there is no obvious reduction of FQMA problems to [[Complexity Zoo:Q#qma|QMA]] problems.<br />
<br />
----<br />
===== <span id="frip" style="color:red">frIP</span>: Function-Restricted [[Complexity Zoo:I#ip|IP]] Proof Systems =====<br />
The class of problems L that have a <i>decider</i> in the following sense. There exists a [[Complexity Zoo:B#bpp|BPP]] machine D such that for all inputs x,<br />
<ol><br />
<li>If the answer is "yes" then D<sup>L</sup>(x) (D with oracle for L) accepts with probability at least 2/3.</li><br />
<li>If the answer is "no" then D<sup>A</sup>(x) accepts with probability at most 1/3 for all oracles A.</li><br />
</ol><br />
<br />
Contains [[Complexity Zoo:C#compip|compIP]] [[zooref#bg94|[BG94]]] and [[#check|Check]] [[zooref#bk89|[BK89]]].<br />
<br />
Contained in [[Complexity Zoo:M#mip|MIP]] = [[Complexity Zoo:N#nexp|NEXP]] [[zooref#frs88|[FRS88]]].<br />
<br />
Assuming [[Complexity Zoo:N#nee|NEE]] is not contained in [[Complexity Zoo:B#bpee|BPEE]], [[Complexity Zoo:N#np|NP]] (and indeed [[Complexity Zoo:N#np|NP]] &#8745; [[Complexity Zoo:C#coh|Coh]]) is not contained in [[#frip|frIP]] [[zooref#bg94|[BG94]]].<br />
<br />
----<br />
===== <span id="ftape" style="color:red">F-TAPE(f(n))</span>: Provable [[Complexity Zoo:D#dspace|DSPACE(f(n))]] For Formal System F =====<br />
The class of decision problems that can be <i>proven</i> to be solvable in O(f(n)) space on a deterministic Turing machine, from the axioms of formal system F.<br />
<br />
Defined in [[zooref#har78|[Har78]]].<br />
<br />
See also [[#ftime|F-TIME(f(n))]]. The results about F-TAPE mirror those about [[#ftime|F-TIME]], but in some cases are sharper.<br />
<br />
----<br />
===== <span id="ftime" style="color:red">F-TIME(f(n))</span>: Provable [[Complexity Zoo:D#dtime|DTIME(f(n))]] For Formal System F =====<br />
The class of decision problems that can be <i>proven</i> to be solvable in O(f(n)) time on a deterministic Turing machine, from the axioms of formal system F.<br />
<br />
Defined in [[zooref#har78|[Har78]]], where the following was also shown:<br />
<ul><br />
<li>If F-TIME(f(n)) = [[Complexity Zoo:D#dtime|DTIME(f(n))]], then [[Complexity Zoo:D#dtime|DTIME(f(n))]] is strictly contained in [[Complexity Zoo:D#dtime|DTIME(f(n)g(n))]] for any nondecreasing, unbounded, recursive g(n).</li><br />
<li>There exist recursive, monotonically increasing f(n) such that F-TIME(f(n)) is strictly contained in [[Complexity Zoo:D#dtime|DTIME(f(n))]].</li><br />
</ul><br />
See also [[#ftape|F-TAPE(f(n))]].</div>LegionMammal978https://complexityzoo.net/index.php?title=Complexity_Zoo_References&diff=6519Complexity Zoo References2017-04-08T21:00:08Z<p>LegionMammal978: fix ordering</p>
<hr />
<div>__NOTOC__<br />
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<br />
===== A =====<br />
<span id="aar02" style="color:maroon">[Aar02]</span><br />
S. Aaronson.<br />
Quantum lower bound for the collision problem,<br />
<i>Proceedings of ACM STOC'2002</i>, pp. 635-642, 2002.<br />
arXiv:[http://arxiv.org/abs/quant-ph/0111102 quant-ph/0111102].<br />
<br />
<span id="aar03" style="color:maroon">[Aar03]</span><br />
S. Aaronson.<br />
Lower bounds for local search by quantum arguments,<br />
<i>Proceedings of ACM STOC 2004</i>.<br />
arXiv:[http://arxiv.org/abs/quant-ph/0307149 quant-ph/0307149],<br />
ECCC [http://eccc.uni-trier.de/eccc-reports/2003/TR03-057/ TR03-057].<br />
<br />
<span id="aar03b" style="color:maroon">[Aar03b]</span><br />
S. Aaronson.<br />
Multilinear formulas and skepticism of quantum computing,<br />
<i>Proceedings of ACM STOC 2004</i>.<br />
arXiv:[http://arxiv.org/abs/quant-ph/0311039 quant-ph/0311039],<br />
ECCC [http://eccc.uni-trier.de/eccc-reports/2003/TR03-079/ TR03-079].<br />
<br />
<span id="aar04b" style="color:maroon">[Aar04b]</span><br />
S. Aaronson.<br />
Limitations of quantum advice and one-way communication,<br />
<i>Proceedings of IEEE Complexity 2004</i>, pp. 320-332, 2004.<br />
arXiv:[http://arxiv.org/abs/quant-ph/0402095 quant-ph/0402095],<br />
ECCC [http://eccc.uni-trier.de/eccc-reports/2004/TR04-026/ TR04-026].<br />
<br />
<span id="aar05" style="color:maroon">[Aar05]</span><br />
S. Aaronson.<br />
Quantum computing and hidden variables,<br />
<i>Physical Review A</i> 71:032325, March 2005.<br />
arXiv:[http://arxiv.org/abs/quant-ph/0408035 quant-ph/0408035].<br />
<br />
<span id="aar05b" style="color:maroon">[Aar05b]</span><br />
S. Aaronson.<br />
Quantum computing, postselection, and probabilistic polynomial-time,<br />
<i>Proceedings of the Royal Society A</i>, 461(2063):3473-3482, 2005.<br />
arXiv:[http://arxiv.org/abs/quant-ph/0412187 quant-ph/0412187].<br />
<br />
<span id="aar05c" style="color:maroon">[Aar05c]</span><br />
S. Aaronson.<br />
NP-complete problems and physical reality.<br />
<i>ACM SIGACT News</i>, March 2005<br />
[http://arxiv.org/abs/quant-ph/0502072 quant-ph/0502072].<br />
<br />
<span id="aar06" style="color:maroon">[Aar06]</span><br />
S. Aaronson.<br />
Oracles are subtle but not malicious,<br />
<i>Proceedings of IEEE Complexity 2006</i>, 2006.<br />
arXiv:[http://arxiv.org/abs/cs.CC/0504048 cs.CC/0504048],<br />
ECCC [http://eccc.uni-trier.de/eccc-reports/2004/TR05-040/ TR05-040].<br />
<br />
<span id="aar06b" style="color:maroon">[Aar06b]</span><br />
S. Aaronson.<br />
QMA/qpoly is contained in PSPACE/poly: de-Merlinizing quantum protocols,<br />
<i>Proceedings of IEEE Complexity 2006</i>, 2006.<br />
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{{Reference<br />
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|title=Are Cook and Karp ever the same?<br />
|journal=Proceedings of the 18th Annual IEEE Conference on Computational Complexity<br />
|srcdetail=333-336<br />
}}<br />
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{{Reference<br />
|tag=BH08<br />
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|journal=Electronic Colloquium on Computational Complexity<br />
|authors=H. Buhrman and J. Hitchcock <br />
|srcdetail=ECCC Report TR08-022, accepted on Mar 11, 2008<br />
|link=[http://eccc.hpi-web.de/eccc-reports/2008/TR08-022/index.html http://eccc.hpi-web.de/eccc-reports/2008/TR08-022/index.html]<br />
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{{Reference<br />
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|title=Average-Case Complexity<br />
|journal=ECCC Report TR06-073<br />
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}}<br />
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<span id="frs88" style="color:maroon">[FRS88]</span><br />
L. Fortnow, J. Rompel, and M. Sipser.<br />
On the power of multiprover interactive protocols,<br />
<i>Proceedings of IEEE Complexity'88</i>, 1988.<br />
[http://people.cs.uchicago.edu/~fortnow/papers/mip.ps http://people.cs.uchicago.edu/~fortnow/papers/mip.ps].<br />
<br />
<span id="fs04" style="color:maroon">[FS04]</span><br />
L. Fortnow and R. Santhanam.<br />
Hierarchy theorems for probabilistic polynomial time,<br />
<i>Proceedings of IEEE FOCS'2004</i>, 2004.<br />
[http://people.cs.uchicago.edu/~fortnow/papers/probhier.ps http://people.cs.uchicago.edu/~fortnow/papers/probhier.ps].<br />
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<span id="fs88" style="color:maroon">[FS88]</span><br />
L. Fortnow and M. Sipser.<br />
Are there interactive protocols for co-NP languages?<br />
Inform. Process. Lett. 28 (1988), no. 5, 249--251.<br />
[http://cs-www.uchicago.edu/~fortnow/papers/conpipl.ps http://cs-www.uchicago.edu/~fortnow/papers/conpipl.ps]<br />
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<span id="fss84" style="color:maroon">[FSS84]</span><br />
M. Furst, J. Saxe, and M. Sipser.<br />
Parity, circuits, and the polynomial hierarchy,<br />
<i>Mathematical Systems Theory</i> 17:13-27, 1984.<br />
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<span id="fur07" style="color:maroon">[Fur07]</span><br />
M. Furer.<br />
Fast Integer Multiplication,<br />
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<span id="fv93" style="color:maroon">[FV93]</span><br />
T. Feder and M. Y. Vardi.<br />
Monotone monadic SNP and constraint satisfaction,<br />
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<br />
===== G =====<br />
<br />
<span id="gas02" style="color:maroon">[Gas02]</span><br />
W. Gasarch.<br />
The P=?NP poll,<br />
<i>SIGACT News Complexity Theory Column 36</i> (L. A. Hemaspaandra, ed.), 2002.<br />
<br />
<span id="gas02" style="color:maroon">[Geff91]</span><br />
V. Geffert.<br />
Nondeterministic computations in sublogarithmic space and space constructibility,<br />
<i>SIAM Journal on Computing</i> v. 20 iss. 3, 1991.<br />
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<span id="gg66" style="color:maroon">[GG66]</span><br />
S. Ginsburg and S. Greibach.<br />
Deterministic context free languages,<br />
<i>Information and Control</i> 9:620-648, 1966.<br />
<br />
<span id="ggk03" style="color:maroon">[GGK03]</span><br />
W. Gasarch, E. Golub, and C. Kruskal. <br />
Constant time parallel sorting: an empirical view,<br />
<i>J. Comput. Syst. Sci.</i> 67:63-91, 2003.<br />
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<span id="ghj91" style="color:maroon">[GHJ+91]</span><br />
J. Goldsmith, L. A. Hemaspaandra, D. Joseph, and P. Young.<br />
Near-testable sets.<br />
SIAM J. Comput. 20 (1991), no. 3, 506--523<br />
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<span id="ghp00" style="color:maroon">[GHP00]</span><br />
F. Green, S. Homer, and C. Pollett.<br />
On the complexity of quantum ACC,<br />
<i>Proceeedings of IEEE Complexity'2000</i>, pp. 250-262, 2000.<br />
See also:<br />
F. Green, S. Homer, C. Moore, and S. Pollett.<br />
Counting, fanout, and the complexity of quantum ACC,<br />
arXiv:[http://arxiv.org/abs/quant-ph/0106017 quant-ph/0106017], 2001.<br />
<br />
<span id="gil77" style="color:maroon">[Gil77]</span><br />
J. Gill.<br />
Computational complexity of probabilistic Turing machines,<br />
<i>SIAM Journal on Computing</i> 6(4):675-695, 1977.<br />
<br />
<span id="gj79" style="color:maroon">[GJ79]</span><br />
M. R. Garey and D. S. Johnson.<br />
<i>Computers and Intractability: A Guide to the Theory of NP-Completeness</i>,<br />
Freeman, 1979.<br />
<br />
<span id="gk14" style="color:maroon">[GK14]</span><br />
S. Gharibian, and J. Kempe.<br />
Hardness of approximation for quantum problems,<br />
<i>Quantum Information & Computation</i> 14(5 &amp; 6): 517-540, 2014.<br />
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<br />
<span id="gy16" style="color:maroon">[GY16]</span><br />
S. Gharibian, and J. Yirka.<br />
The complexity of estimating local physical quantities, <br />
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<br />
<span id="gkr95" style="color:maroon">[GKR+95]</span><br />
F. Green, J. K&ouml;bler, K. W. Regan, T. Schwentick, and J. Tor&aacute;n.<br />
The power of the middle bit of a #P function,<br />
<i>Journal of Computer and System Sciences</i> 50(3):456-467, 1995.<br />
<br />
<span id="gl14" style="color:maroon">[GL14]</span><br />
D. Gavinsky and S. Lovett.<br />
En route to the log-rank conjecture: New reductions and equivalent formulations,<br />
<i>Proceedings of ICALP'14</i>, pp. 514-524, 2014.<br />
<br />
<span id="glm96" style="color:maroon">[GLM96]</span><br />
J. Goldsmith, M. A. Levy, and M. Mundhenk.<br />
Limited nondeterminism,<br />
<i>SIGACT News</i> 27(2):20-29, 1996.<br />
[http://cs.engr.uky.edu/~goldsmit/papers/extended.ps http://cs.engr.uky.edu/~goldsmit/papers/extended.ps]<br />
<br />
<span id="glm+15" style="color:maroon">[GLM+15]</span><br />
M. Göös, S. Lovett, R. Meka, T. Watson, and D. Zuckerman.<br />
Rectangles Are Nonnegative Juntas,<br />
<i>Proceedings of ACM STOC'15</i>, pp. 257-266, 2015.<br />
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<span id="gmr89" style="color:maroon">[GMR89]</span><br />
S. Goldwasser, S. Micali, and C. Rackoff.<br />
The knowledge complexity of interactive proof systems,<br />
<i>SIAM Journal on Computing</i> 18(1):186-208, 1989.<br />
<br />
<span id="gmw91" style="color:maroon">[GMW91]</span><br />
O. Goldreich, S. Micali, and A. Wigderson.<br />
Proofs that yield nothing but their validity or all languages in NP have zero-knowledge proof systems,<br />
<i>Journal of the ACM</i> 38(1):691-729, 1991.<br />
<br />
<span id="gn13" style="color:maroon">[GN13]</span><br />
D. Gosset and D. Nagaj.<br />
Quantum 3-SAT is QMA1-complete,<br />
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{{Reference<br />
|tag=GO95<br />
|title=On a class of <math>O(n^2)</math> problems in computational geometry<br />
|authors=A. Gajentaan and M. Overmars<br />
|journal=Computational Geometry<br />
|srcdetail=Volume 5, Issue 3, October 1995, pages 165-185<br />
}}<br />
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<span id="gol97" style="color:maroon">[Gol97]</span><br />
O. Goldreich.<br />
Notes on Levin's theory of average-case complexity,<br />
ECCC [http://eccc.uni-trier.de/eccc-reports/1997/TR97-058/ TR97-058].<br />
<br />
<span id="gp01" style="color:maroon">[GP01]</span><br />
F. Green and R. Pruim.<br />
Relativized separation of EQP from P^NP,<br />
Information Processing Letters 80 (2001) 257-260.<br />
[http://cs.clarku.edu/~fgreen/papers/eqp.ps http://cs.clarku.edu/~fgreen/papers/eqp.ps]<br />
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<span id="gp86" style="color:maroon">[GP86]</span><br />
L. Goldschlager and I. Parberry.<br />
On the construction of parallel computers from various bases of Boolean functions,<br />
<i>Theoretical Computer Science</i> 43(1):43-58, 1986.<br />
<br />
<span id="gp91" style="color:maroon">[GP91]</span><br />
O. Goldreich and E. Petrank.<br />
Quantifying knowledge complexity,<br />
<i>Proceedings of IEEE FOCS'91</i>, pp. 59-68, 1991.<br />
[http://www.wisdom.weizmann.ac.il/~oded/PS/gp.ps http://www.wisdom.weizmann.ac.il/~oded/PS/gp.ps]<br />
<br />
<span id="gpw15" style="color:maroon">[GPW15]</span><br />
M. Göös, T. Pitassi, and T. Watson.<br />
Deterministic Communication vs. Partition Number,<br />
<i>Proceedings of IEEE FOCS'15</i>, 1077-1088, 2015.<br />
<br />
<span id="gpw16a" style="color:maroon">[GPW16a]</span><br />
M. Göös, T. Pitassi, and T. Watson.<br />
Zero-Information Protocols and Unambiguity in Arthur-Merlin Communication,<br />
<i>Algorithmica</i>, Online First, 2016.<br />
<br />
<span id="gpw16b" style="color:maroon">[GPW16b]</span><br />
M. Göös, T. Pitassi, and T. Watson.<br />
The Landscape of Communication Complexity Classes,<br />
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<span id="gra92" style="color:maroon">[Grä92]</span><br />
E. Grädel<br />
Capturing complexity classes b fragments of second order logic<br />
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<br />
<span id="gre90" style="color:maroon">[Gre90]</span><br />
F. Green.<br />
An oracle separating +P from PP<sup>PH</sup>,<br />
Inform. Process. Lett. 37 (1991), no. 3, 149--153.<br />
<br />
<span id="gre93" style="color:maroon">[Gre93]</span><br />
F. Green.<br />
On the power of deterministic reductions to C<sub>=</sub>P,<br />
Math. Systems Theory 26 (1993), no. 2, 215--233.<br />
<br />
<span id="gs86" style="color:maroon">[GS86]</span><br />
S. Goldwasser and M. Sipser.<br />
Private coins versus public coins in interactive proof systems,<br />
<i>Proceedings of ACM STOC'86</i>, pp. 58-68, 1986.<br />
<br />
<span id="gs88" style="color:maroon">[GS88]</span><br />
J. Grollman and A. L. Selman.<br />
Complexity measures for public-key cryptosystems,<br />
<i>SIAM Journal on Computing</i> 17:309-335, 1988.<br />
<br />
<span id="gs89" style="color:maroon">[GS89]</span><br />
Y. Gurevich and S. Shelah.<br />
Nearly Linear Time,<br />
<i>Proceedings of LFCS'89</i>, Springer LNCS 363, pp. 108-118, 1989.<br />
<br />
<span id="gs90" style="color:maroon">[GS90]</span><br />
M. Grigni and M. Sipser.<br />
Monotone complexity,<br />
<i>Proceedings of LMS Workshop on Boolean Function Complexity</i> (M. S. Paterson, ed.), Durham, Cambridge University Press, 1990.<br />
[http://www.mathcs.emory.edu/~mic/papers/4.ps http://www.mathcs.emory.edu/~mic/papers/4.ps]<br />
<br />
<span id="gs91" style="color:maroon">[GS91]</span><br />
M. Grigni and M. Sipser.<br />
Monotone separation of NC<sup>1</sup> from logspace,<br />
<i>Proceedings of IEEE Complexity'91</i>, pp. 294-298, 1991.<br />
<br />
<span id="gs15" style="color:maroon">[GS15]</span><br />
S. Gharibian, and J. Sikora.<br />
Ground state connectivity of local Hamiltonians, <i>Proceeedings of the 42nd International Colloquium on Automata, Languages, and Programming (ICALP)</i>, volume 9134 of Lecture Notes in Computer Science, pages 617-628, 2015.<br />
<br />
<span id="gri01" style="color:maroon">[Gri01]</span><br />
M. Grigni.<br />
A Sperner lemma complete for PPA,<br />
<i>Information Processing Letters</i> 77:5-6 (2001), pp. 255-259.<br />
<br />
<span id="gss03" style="color:maroon">[GSS+03]</span><br />
C. Gla&szlig;er, A. L. Selman, S. Sengupta, and L. Zhang.<br />
Disjoint NP-pairs,<br />
ECCC [http://eccc.uni-trier.de/eccc-reports/2003/TR03-011/ TR03-011], 2003.<br />
<br />
<span id="gst03" style="color:maroon">[GST03]</span><br />
D. Gutfreund, R. Shaltiel, and A. Ta-Shma.<br />
Uniform hardness vs. randomness tradeoffs for Arthur-Merlin games,<br />
<i>Comput. Complexity</i> 12 (2003), no. 3-4, 85--130.<br />
[http://www.cs.huji.ac.il/~danig/pubs/ccc.ps http://www.cs.huji.ac.il/~danig/pubs/ccc.ps].<br />
<br />
<span id="gsv99" style="color:maroon">[GSV99]</span><br />
O. Goldreich, A. Sahai, and S. Vadhan.<br />
Can statistical zero knowledge be made non-interactive? or on the relationship of SZK and NISZK,<br />
ECCC [http://eccc.uni-trier.de/eccc-reports/1999/TR99-013/ TR99-013], 1999.<br />
Abstract appeared in CRYPTO'99.<br />
<br />
<span id="gtw91" style="color:maroon">[GTW+91]</span><br />
R. Gavald&aacute;, L. Torenvliet, O. Watanabe, and J. Balc&aacute;zar.<br />
Generalized Kolmogorov complexity in relativized separations,<br />
<i>Proceedings of MFCS'91 (Mathematical Foundations of Computer Science)</i>, Springer-Verlag Lecture Notes in Computer Science, vol. 452, pp. 269-276, 1991.<br />
<br />
<span id="gup95" style="color:maroon">[Gup95]</span><br />
S. Gupta.<br />
Closure properties and witness reduction,<br />
<i>Journal of Computer and System Sciences</i> 50(3):412-432, 1995.<br />
[ftp://ftp.cis.ohio-state.edu/pub/tech-report/1993/TR46.ps.gz ftp://ftp.cis.ohio-state.edu/pub/tech-report/1993/TR46.ps.gz]<br />
<br />
<span id="gur87" style="color:maroon">[Gur87]</span><br />
Y. Gurevich.<br />
Complete and incomplete randomized NP problems,<br />
<i>Proceedings of IEEE FOCS'87</i>, pp. 111-117, 1987.<br />
<br />
<span id="gur89" style="color:maroon">[Gur89]</span><br />
E. Gurari.<br />
<i>An Introduction to the Theory of Computation</i>,<br />
Computer Science Press, 1989.<br />
[http://www.cse.ohio-state.edu/~gurari/theory-bk/theory-bk.html http://www.cse.ohio-state.edu/~gurari/theory-bk/theory-bk.html].<br />
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<span id="gut05" style="color:maroon">[Gut05]</span><br />
G. Gutoski.<br />
Upper bounds for quantum interactive proofs with competing provers,<br />
<i>Proceedings of IEEE Complexity'2005</i>, pp. 334-343, 2005.<br />
[http://www.cs.uwaterloo.ca/~gmgutosk/gutoskig_competing.pdf http://www.cs.uwaterloo.ca/~gmgutosk/gutoskig_competing.pdf].<br />
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<span id="gv02" style="color:maroon">[GV02]</span><br />
M. de Graaf and P. Valiant.<br />
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<span id="gv99" style="color:maroon">[GV99]</span><br />
O. Goldreich and S. Vadhan.<br />
Comparing entropies in statistical zero-knowledge with applications to the structure of SZK,<br />
<i>Proceedings of IEEE Complexity'99</i>, pp. 54-73, 1999.<br />
ECCC [http://eccc.uni-trier.de/eccc-reports/1998/TR98-063/ TR98-063].<br />
<br />
<span id="gw05" style="color:maroon">[GW05]</span><br />
G. Gutoski and J. Watrous.<br />
Quantum interactive proofs with competing provers,<br />
<i>Proceedings of STACS'2005</i>, pp. 605-616, Springer-Verlag, 2005.<br />
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<span id="gw07" style="color:maroon">[GW07]</span><br />
G. Gutoski and J. Watrous.<br />
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<span id="gw10" style="color:maroon">[GW10]</span><br />
G. Gutoski and X. Wu.<br />
Short quantum games characterize PSPACE,<br />
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<span id="gw96" style="color:maroon">[GW96]</span><br />
A. G&aacute;l and A. Wigderson.<br />
Boolean complexity classes vs. their arithmetic analogs,<br />
<i>Random Structures and Algorithms</i> 9:1-13, 1996.<br />
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<span id="gw14" style="color:maroon">[GW14]</span><br />
M. Göös and T. Watson.<br />
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O. Goldreich and D. Zuckerman.<br />
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<br />
===== H =====<br />
<br />
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S. Hallgren.<br />
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<span id="har78" style="color:maroon">[Har78]</span><br />
J. Hartmanis.<br />
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<span id="har87" style="color:maroon">[Har87]</span><br />
J. Hartmanis.<br />
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<i>Bulletin of the EATCS</i> 33, October 1987.<br />
[http://external.nj.nec.com/homepages/fortnow/beatcs/column33.ps http://external.nj.nec.com/homepages/fortnow/beatcs/column33.ps].<br />
<br />
<span id="har87b" style="color:maroon">[Har87b]</span><br />
J. Hartmanis.<br />
Sparse complete sets for NP and the optimal collapse of the polynomial hierarchy,<br />
<i>Bulletin of the EATCS</i> 32, June 1987.<br />
[http://external.nj.nec.com/homepages/fortnow/beatcs/column32.ps http://external.nj.nec.com/homepages/fortnow/beatcs/column32.ps].<br />
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<span id="has87" style="color:maroon">[Has87]</span><br />
J. H&aring;stad.<br />
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<br />
<span id="has88" style="color:maroon">[Has88]</span><br />
J. H&aring;stad.<br />
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<span id="hcc92" style="color:maroon">[HCC+92]</span><br />
J. Hartmanis, R. Chang, S. Chari, D. Ranjan, and P. Rohatgi.<br />
Relativization: a revisionistic retrospective,<br />
<i>Bulletin of the EATCS</i> 47, June 1992.<br />
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<span id="hck88" style="color:maroon">[HCK+88]</span><br />
J. Hartmanis, R. Chang, J. Kadin, and S. G. Mitchell.<br />
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<i>Bulletin of the EATCS</i> 35, June 1988.<br />
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<span id="hel84" style="color:maroon">[Hel84a]</span><br />
H. Heller.<br />
Relativized polynomial hierarchies extending two levels,<br />
<i>Mathematical Systems Theory</i> 17(2):71-84, 1984.<br />
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H. Heller.<br />
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<i>SIAM Journal on Computing</i> 13(4):717-725, 1984.<br />
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H. Heller.<br />
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U. Hertrampf.<br />
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J. Hartmanis and J. Hopcroft.<br />
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J. Hartmanis and L. Hemachandra.<br />
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L. Hemaspaandra, A. Hoene, A. Naik, M. Ogihara, A. Selman, T. Thierauf, and J. Wang.<br />
Nondeterministically selective sets,<br />
<i>International Journal of Foundations of Computer Science (IJFCS)</i>, 6(4):403-416, 1995.<br />
<br />
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E. Hemaspaandra, L. Hemaspaandra, and J. Rothe.<br />
Exact analysis of Dodgson elections: Lewis Carroll's 1876 voting system is complete for parallel access to NP,<br />
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Threshold computation and cryptographic security,<br />
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<i>International Journal of Foundations of Computer Science</i>, 3-4:245-265, 1993.<br />
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L. Hemaspaandra, S. Kosub, and K. Wagner.<br />
The complexity of computing the size of an interval,<br />
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Computing solutions uniquely collapses the polynomial hierarchy,<br />
<i>SIAM Journal on Computing</i> 25(4):697-708, 1996.<br />
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<br />
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<br />
===== I =====<br />
<br />
<span id="iba72" style="color:maroon">[Iba72]</span><br />
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No better ways to generate hard NP instances than picking uniformly at random,<br />
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{{Reference<br />
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{{Reference<br />
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|journal=Proceedings of the 10th Conference on Structure in Complexity Theory<br />
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N. Immerman.<br />
Relational queries computable in in polynomial time.<br />
<i>14th ACM STOC Symp. (1987), 86-104</i><br />
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<br />
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N. Immerman.<br />
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N. Immerman.<br />
Nondeterministic space is closed under complement,<br />
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<br />
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N. Immerman.<br />
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<span id="in96" style="color:maroon">[IN96]</span><br />
R. Impagliazzo and M. Naor.<br />
Efficient cryptographic schemes provably as secure as subset sum,<br />
<i>Journal of Cryptology</i> 9(4):199-216, 1996.<br />
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R. Impagliazzo and M. Sudan.<br />
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R. Impagliazzo and G. Tardos.<br />
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T. Ito and T. Vidick.<br />
A multi-prover interactive proof for NEXP sound against entangled provers,<br />
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<span id="iw97" style="color:maroon">[IW97]</span><br />
R. Impagliazzo and A. Wigderson.<br />
P=BPP if E requires exponential circuits: derandomizing the XOR lemma,<br />
<i>Proceedings of ACM STOC'97</i>, pp. 220-229, 1997.<br />
<br />
===== J =====<br />
<br />
<span id="jer07" style="color:maroon">[Jeř07]</span><br />
E. Jeřábek.<br />
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<br />
===== K =====<br />
<br />
<span id="kad88" style="color:maroon">[Kad88]</span><br />
J. Kadin.<br />
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===== O =====<br />
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===== P =====<br />
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|journal=Lecture Notes in Computer Science<br />
|authors=A. Szepietowski<br />
|srcdetail=volume 843<br />
}}<br />
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===== Y =====<br />
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{{Reference<br />
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|srcdetail=(1983), 26, 287-300<br />
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<br />
[[Category:Computational Complexity]]</div>LegionMammal978https://complexityzoo.net/index.php?title=Complexity_Zoo_References&diff=6518Complexity Zoo References2017-04-08T20:59:27Z<p>LegionMammal978: AW09</p>
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<br />
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{{Reference<br />
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|journal=Proceedings of the 18th Annual IEEE Conference on Computational Complexity<br />
|srcdetail=333-336<br />
}}<br />
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|srcdetail=ECCC Report TR08-022, accepted on Mar 11, 2008<br />
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===== G =====<br />
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{{Reference<br />
|tag=GO95<br />
|title=On a class of <math>O(n^2)</math> problems in computational geometry<br />
|authors=A. Gajentaan and M. Overmars<br />
|journal=Computational Geometry<br />
|srcdetail=Volume 5, Issue 3, October 1995, pages 165-185<br />
}}<br />
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<br />
===== J =====<br />
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===== L =====<br />
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===== M =====<br />
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===== O =====<br />
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===== P =====<br />
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<span id="tor90" style="color:maroon">[Tor90]</span><br />
J. Tor&aacute;n.<br />
Counting the number of solutions,<br />
<i>Proceedings of 15th Conference on Mathematical Foundations of Computer Science (MFCS)</i>, pp. 121-135, Springer-Verlag Lecture Notes in Computer Science 452, 1990.<br />
<br />
<span id="tor91" style="color:maroon">[Tor91]</span><br />
J. Tor&aacute;n.<br />
Complexity classes defined by counting quantifiers,<br />
<i>Journal of the ACM</i> 38:753-774, 1991.<br />
<br />
<span id="tur36" style="color:maroon">[Tur36]</span><br />
A. M. Turing.<br />
On computable numbers, with an application to the <i>Entscheidungsproblem</i>,<br />
<i>Proceedings of the London Mathematical Society</i> 2(42):230-265, 1936; 2(43):544-546, 1937.<br />
<br />
<span id="tv02" style="color:maroon">[TV02]</span><br />
L. Trevisan and S. Vadhan.<br />
Pseudorandomness and average-case complexity via uniform reductions,<br />
<i>Proceedings of CCC'2002</i>, pp. 129-138, 2002.<br />
<br />
===== U =====<br />
<br />
<span id="uma98" style="color:maroon">[Uma98]</span><br />
C. Umans.<br />
The minimum equivalent DNF problem and shortest implicants,<br />
<i>Proceedings of IEEE FOCS'98</i>, pp. 556-563, 1998.<br />
<br />
===== V =====<br />
<br />
<span id="vad06" style="color:maroon">[Vad06]</span><br />
S. Vadhan.<br />
An Unconditional Study of Computational Zero Knowledge,<br />
ECCC [http://eccc.hpi-web.de/eccc-reports/2006/TR06-056/ TR06-056].<br />
<br />
<span id="val03" style="color:maroon">[Val03]</span><br />
L. G. Valiant.<br />
Three problems in computer science,<br />
<i>Journal of the ACM</i> 50(1):96-99, 2003.<br />
<br />
<span id="val76" style="color:maroon">[Val76]</span><br />
L. G. Valiant.<br />
Relative complexity of checking and evaluating,<br />
<i>Information Processing Letters</i>, 5:20-23, 1976.<br />
<br />
<span id="val79" style="color:maroon">[Val79]</span><br />
L. G. Valiant.<br />
The complexity of computing the permanent,<br />
<i>Theoretical Computer Science</i>, 8:189-201, 1979.<br />
<br />
<span id="val79b" style="color:maroon">[Val79b]</span><br />
L. G. Valiant.<br />
Completeness classes in algebra,<br />
<i>Proceedings of ACM STOC'79</i>, pp. 249-261, 1979.<br />
<br />
<span id="var82" style="color:maroon">[Var82]</span><br />
M. Vardi.<br />
Complexity of relational query languages,<br />
<i>Proceedings of ACM STOC'82</i>, pp. 137-146, 1982.<br />
<br />
<span id="ven91" style="color:maroon">[Ven91]</span><br />
H. Venkateswaran.<br />
Properties that characterize LOGCFL,<br />
<i>Journal of Computer and System Sciences</i> 43(2):380-404, 1991.<br />
<br />
<span id="ver92" style="color:maroon">[Ver92]</span><br />
N. K. Vereshchagin.<br />
On the power of PP,<br />
<i>Proceedings of IEEE Complexity'92</i>, pp. 138-143, 1992.<br />
<br />
<span id="ver95" style="color:maroon">[Ver95]</span><br />
N. K. Vereshchagin.<br />
Oracle separation of complexity classes and lower bounds for perceptrons solving separation problems,<br />
<i>Izvestiya Mathematics</i> 59(6):1103-1122, 1995.<br />
<br />
<span id="vid03" style="color:maroon">[Vid03]</span><br />
G. Vidal.<br />
Efficient classical simulation of slightly entangled quantum computations,<br />
<i>Physical Review Letters</i> 91:147902, 2003.<br />
arXiv:[http://arxiv.org/abs/quant-ph/0301063 quant-ph/0301063].<br />
<br />
<span id="vin04" style="color:maroon">[Vin04]</span><br />
N. V. Vinodchandran.<br />
Counting complexity of solvable group problems,<br />
<i>SIAM Journal on Computing</i> 33(4):852-869, 2004,<br />
[http://www.cse.unl.edu/~vinod/papers/SIAMFinal.ps http://www.cse.unl.edu/~vinod/papers/SIAMFinal.ps].<br />
<br />
<span id="vin04b" style="color:maroon">[Vin04b]</span><br />
N. V. Vinodchandran.<br />
A note on the circuit complexity of PP,<br />
ECCC [http://eccc.uni-trier.de/eccc-reports/2004/TR04-056/ TR04-056], 2004.<br />
<br />
<span id="vsb83" style="color:maroon">[VSB+83]</span><br />
L. G. Valiant, S. Skyum, S. Berkowitz, and C. Rackoff.<br />
Fast parallel computation of polynomials using few processors,<br />
<i>SIAM Journal on Computing</i> 12(4):641-644, 1983.<br />
<br />
<span id="vv85" style="color:maroon">[VV85]</span><br />
U. V. Vazirani and V. V. Vazirani.<br />
Random polynomial time equals semi-random polynomial time,<br />
<i>Proceedings of IEEE FOCS'85</i>, pp. 417-428, 1985.<br />
<br />
<span id="vv86" style="color:maroon">[VV86]</span><br />
L. G. Valiant and V. V. Vazirani.<br />
NP is as easy as detecting unique solutions,<br />
<i>Theoretical Computer Science</i> 47(3):85-93, 1986.<br />
<br />
<span id="vya03" style="color:maroon">[Vya03]</span><br />
M. Vyalyi.<br />
QMA=PP implies that PP contains PH,<br />
ECCC [http://eccc.uni-trier.de/eccc-reports/2003/TR03-021/ TR03-021], 2003.<br />
<br />
===== W =====<br />
<br />
<span id="wag86" style="color:maroon">[Wag86]</span><br />
K. W. Wagner.<br />
The complexity of combinatorial problems with succinct input representation,<br />
<i>Acta Informatica</i> 23:325-356, 1986.<br />
<br />
<span id="wag88" style="color:maroon">[Wag88]</span><br />
K. W. Wagner.<br />
Bounded query computation,<br />
<i>Proceedings of IEEE Complexity'88</i>, pp. 260-277, 1988.<br />
<br />
<span id="ww85" style="color:maroon">[WW85]</span><br />
G. Wechsung.<br />
On the Boolean closure of NP,<br />
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<br />
<span id="wat00" style="color:maroon">[Wat00]</span><br />
J. Watrous.<br />
Succinct quantum proofs for properties of finite groups,<br />
<i>Proceedings of IEEE FOCS'2000</i>, pp. 537-546, 2000.<br />
arXiv:[http://arxiv.org/abs/cs.CC/0009002 cs.CC/0009002].<br />
<br />
<span id="wat02" style="color:maroon">[Wat02]</span><br />
J. Watrous.<br />
Limits on the power of quantum statistical zero-knowledge,<br />
to appear in <i>Proceedings of IEEE FOCS'2002</i>.<br />
arXiv:[http://arxiv.org/abs/quant-ph/0202111 quant-ph/0202111].<br />
<br />
<span id="wat09" style="color:maroon">[Wat09]</span><br />
J. Watrous.<br />
Quantum Computational Complexity, <i>Encyclopedia of Complexity and Systems Science</i>, Springer, pp. 7174-7201, 2009.<br />
arXiv:[http://arxiv.org/abs/0804.3401 quant-ph/0804.3401].<br />
<br />
<span id="wat87" style="color:maroon">[Wat87]</span><br />
O. Watanabe.<br />
Comparison of polynomial time completeness notions,<br />
<i>Theoretical Computer Science</i> 53:249-265, 1987.<br />
<br />
<span id="wat99" style="color:maroon">[Wat99]</span><br />
J. Watrous.<br />
PSPACE has constant-round quantum interactive proof systems,<br />
<i>Proceedings of IEEE FOCS'99</i>, pp. 112-119, 1999.<br />
arXiv:[http://arxiv.org/abs/cs.CC/9901015 cs.CC/9901015].<br />
<br />
<span id="wat99b" style="color:maroon">[Wat99b]</span><br />
J. Watrous.<br />
Space-bounded quantum complexity,<br />
<i>Journal of Computer and System Sciences</i> 59(2):281-326, 1999.<br />
[http://www.cpsc.ucalgary.ca/%7Ejwatrous/papers/jcss_space.ps http://www.cpsc.ucalgary.ca/%7Ejwatrous/papers/jcss_space.ps].<br />
<br />
<span id="wat15" style="color:maroon">[Wat15]</span><br />
T. Watson.<br />
The complexity of deciding statistical properties of samplable distributions,<br />
<i>Theory of Computing</i>, 11:1-34, 2015.<br />
<br />
<span id="weg87" style="color:maroon">[Weg87]</span><br />
I. Wegener.<br />
The Complexity of Boolean Functions, New York: Wiley 1987.<br />
<br />
<span id="weg88" style="color:maroon">[Weg88]</span><br />
I. Wegener.<br />
On the Complexity of Branching Programs and Decision Trees for Clique Functions,<br />
<i>Journal of the ACM</i> 35(2):461-471, 1988.<br />
DOI:[http://doi.acm.org/10.1145/42282.46161 10.1145/42282.46161].<br />
<br />
<span id="weh06" style="color:maroon">[Weh06]</span><br />
S. Wehner.<br />
Entanglement in interactive proof systems with binary answers, In <i>Proceedings of<br />
the 23rd Annual Symposium on Theoretical Aspects of Computer Science</i>, volume 3884 of <i>Lecture<br />
Notes in Computer Science</i>, pages 162–171. Springer, 2006<br />
<br />
<span id="wig06" style="color:maroon">[Wig06]</span><br />
A. Wigderson<br />
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[www.math.ias.edu/~avi/PUBLICATIONS/MYPAPERS/W06/W06.pdf].<br />
<br />
<span id="wil85" style="color:maroon">[Wil85]</span><br />
C. Wilson.<br />
Relativized circuit complexity,<br />
<i>Journal of Computer and System Sciences</i> 31:169-181, 1985.<br />
<br />
<span id="wil11" style="color:maroon">[Wil11]</span><br />
R. Williams. Non-uniform ACC circuit lower bounds,<br />
<i>To appear in IEEE Conference on Computational Complexity</i> 2011.<br />
<br />
<span id="wol94" style="color: maroon;">[Wol94]</span><br />
M. J. Wolf.<br />
Nondeterministic circuits, space complexity and quasigroups,<br />
<i>Theoretical Computer Science</i> 125:295–313, 1994.<br />
<br />
===== Y =====<br />
<br />
{{Reference<br />
|tag=Yap83<br />
|authors=C. Yap<br />
|title=Some consequences of non-uniform conditions on uniform classes<br />
|journal=Theoretical Computer Science<br />
|srcdetail=(1983), 26, 287-300<br />
}}<br />
<br />
<span id="yam99" style="color:maroon">[Yam99]</span><br />
T. Yamakami.<br />
Polynomial time samplable distributions,<br />
J. Complexity 15 (1999), no. 4, 557-574.<br />
ECCC [http://eccc.hpi-web.de/eccc-reports/1995/TR95-039/ TR95-039].<br />
<br />
<span id="yan81" style="color:maroon">[Yan81]</span><br />
M. Yannakakis.<br />
Algorithms for acyclic database schemas,<br />
<i>Proceedings of VLDB</i> (Very Large Databases), 1981.<br />
<br />
<span id="yan91" style="color:maroon">[Yan91]</span><br />
M. Yannakakis.<br />
Expressing combinatorial optimization problems by linear programs,<br />
<i>Journal of Computer and System Sciences</i>, 43(3):441-466, 1991.<br />
<br />
<span id="yao85" style="color:maroon">[Yao85]</span><br />
A. C.-C. Yao.<br />
Separating the polynomial hierarchy by oracles,<br />
<i>Proceedings of IEEE FOCS'85</i>, pp. 1-10, 1985.<br />
<br />
<span id="yao89" style="color:maroon">[Yao89]</span><br />
A. C.-C. Yao.<br />
Circuits and local computation,<br />
<i>Proceedings of ACM STOC'89</i>, pp. 186-196, 1989.<br />
<br />
<span id="yao90" style="color:maroon">[Yao90]</span><br />
A. C.-C. Yao.<br />
On ACC and threshold circuits,<br />
<i>Proceedings of IEEE FOCS'90</i>, pp. 619-627, 1990.<br />
<br />
<span id="yao90b" style="color:maroon">[Yao90b]</span><br />
A. C.-C. Yao.<br />
Coherent functions and program checkers,<br />
<i>Proceedings of ACM STOC'90</i>, 1990.<br />
<br />
<span id="yao93" style="color:maroon">[Yao93]</span><br />
A. C.-C. Yao.<br />
Quantum circuit complexity,<br />
<i>Proceedings of IEEE FOCS'93</i>, pp. 352-361, 1993.<br />
<br />
<span id="yes83" style="color:maroon">[Yes83]</span><br />
Y. Yesha.<br />
On certain polynomial-time truth-table reducibilities of complete sets to sparse sets,<br />
<i>SIAM Journal on Computing</i>, 12(3):411-425, 1983.<br />
DOI:[http://dx.doi.org/10.1137/0212027 10.1137/0212027]<br />
<br />
===== Z =====<br />
<br />
<span id="zac88" style="color:maroon">[Zac88]</span><br />
S. Zachos.<br />
Probabilistic quantifiers and games,<br />
<i>Journal of Computer and System Sciences</i> 36(3):433-451, 1988.<br />
<br />
<span id="zh86" style="color:maroon">[ZH86]</span><br />
S. Zachos and H. Heller.<br />
A decisive characterization of BPP.<br />
''Information and Control'', 69(1&ndash;3):125&ndash;135, 1986.<br />
<br />
<span id="zuc91" style="color:maroon">[Zuc91]</span><br />
D. Zuckerman.<br />
Simulating BPP using a general weak random source,<br />
<i>Algorithmica</i> 16 (1996), no. 4-5, 367--391<br />
[http://www.cs.utexas.edu/users/diz/pubs/bpp.ps http://www.cs.utexas.edu/users/diz/pubs/bpp.ps].<br />
<br />
[[Category:Computational Complexity]]</div>LegionMammal978https://complexityzoo.net/index.php?title=Complexity_Zoo:B&diff=6517Complexity Zoo:B2017-04-08T20:58:34Z<p>LegionMammal978: BQP_CTC = PSPACE</p>
<hr />
<div>__NOTOC__<br />
{{CZ-Letter-Menu|B}}<br />
<br />
<br />
===== <span id="betap" style="color:red">&#946;P</span>: Limited-Nondeterminism [[Complexity Zoo:N#np|NP]] =====<br />
&#946;<sub>k</sub>P is the class of decision problems solvable by a polynomial-time Turing machine that makes O(log<sup>k</sup>n) nondeterministic transitions, with the same acceptance mechanism as [[Complexity Zoo:N#np|NP]]. Equivalently, the machine receives a purported proof of size O(log<sup>k</sup>n) that the answer is 'yes.'<br />
<br />
Then &#946;P is the union of &#946;<sub>k</sub>P over all constant k.<br />
<br />
Defined in [[zooref#kf84|[KF84]]]. See also the survey [[zooref#glm96|[GLM96]]].<br />
<br />
There exist oracles relative to which basically any consistent inclusion structure among the &#946;<sub>k</sub>P's can be realized [[zooref#bg98|[BG98]]].<br />
<br />
&#946;<sub>2</sub>P contains [[Complexity Zoo:L#lognp|LOGNP]] and [[Complexity Zoo:L#logsnp|LOGSNP]].<br />
<br />
----<br />
<br />
===== <span id="bcequalsp" style="color:red">BC<sub>=</sub>P</span>: Bounded-Error [[Complexity Zoo:C#cequalsp|C<sub>=</sub>P]] =====<br />
The class of decision problems solvable by an [[Complexity Zoo:N#np|NP]] machine such that<br />
<ol><br />
<li>If the answer is 'yes' then exactly 1/2 of the computation paths accept.</li><br />
<li>If the answer is 'no' then either at most 1/4 or at least 3/4 of the computation paths accept.</li><br />
</ol><br />
(Here all computation paths have the same length.)<br />
<br />
Defined in [[zooref#wat15|[Wat15]]], where it was shown that BC<sub>=</sub>P admits efficient amplification and is closed under union, intersection, disjunction, and conjunction, and that [[Complexity Zoo:C#corp|coRP]] &sube; BC<sub>=</sub>P &sube; [[Complexity Zoo:B#bpp|BPP]].<br />
<br />
----<br />
<br />
===== <span id="bh" style="color:red">BH</span>: Boolean Hierarchy Over [[Complexity Zoo:N#np|NP]] =====<br />
The smallest class that contains [[Complexity Zoo:N#np|NP]] and is closed under union, intersection, and complement.<br />
<br />
The levels are defined as follows:<br />
<ul><br />
<li>BH<sub>1</sub> = [[Complexity Zoo:N#np|NP]].</li><br />
<li>BH<sub>2i</sub> is the class of languages that are the intersection of a BH<sub>2i-1</sub> language with a [[Complexity Zoo:C#conp|coNP]] language.</li><br />
<li>BH<sub>2i+1</sub> is the class of languages that are the union of a BH<sub>2i</sub> language with an [[Complexity Zoo:N#np|NP]] language.</li><br />
</ul><br />
Then BH is the union over all i of BH<sub>i</sub>.<br />
<br />
Defined in [[zooref#ww85|[WW85]]].<br />
<br />
For more detail see [[zooref#cgh88|[CGH+88]]].<br />
<br />
Contained in [[Complexity Zoo:D#delta2p|&#916;<sub>2</sub>P]] and indeed in [[Complexity Zoo:P#pnplog|P<sup>NP[log]</sup>]].<br />
<br />
If BH collapses at any level, then [[Complexity Zoo:P#ph|PH]] collapses to &#931;<sub>3</sub>P [[zooref#kad88|[Kad88]]].<br />
<br />
''See also'': {{zcls|d|dp}}, [[Complexity Zoo:Q#qh|QH]].<br />
<br />
----<br />
<br />
===== <span id="bpdp" style="color:red">BP<sub>d</sub>(P)</span>: Polynomial Size d-Times-Only Branching Program =====<br />
Defined in [[zooref#weg88|[Weg88]]].<br />
<br />
The class of decision problems solvable by a family of polynomial size branching programs, with the additional condition that each bit of the input is tested at most d times.<br />
<br />
BP<sub>d</sub>(P) strictly contains BP<sub>d-1</sub>(P), for every d &gt; 1 [[zooref#tha98|[Tha98]]].<br />
<br />
Contained in [[Complexity Zoo:P#pbp|PBP]].<br />
<br />
See also: [[Complexity Zoo:P#pobdd|P-OBDD]], [[Complexity Zoo:P#kpbp|k-PBP]].<br />
----<br />
<br />
===== <span id="bpe" style="color:red">BPE</span>: Bounded-Error Probabilistic [[Complexity Zoo:E#e|E]] =====<br />
Has the same relation to [[Complexity Zoo:E#e|E]] as [[#bpp|BPP]] does to [[Complexity Zoo:P#p|P]].<br />
<br />
[[Complexity Zoo:E#ee|EE]] = BPE if and only if [[Complexity Zoo:E#exp|EXP]] = [[#bpp|BPP]] [[zooref#ikw01|[IKW01]]].<br />
<br />
----<br />
===== <span id="bpee" style="color:red">BPEE</span>: Bounded-Error Probabilistic [[Complexity Zoo:E#ee|EE]] =====<br />
Has the same relation to [[Complexity Zoo:E#ee|EE]] as [[#bpp|BPP]] does to [[Complexity Zoo:P#p|P]].<br />
<br />
----<br />
===== <span id="bphspace" style="color:red">BP<sub>H</sub>SPACE(f(n))</span>: Bounded-Error Halting Probabilistic f(n)-Space =====<br />
The class of decision problems solvable in O(f(n))-space with error probability at most 1/3, by a Turing machine that halts on every input <i>and</i> every random tape setting.<br />
<br />
Contains [[Complexity Zoo:R#rhspace|R<sub>H</sub>SPACE]](f(n)).<br />
<br />
Is contained in [[Complexity Zoo:D#dspace|DSPACE]](f(n)<sup>3/2</sup>) [[zooref#sz95|[SZ95]]].<br />
<br />
----<br />
===== <span id="bpl" style="color:red">BPL</span>: Bounded-Error Probabilistic [[Complexity Zoo:L#l|L]] =====<br />
Has the same relation to [[Complexity Zoo:L#l|L]] as [[#bpp|BPP]] does to [[Complexity Zoo:P#p|P]]. The Turing machine has to halt for every input and every randomness.<br />
<br />
Contained in [[Complexity Zoo:S#sc|SC]] [[zooref#nis92|[Nis92]]] and in [[Complexity Zoo:P#pl|PL]].<br />
<br />
----<br />
<br />
===== <span id="bpnp" style="color:red">BP&#149;NP</span>: Probabilistic [[Complexity Zoo:N#np|NP]] =====<br />
Equals [[Complexity Zoo:A#am|AM]].<br />
<br />
----<br />
<br />
===== <span id="bpp" style="color:red">BPP</span>: Bounded-Error Probabilistic Polynomial-Time =====<br />
The class of decision problems solvable by an [[Complexity Zoo:N#np|NP]] machine such that<br />
<ol><br />
<li>If the answer is 'yes' then at least 2/3 of the computation paths accept.</li><br />
<li>If the answer is 'no' then at most 1/3 of the computation paths accept.</li><br />
</ol><br />
(Here all computation paths have the same length.)<br />
<br />
Often identified as the class of feasible problems for a computer with access to a genuine random-number source.<br />
<br />
Defined in [[zooref#gil77|[Gil77]]].<br />
<br />
Contained in [[Complexity Zoo:S#sigma2p|&#931;<sub>2</sub>P]] &#8745; [[Complexity Zoo:P#pi2p|&#928;<sub>2</sub>P]] [[zooref#lau83|[Lau83]]], and indeed in [[Complexity Zoo:Z#zpp|ZPP]]<sup>[[Complexity Zoo:N#np|NP]]</sup> [[zooref#gz97|[GZ97]]].<br />
<br />
If BPP contains [[Complexity Zoo:N#np|NP]], then [[Complexity Zoo:R#rp|RP]] = [[Complexity Zoo:N#np|NP]] [[zooref#ko82|[Ko82,]][[zooref#gil77|Gil77]]] and [[Complexity Zoo:P#ph|PH]] is contained in BPP [[zooref#zac88|[Zac88]]].<br />
<br />
If any problem in [[Complexity Zoo:E#e|E]] requires circuits of size 2<sup>&#937;(n)</sup>, then BPP = [[Complexity Zoo:P#p|P]] [[zooref#iw97|[IW97]]] (in other words, BPP can be derandomized).<br />
<br />
Indeed, <i>any</i> proof that BPP = [[Complexity Zoo:P#p|P]] requires showing either that [[Complexity Zoo:N#nexp|NEXP]] is not in [[Complexity Zoo:P#ppoly|P/poly]], or else that [[Complexity Zoo:Symbols#sharpp|#P]] requires superpolynomial-size arithmetic circuits [[zooref#ki02|[KI02]]].<br />
<br />
BPP is not known to contain complete problems. [[zooref#sip82|[Sip82]]], [[zooref#hh86|[HH86]]] give oracles relative to which BPP has no complete problems.<br />
<br />
There exist oracles relative to which [[Complexity Zoo:P#p|P]] = [[Complexity Zoo:R#rp|RP]] but still [[Complexity Zoo:P#p|P]] is not equal to BPP [[zooref#bf99|[BF99]]].<br />
<br />
In contrast to the case of [[Complexity Zoo:P#p|P]], it is unknown whether BPP collapses to [[#bptime|BPTIME]](n<sup>c</sup>) for some fixed constant c. However, [[zooref#bar02|[Bar02]]] and [[zooref#fs04|[FS04]]] have shown hierarchy theorems for BPP with a small amount of advice.<br />
<br />
A [[Zoo Glossary#zeroone-law|zero-one law]] exists stating that BPP has [[Zoo Glossary#p-measure|p-measure]] zero unless BPP = {{zcls|e|exp}} {{zcite|Mel00}}.<br />
<br />
Equals [[Complexity Zoo:A#almostp|Almost-P]].<br />
<br />
See also: [[#bpppath|BPP<sub>path</sub>]].<br />
<br />
----<br />
<br />
===== <span id="bppcc" style="color:red">BPP<sup>cc</sup></span>: Communication Complexity [[#bpp|BPP]] =====<br />
The analogue of [[Complexity Zoo:P#pcc|P<sup>cc</sup>]] for bounded-error probabilistic communication complexity.<br />
<br />
Does not equal [[Complexity Zoo:P#pcc|P<sup>cc</sup>]], and is not contained in [[Complexity Zoo:N#npcc|NP<sup>cc</sup>]], because of the [[Complexity Garden#equality|EQUALITY]] problem.<br />
<br />
If the classes are defined in terms of partial functions, then BPP<sup>cc</sup><br />
<ul><br />
<li>is not contained in [[Complexity Zoo:P#pnpcc|P<sup>NPcc</sup>]] [[zooref#pps14|[PPS14]]];<br />
<li>does not contain [[Complexity Zoo:N#npcc|NP<sup>cc</sup>]] &#8745; [[Complexity Zoo:N#npcc|coNP<sup>cc</sup>]] [[zooref#kla03|[Kla03]]].<br />
</ul><br />
<br />
----<br />
<br />
===== <span id="bppkcc" style="color:red">BPP<sub><math>k</math></sub><sup>cc</sup></span>: [[#bppcc|BPP<sup>cc</sup>]] in NOF model, <math>k</math> players =====<br />
<br />
Has the same relation to [[#bppcc|BPP<sup>cc</sup>]] and [[#bpp|BPP]] as {{zcls|p|pkcc|P<sub><math>k</math></sub><sup>cc</sup>}} does to {{zcls|p|pcc|P<sup>cc</sup>}} and {{zcls|p|p}}.<br />
<br />
{{zcls|n|npkcc|NP<sub><math>k</math></sub><sup>cc</sup>}} is not contained in {{zcls|b|bppkcc|BPP<sub><math>k</math></sub><sup>cc</sup>}} for <math>k\le(1-\delta)\cdot\log n</math> players, for any constant <math>\delta>0</math> {{zcite|DP08}}.<br />
----<br />
<br />
===== <span id="bppkt" style="color:red">BPP<sup>KT</sup></span>: [[#bpp|BPP]] With Time-Bounded Kolmogorov Complexity Oracle =====<br />
[[#bpp|BPP]] with an oracle that, given a string x, returns the minimum over all programs P that output x<sub>i</sub> on input i, of the length of P plus the maximum time taken by P on any input.<br />
<br />
A similar class was defined in [[zooref#abk02|[ABK+02]]], where it was also shown that in BPP<sup>KT</sup> one can [[Complexity_Garden#integer_factorization|factor]], compute [[Complexity_Garden#discrete_logarithm|discrete logarithms]], and more generally invert any one-way function on a non-negligible fraction of inputs.<br />
<br />
See also: [[Complexity Zoo:P#pk|P<sup>K</sup>]].<br />
<br />
----<br />
<br />
===== <span id="bpplog" style="color:red">BPP/log</span>: [[#bpp|BPP]] With Logarithmic Karp-Lipton Advice =====<br />
The class of problems solvable by a semantic [[#bpp|BPP]] machine with O(log n) advice bits that depend only on the input length n. If the advice is good, the output must be correct with probability at least 2/3. If it is bad, the machine must provide some answer with probability at least 2/3. See the discussion for [[#bqppoly|BQP/poly]].<br />
<br />
Contained in [[#bppmlog|BPP/mlog]].<br />
<br />
----<br />
===== <span id="bppmlog" style="color:red">BPP/mlog</span>: [[#bpp|BPP]] With Logarithmic Deterministic Merlin-Like Advice =====<br />
The class of problems solvable by a syntactic [[#bpp|BPP]] machine with O(log n) advice bits that depend only on the input length n. If the advice is good, the output must be correct with probability at least 2/3. If it is bad, it need not be.<br />
<br />
Contained in [[#bpprlog|BPP/rlog]].<br />
<br />
----<br />
===== <span id="bppsslog" style="color:red">BPP//log</span>: [[#bpp|BPP]] With Logarithmic Randomness-Dependent Advice =====<br />
The class of problems solvable by a [[#bpp|BPP]] machine that is given O(log n) advice bits, which can depend on both the machine's random coin flips and the input length n, but not on the input itself.<br />
<br />
Defined in [[zooref#tv02|[TV02]]], where it was also shown that if [[Complexity Zoo:E#exp|EXP]] is in BPP//log then<br />
[[Complexity Zoo:E#exp|EXP]] = [[#bpp|BPP]], and if [[Complexity Zoo:P#pspace|PSPACE]] is in BPP//log then [[Complexity Zoo:P#pspace|PSPACE]] = [[#bpp|BPP]].<br />
<br />
----<br />
===== <span id="bpprlog" style="color:red">BPP/rlog</span>: [[#bpp|BPP]] With Logarithmic Deterministic Merlin-Like Advice =====<br />
The class of problems solvable by a syntactic [[#bpp|BPP]] machine with O(log n) random advice bits whose probability distribution depends only on the input length n. For each n, there exists good advice such that the output is correct with probability at least 2/3.<br />
<br />
Contains [[#bppmlog|BPP/mlog]]. The inclusion is strict, because BPP/rlog contains any finitely sparse language by fingerprinting; see the discussion for [[Complexity Zoo:A#all|ALL]].<br />
<br />
Contained in [[#bppsslog|BPP//log]].<br />
<br />
----<br />
===== <span id="bppobdd" style="color:red">BPP-OBDD</span>: Polynomial-Size Bounded-Error Ordered Binary Decision Diagram =====<br />
Same as [[Complexity Zoo:P#pobdd|P-OBDD]], except that probabilistic transitions are allowed and the OBDD need only accept with probability at least 2/3.<br />
<br />
Does not contain the integer multiplication problem [[zooref#ak96|[AK96]]].<br />
<br />
Strictly contained in [[#bqpobdd|BQP-OBDD]] [[zooref#nhk00|[NHK00]]].<br />
<br />
----<br />
===== <span id="bpppath" style="color:red">BPP<sub>path</sub></span>: Threshold [[#bpp|BPP]] =====<br />
Same as [[#bpp|BPP]], except that now the computation paths need not all have the same length.<br />
<br />
Defined in [[zooref#hht97|[HHT97]]], where the following was also shown:<br />
<ul><br />
<li>BPP<sub>path</sub> contains [[Complexity Zoo:M#ma|MA]] and [[Complexity Zoo:P#pnplog|P<sup>NP[log]</sup>]], and is contained in [[Complexity Zoo:P#pp|PP]] and [[#bpp|BPP]]<sup>[[Complexity Zoo:N#np|NP]]</sup>.</li><br />
<li>BPP<sub>path</sub> is closed under complementation, intersection, and union.</li><br />
<li>If BPP<sub>path</sub> = BPP<sub>path</sub><sup>BPP<sub>path</sub></sup>, then [[Complexity Zoo:P#ph|PH]] collapses to BPP<sub>path</sub>.</li><br />
<li>If BPP<sub>path</sub> contains [[Complexity Zoo:S#sigma2p|&#931;<sub>2</sub>P]], then [[Complexity Zoo:P#ph|PH]] collapses to [[#bpp|BPP]]<sup>[[Complexity Zoo:N#np|NP]]</sup>.</li><br />
</ul><br />
There exists an oracle relative to which BPP<sub>path</sub> is not contained in [[Complexity Zoo:S#sigma2p|&#931;<sub>2</sub>P]] [[zooref#bgm02|[BGM02]]].<br />
<br />
An alternate characterization of BPP<sub>path</sub> uses the idea of post-selection. That is, BPP<sub>path</sub> is the class of languages <math>L</math> for which there exists a pair of polynomial-time Turing machines <math>A</math> and <math>B</math> such that the following conditions hold for all <math>x</math>:<br />
* If <math>x\in L</math>, <math>\Pr_{r\in\{0,1\}^{\mathrm{poly}(\left\vert x\right\vert)}}\left[A(x,r)\mid B(x,r)\right]\ge \frac23</math>.<br />
* If <math>x\notin L</math>, <math>\Pr_{r\in\{0,1\}^{\mathrm{poly}(\left\vert x\right\vert)}}\left[A(x,r)\mid B(x,r)\right]< \frac13</math>.<br />
* <math>\Pr_{r\in\{0,1\}^{\mathrm{poly}(n)}}\left[B(x,r)\right]>0</math>.<br />
We say that <math>B</math> is the post-selector. Intuitively, this characterization allows a BPP machine to require that its random bits have some special but easily verifiable property. This characterization makes the inclusion NP &sube; BPP<sub>path</sub> nearly trivial.<br />
<br />
''See Also'': {{zcls|p|postbqp|PostBQP}} (quantum analogue).<br />
<br />
----<br />
<br />
===== <span id="bpqp" style="color:red">BPQP</span>: Bounded-Error Probabilistic [[Complexity Zoo:Q#qp|QP]] =====<br />
Equals [[#bptime|BPTIME]](2<sup>O((log n)^k)</sup>); that is, the class of problems solvable in quasipolynomial-time on a bounded-error machine.<br />
<br />
Defined in [[zooref#cns99|[CNS99]]], where the following was also shown:<br />
<ul><br />
If either (1) [[Complexity Zoo:Symbols#sharpp|#P]] does not have a subexponential-time bounded-error algorithm, or (2) [[Complexity Zoo:E#exp|EXP]] does not have subexponential-size circuits, then the BPQP hierarchy is strict -- that is, for all a &lt; b at least 1, [[#bptime|BPTIME]](2<sup>(log n)^a</sup>) is strictly contained in [[#bptime|BPTIME]](2<sup>(log n)^b</sup>).<br />
</ul><br />
<br />
----<br />
===== <span id="bpspace" style="color:red">BPSPACE(f(n))</span>: Bounded-Error Probabilistic f(n)-Space =====<br />
The class of decision problems solvable in O(f(n))-space with error probability at most 1/3, by a Turing machine that halts with probability 1 on every input.<br />
<br />
Contains [[Complexity Zoo:R#rspace|RSPACE(f(n))]] and [[#bphspace|BP<sub>H</sub>SPACE(f(n))]].<br />
<br />
----<br />
===== <span id="bptime" style="color:red">BPTIME(f(n))</span>: Bounded-Error Probabilistic f(n)-Time =====<br />
Same as [[#bpp|BPP]], but with f(n)-time (for some constructible function f) rather than polynomial-time machines.<br />
<br />
Defined in [[zooref#gil77|[Gil77]]].<br />
<br />
BPTIME(n<sup>log n</sup>) does not equal BPTIME(2<sup>n^&epsilon;</sup>) for any &epsilon;>0 [[zooref#kv88|[KV88]]]. Proving a stronger time hierarchy theorem for BPTIME is a longstanding open problem; see [[zooref#bh97|[BH97]]] for details.<br />
<br />
[[zooref#bar02|[Bar02]]] has shown the following:<br />
<ul><br />
<li>If we allow a small number of advice bits (say log n), then there is a strict hierarchy: for every d at least 1, BPTIME(n<sup>d</sup>)/(log n) does not equal BPTIME(n<sup>d+1</sup>)/(log n).</li><br />
<li>In the uniform setting, if [[#bpp|BPP]] has complete problems then BPTIME(n<sup>d</sup>) does not equal BPTIME(n<sup>d+1</sup>).</li><br />
<li>BPTIME(n) does not equal [[Complexity Zoo:N#np|NP]].</li><br />
</ul><br />
Subsequently, [[zooref#fs04|[FS04]]] managed to reduce the number of advice bits to only 1: BPTIME(n<sup>d</sup>)/1 does not equal BPTIME(n<sup>d+1</sup>)/1. They also proved a hierarchy theorem for [[#heurbptime|HeurBPTIME]].<br />
<br />
For another bounded-error hierarchy result, see [[#bpqp|BPQP]].<br />
<br />
----<br />
===== <span id="bqnc" style="color:red">BQNC</span>: Alternate Name for [[Complexity Zoo:Q#qnc|QNC]] =====<br />
<br />
----<br />
===== <span id="bqnp" style="color:red">BQNP</span>: Alternate Name for [[Complexity Zoo:Q#qma|QMA]] =====<br />
<br />
----<br />
===== <span id="bqp" style="color:red">BQP</span>: Bounded-Error Quantum Polynomial-Time =====<br />
The class of decision problems solvable in polynomial time by a quantum Turing machine, with at most 1/3 probability of error.<br />
<br />
One can equivalently define BQP as the class of decision problems solvable by a uniform family of polynomial-size quantum circuits, with at most 1/3 probability of error [[zooref#yao93|[Yao93]]]. Any universal gate set can be used as a basis; however, a technicality is that the transition amplitudes must be efficiently computable, since otherwise one could use them to encode the solutions to hard problems (see [[zooref#adh97|[ADH97]]]).<br />
<br />
BQP is often identified as the class of feasible problems for quantum computers.<br />
<br />
Contains the [[Complexity_Garden#integer_factorization|factoring]] and [[Complexity_Garden#discrete_logarithm|discrete logarithm]] problems [[zooref#sho97|[Sho97]]], the hidden Legendre symbol problem [[zooref#dhi02|[DHI02]]], the Pell's equation and principal ideal problems [[zooref#hal02|[Hal02]]], and some other problems not thought to be in [[#bpp|BPP]].<br />
<br />
Defined in [[zooref#bv97|[BV97]]], where it is also shown that BQP contains [[#bpp|BPP]] and is contained in [[Complexity Zoo:P#p|P]] with a [[Complexity Zoo:Symbols#sharpp|#P]] oracle.<br />
<br />
BQP<sup>BQP</sup> = BQP [[zooref#bv97|[BV97]]].<br />
<br />
[[zooref#adh97|[ADH97]]] showed that BQP is contained in [[Complexity Zoo:P#pp|PP]], and [[zooref#fr98|[FR98]]] showed that BQP is contained in [[Complexity Zoo:A#awpp|AWPP]].<br />
<br />
There exist oracles relative to which:<br />
<ul><br />
<li>BQP does not equal to [[#bpp|BPP]] [[zooref#bv97|[BV97]]] (and by similar arguments, is not in [[Complexity Zoo:P#ppoly|P/poly]]).</li><br />
<li>BQP is not contained in [[Complexity Zoo:M#ma|MA]] [[zooref#wat00|[Wat00]]].</li><br />
<li>BQP is not contained in [[Complexity Zoo:M#modkp|Mod<sub>p</sub>P]] for prime p [[zooref#gv02|[GV02]]].</li><br />
<li>[[Complexity Zoo:N#np|NP]], and indeed [[Complexity Zoo:N#npiconp|NP &#8745; coNP]], are not contained in BQP with probability 1 relative to a random oracle and a random permutation oracle, respectively [[zooref#bbb97|[BBB+97]]].</li><br />
<li>[[Complexity Zoo:S#szk|SZK]] is not contained in BQP [[zooref#aar02|[Aar02]]].</li><br />
<li>BQP is not contained in [[Complexity Zoo:S#szk|SZK]] (follows easily using the quantum walk problem in [[zooref#ccd03|[CCD+03]]]).</li><br />
<li>[[Complexity Zoo:P#ppad|PPAD]] is not contained in BQP [[zooref#li11|[Li11]]].</li><br />
</ul><br />
<br />
----<br />
<br />
===== <span id="bqplog" style="color:red">BQP/log</span>: [[#bqp|BQP]] With Logarithmic-Size Karp-Lipton Advice =====<br />
Same as [[#bqppoly|BQP/poly]] except that the advice is O(log n) bits instead of a polynomial number.<br />
<br />
Contained in [[#bqpmlog|BQP/mlog]].<br />
<br />
----<br />
===== <span id="bqppoly" style="color:red">BQP/poly</span>: [[#bqp|BQP]] With Polynomial-Size Karp-Lipton Advice =====<br />
Is to [[#bqpmpoly|BQP/mpoly]] as [[#existsbpp|&#8707;BPP]] is to [[Complexity Zoo:M#ma|MA]]. Namely, the [[#bqp|BQP]] machine is required to give some answer with probability at least 2/3 even if the advice is bad. Even though [[#bqpmpoly|BQP/mpoly]] is a more natural class, BQP/poly follows the standard definition of advice as a class operator [[zooref#kl82|[KL82]]].<br />
<br />
Contained in [[#bqpmpoly|BQP/mpoly]] and contains [[#bqplog|BQP/log]].<br />
<br />
----<br />
===== <span id="bqpmlog" style="color:red">BQP/mlog</span>: [[#bqp|BQP]] With Logarithmic-Size Deterministic Merlin-Like Advice =====<br />
Same as [[#bqpmpoly|BQP/mpoly]] except that the advice is O(log n) bits instead of a polynomial number.<br />
<br />
Strictly contained in [[#bqpqlog|BQP/qlog]] [[zooref#ny03|[NY03]]].<br />
<br />
----<br />
===== <span id="bqpmpoly" style="color:red">BQP/mpoly</span>: [[#bqp|BQP]] With Polynomial-Size Deterministic Merlin-Like Advice =====<br />
The class of languages recognized by a syntactic BQP machine with deterministic polynomial advice that depends only on the input length, such that the output is correct with probability 2/3 when the advice is good.<br />
<br />
Can also be defined as the class of problems solvable by a nonuniform family of polynomial-size quantum circuits, just as [[#ppoly|P/poly]] is the class solvable by a nonuniform family of polynomial-size classical circuits.<br />
<br />
Referred to with a variety of other ad hoc names, including [[#bqppoly|BQP/poly]] on occassion.<br />
<br />
Contains [[#bqpqlog|BQP/qlog]], and is contained in [[#bqpqpoly|BQP/qpoly]].<br />
<br />
Does not contain [[Complexity Zoo:E#espace|ESPACE]] [[zooref#ny03|[NY03]]].<br />
<br />
----<br />
===== <span id="bqpqlog" style="color:red">BQP/qlog</span>: [[#bqp|BQP]] With Logarithmic-Size Quantum Advice =====<br />
Same as [[#bqpmlog|BQP/mlog]] except that the advice is quantum instead of classical.<br />
<br />
Strictly contains [[#bqpmlog|BQP/mlog]] [[zooref#ny03|[NY03]]].<br />
<br />
Contained in [[#bqpmpoly|BQP/mpoly]].<br />
<br />
----<br />
===== <span id="bqpqpoly" style="color:red">BQP/qpoly</span>: [[#bqp|BQP]] With Polynomial-Size Quantum Advice =====<br />
The class of problems solvable by a [[#bqp|BQP]] machine that receives a quantum state &psi;<sub>n</sub> as advice, which depends only on the input length n.<br />
<br />
As with [[#bqpmpoly|BQP/mpoly]], the acceptance probability does not need to be bounded away from 1/2 if the machine is given bad advice. (Thus, we are discussing the class that [[zooref#ny03|[NY03]]] call BQP/*Qpoly.) Indeed, such a condition would make ''quantum'' advice unusable, by a continuity argument.<br />
<br />
Does not contain [[Complexity Zoo:E#eespace|EESPACE]] [[zooref#ny03|[NY03]]].<br />
<br />
[[zooref#aar04b|[Aar04b]]] shows the following:<br />
<ul><br />
<li>There exists an oracle relative to which BQP/qpoly does not contain [[Complexity Zoo:N#np|NP]].</li><br />
<li>BQP/qpoly is contained in [[Complexity Zoo:P#pppoly|PP/poly]].</li><br />
</ul><br />
A ''classical'' oracle separation between BQP/qpoly and [[#bqpmpoly|BQP/mpoly]] is presently unknown, but there is a ''quantum'' oracle separation [[zooref#ak06|[AK06]]]. An unrelativized separation is too much to hope for, since it would imply that [[Complexity Zoo:P#pp|PP]] is not contained in [[Complexity Zoo:P#ppoly|P/poly]].<br />
<br />
Contains [[#bqpmpoly|BQP/mpoly]].<br />
<br />
----<br />
<br />
===== <span id="bqpobdd" style="color:red">BQP-OBDD</span>: Polynomial-Size Bounded-Error Quantum Ordered Binary Decision Diagram =====<br />
Same as [[Complexity Zoo:P#pobdd|P-OBDD]], except that unitary (quantum) transitions are allowed and the OBDD need only accept with probability at least 2/3.<br />
<br />
Strictly contains [[#bppobdd|BPP-OBDD]] [[zooref#nhk00|[NHK00]]].<br />
<br />
----<br />
===== <span id="bqpspace" style="color:red">BQPSPACE</span>: Bounded-Error Quantum [[Complexity Zoo:P#pspace|PSPACE]] =====<br />
Equals [[Complexity Zoo:P#pspace|PSPACE]] and [[Complexity Zoo:P#ppspace|PPSPACE]].<br />
<br />
----<br />
===== <span id="bqtime" style="color:red">BQTIME(f(n))</span>: Bounded-Error Quantum f(n)-Time =====<br />
Same as [[#bqp|BQP]], but with f(n)-time (for some constructible function f) rather than polynomial-time machines.<br />
<br />
Defined in [[zooref#bv97|[BV97]]].<br />
<br />
----<br />
===== <span id="bqpctc" style="color: red">BQP<sub>CTC</sub></span>: [[#bqp|BQP]] With Closed Time Curves =====<br />
Same as [[#bqp|BQP]] with access to two sets of qubits: causality-respecting qubits and CTC qubits.<br />
<br />
Defined in [[zooref#aar05c|[Aar05c]]], where it was shown that [[Complexity Zoo:P#pspace|PSPACE]] is contained in BQP<sub>CTC</sub>, which in turn is contained in [[Complexity Zoo:S#sqg|SQG]] = PSPACE. Therefore, BQP<sub>CTC</sub> = PSPACE; this was also shown in [[zooref#aw09|[AW09]]].<br />
<br />
See also [[Complexity Zoo:P#pctc|P<sub>CTC</sub>]].<br />
<br />
----<br />
<br />
===== <span id="bqpttpoly" style="color:red">BQP<sub>tt</sub>/poly</span>: [[#bqpmpoly|BQP/mpoly]] With Truth-Table Queries =====<br />
Same as [[#bqpmpoly|BQP/mpoly]], except that the machine only gets to make <i>nonadaptive</i> queries to whatever oracle it might have.<br />
<br />
Defined in [[zooref#ny03b|[NY03b]]], where it was also shown that [[Complexity Zoo:P#p|P]] is not contained in BQP<sub>tt</sub>/poly relative to an oracle.<br />
<br />
----<br />
===== <span id="bwbp" style="color:red">k-BWBP</span>: Bounded-Width Branching Program =====<br />
Alternate name for k-[[Complexity Zoo:P#kpbp|PBP]].</div>LegionMammal978