https://complexityzoo.net/api.php?action=feedcontributions&user=Vprusso&feedformat=atomComplexity Zoo - User contributions [en]2024-03-29T08:57:49ZUser contributionsMediaWiki 1.35.0https://complexityzoo.net/index.php?title=Complexity_Zoo&diff=6612Complexity Zoo2019-07-03T17:59:05Z<p>Vprusso: Updated Vincent Russo website.</p>
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{{CZ-Categories}}</div>Vprussohttps://complexityzoo.net/index.php?title=Complexity_Zoo:Q&diff=6344Complexity Zoo:Q2015-11-03T20:49:42Z<p>Vprusso: /* QMAM: Quantum Merlin-Arthur-Merlin Public-Coin Interactive Proofs */</p>
<hr />
<div>__NOTOC__<br />
{{CZ-Letter-Menu|Q}}<br />
<br />
<br />
===== <span id="q" style="color:red">Q</span>: Quasi-Realtime Languages =====<br />
The class of problems solvable by a nondeterministic multitape Turing machine in linear time. Defined in [[zooref#bg69|[BG69]]], where it was shown that Q equals the class of problems solvable by a nondeterministic multitape Turing machine in exactly n steps (as opposed to O(n) steps).<br />
<br />
Contains [[Complexity Zoo:G#gcsl|GCSL]].<br />
<br />
----<br />
===== <span id="qac0" style="color:red">QAC<sup>0</sup></span>: Quantum [[Complexity Zoo:A#ac0|AC<sup>0</sup>]] =====<br />
The class of decision problems solvable by a family of constant-depth, polynomial-size quantum circuits. Here each layer of the circuit is a tensor product of one-qubit gates and Toffoli gates, or is a tensor product of controlled-NOT gates.<br />
<br />
A uniformity condition may also be imposed.<br />
<br />
Defined in [[zooref#moo99|[Moo99]]].<br />
<br />
It is contained in [[#qacwf0|QAC<sub>f</sub><sup>0</sup>]], but it is not known if it contains [[Complexity Zoo:A#ac0|AC<sup>0</sup>]]. Notice that the latter containment is not obvious, since the set of gates in QAC<sup>0</sup> do not allow to implement fanout in any way we may take for granted.<br />
<br />
<br />
----<br />
<br />
===== <span id="qac0m" style="color:red">QAC<sup>0</sup>[m]</span>: Quantum [[Complexity Zoo:A#ac0m|AC<sup>0</sup>[m]]] =====<br />
Same as [[#qac0|QAC<sup>0</sup>]], except that now Mod-m gates are also allowed. A Mod-m gate computes whether the sum of a given set of bits is congruent to 0 modulo m, and exclusive-OR's the answer into another bit.<br />
<br />
Defined in [[zooref#moo99|[Moo99]]].<br />
<br />
----<br />
===== <span id="qacc0" style="color:red">QACC<sup>0</sup></span>: Quantum [[Complexity Zoo:A#acc0|ACC<sup>0</sup>]] =====<br />
Same as [[#qac0m|QAC<sup>0</sup>[m]]], except that Mod-m gates are allowed for any m.<br />
<br />
Defined in [[zooref#moo99|[Moo99]]].<br />
<br />
[[zooref#ghp00|[GHP00]]] showed that QACC<sup>0</sup> equals [[#qac0m|QAC<sup>0</sup>[p]]] for any prime p.<br />
<br />
----<br />
===== <span id="qacwf0" style="color:red">QAC<sub>f</sub><sup>0</sup></span>: [[#qac0|QAC<sup>0</sup>]] With Fanout =====<br />
Same as [[#qac0|QAC<sup>0</sup>]], except that an additional "quantum fanout" gate is available, which CNOT's a qubit into arbitrarily many target qubits in a single step.<br />
<br />
Defined in [[zooref#moo99|[Moo99]]], where it was also shown that QAC<sub>f</sub><sup>0</sup> =<br />
[[#qac0m|QAC<sup>0</sup>]][2] = [[#qacc0|QACC<sup>0</sup>]].<br />
<br />
----<br />
===== <span id="qam" style="color:red">QAM</span>: Quantum [[Complexity Zoo:A#am|AM]] =====<br />
The class of decision problems for which a "yes" answer can be verified by a public-coin quantum [[Complexity Zoo:A#am|AM]] protocol, as follows. Arthur generates a uniformly random (classical) string and sends it to Merlin. Merlin responds with a polynomial-size quantum certificate, on which Arthur can perform any [[Complexity Zoo:B#bqp|BQP]] operation. The completeness and soundness requirements are the same as for [[Complexity Zoo:A#am|AM]].<br />
<br />
Defined by Marriott and Watrous [[zooref#mw05|[MW05]]].<br />
<br />
Contains [[#qma|QMA]] and is contained in [[#qip2|QIP[2]]] and [[Zoo Operators#bp|BP&#149;]][[Complexity Zoo:P#pp|PP]] (and therefore [[Complexity Zoo:P#pspace|PSPACE]]).<br />
<br />
----<br />
<br />
===== <span id="qcfl" style="color:red">QCFL</span>: Quantum [[Complexity Zoo:C#cfl|CFL]] =====<br />
The class of decision problems recognized by quantum context-free languages, which are defined in [[zooref#mc00|[MC00]]]. The authors also showed that QCFL does not equal [[Complexity Zoo:C#cfl|CFL]].<br />
<br />
----<br />
===== <span id="qcma" style="color:red">QCMA</span>: Quantum Classical [[Complexity Zoo:C#ma|MA]] =====<br />
The class of decision problems for which a "yes" answer can be verified by a <i>quantum</i> computer with access to a <i>classical</i> proof. Also known as the subclass of of QMA with classical witnesses.<br />
<br />
Called '''MQA''' by Watrous [[zooref#wat09|[Wat09]]]. <br />
<br />
Contains [[Complexity Zoo:M#ma|MA]], and is contained in [[Complexity Zoo:Q#qma|QMA]].<br />
<br />
Given a black-box group G and a subgroup H, the problem of testing non-membership in H has polynomial QCMA query complexity [[zooref#ak06|[AK06]]].<br />
<br />
See [[zooref#ak06|[AK06]]] for a "quantum oracle separation" between QCMA and [[#qma|QMA]]. No classical oracle separation between QCMA and [[#qma|QMA]] is currently known.<br />
<br />
----<br />
<br />
===== <span id="qh" style="color:red">QH</span>: Query Hierarchy Over [[Complexity Zoo:N#np|NP]] =====<br />
QH<sub>k</sub> is defined to be P<sup>NP[k]</sup>; that is, [[Complexity Zoo:P#p|P]] with k queries to an [[Complexity Zoo:N#np|NP]] oracle (where k is a constant). Then QH is the union of QH<sub>k</sub> over all nonnegative k.<br />
<br />
QH = [[Complexity Zoo:B#bh|BH]] [[zooref#wag88|[Wag88]]]; thus, either both hierarchies are infinite or both collapse to some finite level.<br />
<br />
----<br />
<br />
===== <span id="qip" style="color:red">QIP</span>: Quantum [[Complexity Zoo:I#ip|IP]] =====<br />
The class of decision problems such that a "yes" answer can be verified by a <i>quantum interactive proof</i>. Here the verifier is a [[Complexity Zoo:B#bqp|BQP]] (i.e. quantum polynomial-time) algorithm, while the prover has unbounded computational resources (though cannot violate the linearity of quantum mechanics). The prover and verifier exchange a polynomial number of messages, which can be quantum states. Thus, the verifier's and prover's states may become entangled during the course of the protocol. Given the verifier's algorithm, we require that<br />
<ol><br />
<li>If the answer is "yes," then the prover can behave in such a way that the verifier accepts with probability at least 2/3.</li><br />
<li>If the answer is "no," then however the prover behaves, the verifier rejects with probability at least 2/3.</li><br />
</ol><br />
Let QIP[k] be QIP where the prover and verifier are restricted to exchanging k messages (with the prover going last).<br />
<br />
Defined in [[zooref#wat99|[Wat99]]], where it was also shown that [[Complexity Zoo:P#pspace|PSPACE]] is in QIP[3].<br />
<br />
Subsequently [[zooref#kw00|[KW00]]] showed that for all k&gt;3, QIP[k] = QIP[3] = QIP.<br />
<br />
QIP is contained in [[Complexity Zoo:E#exp|EXP]] [[zooref#kw00|[KW00]]].<br />
<br />
QIP = IP = PSPACE [[zooref#JJUW09|[JJUW09]]]; thus quantum computing adds no power to single-prover interactive proofs.<br />
<br />
QIP(1) is more commonly known as [[#qma|QMA]].<br />
<br />
See also: [[#qip2|QIP[2]]], [[#qszk|QSZK]].<br />
<br />
----<br />
<br />
===== <span id="qip2" style="color:red">QIP[2]</span>: 2-Message Quantum [[Complexity Zoo:I#ip|IP]] =====<br />
See [[#qip|QIP]] for definition.<br />
<br />
Contains [[#qszk|QSZK]] [[zooref#wat02|[Wat02]]].<br />
<br />
--------<br />
<br />
===== <span id="ql" style="color:red">QL</span>: Quasi-Linear =====<br />
The class of problems that can be decided in quasi-linear time by a multitape deterministic Turing machine. Quasi-linear time here means n(log n)<sup>k</sup> + k, for some k.<br />
<br />
Defined in [[zooref#sch78|[Sch78]]].<br />
<br />
See also: [[#q|Q]], [[Complexity Zoo:N#nql|NQL]].<br />
<br />
--------<br />
<br />
===== <span id="qma" style="color:red">QMA</span>: Quantum [[Complexity Zoo:M#ma|MA]] =====<br />
The class of decision problems such that a "yes" answer can be verified by a 1-message quantum interactive proof. That is, a [[Complexity Zoo:B#bqp|BQP]] (i.e. quantum polynomial-time) verifier is given a quantum state (the "proof"). We require that<br />
<ol><br />
<li>If the answer is "yes," then there exists a state such that verifier accepts with probability at least 2/3.</li><br />
<li>If the answer is "no," then for all states the verifier rejects with probability at least 2/3.</li><br />
</ol><br />
QMA = [[#qip|QIP]](1).<br />
<br />
Defined in [[zooref#wat00|[Wat00]]], where it is also shown that [[Complexity Garden#group-nonmembership|''group non-membership'']] is in QMA.<br />
<br />
Based on this, [[zooref#wat00|[Wat00]]] gives an oracle relative to which [[Complexity Zoo:M#ma|MA]] is strictly contained in QMA.<br />
<br />
Kitaev and Watrous (unpublished) showed QMA is contained in [[Complexity Zoo:P#pp|PP]] (see [[zooref#mw05|[MW05]]] for a proof). Combining that result with [[zooref#ver92|[Ver92]]], one can obtain an oracle relative to which [[Complexity Zoo:A#am|AM]] is not in QMA.<br />
<br />
Kitaev ([[zooref#ksv02|[KSV02]]], see also [[zooref#an02|[AN02]]]) showed that the [[Complexity Garden#k-local-ham|5-Local Hamiltonians]] Problem is QMA-complete. Subsequently, Kempe and Regev [[zooref#kr03|[KR03]]] showed that even 3-Local Hamiltonians is QMA-complete. A subsequent paper by Kempe, Kitaev, and Regev [[zooref#kkr04|[KKR04]]], has hit rock bottom (assuming [[#p|P]] does not equal QMA), by showing '''<font color="red">2</font>'''-local Hamiltonians QMA-complete.<br />
<br />
Compare to [[Complexity Zoo:N#nqp|NQP]].<br />
<br />
If QMA = [[Complexity Zoo:P#pp|PP]] then [[Complexity Zoo:P#pp|PP]] contains [[Complexity Zoo:P#ph|PH]] [[zooref#vya03|[Vya03]]]. This result uses the fact that QMA is contained in [[Complexity Zoo:A#a0pp|A<sub>0</sub>PP]].<br />
<br />
Approximating the ground state energy of a system composed of a line of quantum particles is QMA-complete [[zooref#agk07|[AGK07]]].<br />
<br />
See also: [[#qcma|QCMA]], [[#qmaqpoly|QMA/qpoly]], [[#qszk|QSZK]], [[#qma2|QMA(2)]], [[#qma-plus|QMA-plus]].<br />
<br />
----<br />
<br />
===== <span id="qma-plus" style="color:red">QMA-plus</span>: [[#qma|QMA]] With Super-Verifier =====<br />
Same as [[#qma|QMA]], except now the verifier can directly obtain the <i>probability</i> that a given observable of the certificate state, if measured, would equal 1. (In the usual model, by contrast, one can only sample an observable.)<br />
<br />
Defined in [[zooref#ar03|[AR03]]], where it was also shown that QMA-plus = [[#qma|QMA]].<br />
<br />
----<br />
===== <span id="qma2" style="color:red">QMA(2)</span>: Quantum [[#ma|MA]] With Multiple Certificates =====<br />
Same as [[#qma|QMA]], except that now the verifier is given <i>two</i> polynomial-size quantum certificates, which are guaranteed to be unentangled.<br />
<br />
Defined in [[zooref#kmy01|[KMY01]]]. <br />
<br />
It was shown in {{zcite|ABD+08|abd08}} that a conjecture they call the ''Strong Amplification Conjecture'' implies that QMA(2) is contained in {{zcls|p|pspace}}. The authors also show that there exists no ''perfectly disentangler'' that can be used to simulate QMA(2) in QMA, though other approaches to showing QMA = QMA(2) may still exist.<br />
<br />
It was shown in {{zcite|HM13|hm13}} that QMA(k) = QMA(2) for k >= 2. However we still do not know if QMA(2) = [[#qma|QMA]] and we also do not know any upper bound for QMA(2) better than {{zcls|n|nexp}}. <br />
----<br />
<br />
===== <span id="qma1" style="color:red">QMA<sub>1</sub></span>: One Sided [[#qma|QMA]] =====<br />
Same as [[#qma|QMA]] except that for a "yes" instance, there exists a state that is accepted with probability 1.<br />
<br />
Defined in [[zooref#bra06|[Bra06]]]. It was shown there that [[Complexity Garden#qksat|Quantum k-SAT]] is QMA<sub>1</sub>-complete for any <math> k \geq 4</math>. It was also shown there that Quantum 2-SAT is in [[Complexity Zoo:P#p|P]]. <br />
<br />
This result was later improved in [[zooref#gn13|[GN13]]] where it was shown that Quantum 3-SAT is QMA<sub>1</sub>-complete. <br />
<br />
<br />
----<br />
<br />
===== <span id="qmalog" style="color:red">QMA<sub>log</sub></span>: [[#qma|QMA]] With Logarithmic-Size Proofs =====<br />
Same as [[#qma|QMA]] except that the quantum proof has O(log n) qubits instead of a polynomial number.<br />
<br />
Equals [[#bqp|BQP]] [[zooref#mw05|[MW05]]].<br />
<br />
----<br />
===== <span id="qmam" style="color:red">QMAM</span>: Quantum Merlin-Arthur-Merlin Public-Coin Interactive Proofs =====<br />
The class of decision problems for which a "yes" answer can be verified by a public-coin quantum MAM protocol, as follows. Merlin sends a polynomial-size quantum state to Arthur. Arthur then flips some classical coins (in fact, he only has to flip <i>one</i> without loss of generality) and sends the outcome to Merlin. At this stage Arthur is not yet allowed to perform any quantum operations. Merlin then sends Arthur another quantum state. Finally, Arthur performs a [[Complexity Zoo:B#bqp|BQP]] operation on both of the states simultaneously, and either accepts or rejects. The completeness and soundness requirements are the same as for [[Complexity Zoo:A#am|AM]]. Also, Merlin's messages might be entangled.<br />
<br />
Defined by Marriott and Watrous [[zooref#mw05|[MW05]]], who also showed that QMAM = [[#qip|QIP]](3) = [[#qip|QIP]].<br />
<br />
Hence QMAM equals [[Complexity Zoo:P#pspace|PSPACE]].<br />
<br />
----<br />
<br />
===== <span id="qmaqpoly" style="color:red">QMA/qpoly</span>: [[#qma|QMA]] With Polynomial-Size Quantum Advice =====<br />
Is contained in [[Complexity Zoo:P#pspacepoly|PSPACE/poly]] [[zooref#aar06b|[Aar06b]]].<br />
<br />
----<br />
===== <span id="qmip" style="color:red">QMIP</span>: Quantum Multi-Prover Interactive Proofs =====<br />
The quantum generalization of [[Complexity Zoo:M#mip|MIP]], and the multi-prover generalization of [[#qip|QIP]].<br />
<br />
A quantum multi-prover interactive proof system is the same as a classical one, except that all messages and verifier computations are quantum. As in [[Complexity Zoo:M#mip|MIP]], there is no communication among the provers; however, the provers share unlimited prior entanglement. The number of provers and number of rounds can both be polynomial in n.<br />
<br />
Defined in [[zooref#km02|[KM02]]].<br />
<br />
Shown to be equal to [[Complexity Zoo:M#mipstar|MIP*]] in [[zooref#ruv12|[RUV12]]].<br />
<br />
QMIP contains [[Complexity Zoo:N#nexp|NEXP]] simply because [[Complexity Zoo:M#mipstar|MIP*]] contains [[Complexity Zoo:N#nexp|NEXP]] [[zooref#iv12|[IV12]]]. Since we know that [[Complexity Zoo:N#nexp|NEXP]] = [[#qmipne|QMIP<sub>ne</sub>]], this tells us that giving the provers unlimited prior entanglement does not make the class less powerful.<br />
<br />
----<br />
<br />
===== <span id="qmiple" style="color:red">QMIP<sub>le</sub></span>: Quantum Multi-Prover Interactive Proofs With Limited Prior Entanglement =====<br />
Same as [[#qmip|QMIP]], except that now the provers share only a polynomial number of EPR pairs, instead of an unlimited number.<br />
<br />
Defined in [[zooref#km02|[KM02]]], where it was also shown that QMIP<sub>le</sub> is contained in [[Complexity Zoo:N#nexp|NEXP]] = [[#qmipne|QMIP<sub>ne</sub>]].<br />
<br />
----<br />
===== <span id="qmipne" style="color:red">QMIP<sub>ne</sub></span>: Quantum Multi-Prover Interactive Proofs With No Prior Entanglement =====<br />
Same as [[#qmip|QMIP]], except that now the provers have no prior entanglement.<br />
<br />
Defined in [[zooref#km02|[KM02]]], where it was also shown that QMIP<sub>ne</sub> = [[Complexity Zoo:N#nexp|NEXP]]. Thus, QMIP<sub>ne</sub> contains [[#qmiple|QMIP<sub>le</sub>]].<br />
<br />
----<br />
===== <span id="qnc" style="color:red">QNC</span>: Quantum [[Complexity Zoo:N#nc|NC]] =====<br />
The class of decision problems solvable by polylogarithmic-depth quantum circuits with bounded probability of error. (A uniformity condition may also be imposed.)<br />
<br />
Has the same relation to [[Complexity Zoo:N#nc|NC]] as [[Complexity Zoo:B#bqp|BQP]] does to [[Complexity Zoo:P#p|P]].<br />
<br />
[[zooref#cw00|[CW00]]] showed that [[Complexity_Garden#integer_factorization|factoring]] is in [[Complexity Zoo:Z#zpp|ZPP]] with a QNC oracle.<br />
<br />
Is incomparable with [[Complexity Zoo:B#bpp|BPP]] as far as anyone knows.<br />
<br />
See also: [[Complexity Zoo:R#rnc|RNC]].<br />
<br />
----<br />
<br />
===== <span id="qnc0" style="color:red">QNC<sup>0</sup></span>: Quantum [[Complexity Zoo:N#nc0|NC<sup>0</sup>]] =====<br />
Constant-depth quantum circuits without fanout gates.<br />
<br />
Defined in [[zooref#spa02|[Spa02]]].<br />
<br />
Contained in [[#qncf0|QNC<sub>f</sub><sup>0</sup>]].<br />
<br />
----<br />
===== <span id="qncf0" style="color:red">QNC<sub>f</sub><sup>0</sup></span>: Quantum [[Complexity Zoo:N#nc0|NC<sup>0</sup>]] With Unbounded Fanout =====<br />
Constant-depth quantum circuits with unbounded-fanout gates.<br />
<br />
Defined in [[zooref#spa02|[Spa02]]].<br />
<br />
Contains [[#qnc0|QNC<sup>0</sup>]], and is contained in [[#qacc0|QACC<sup>0</sup>]].<br />
<br />
----<br />
===== <span id="qnc1" style="color:red">QNC<sup>1</sup></span>: Quantum [[Complexity Zoo:N#nc1|NC<sup>1</sup>]] =====<br />
Same as [[#qnc|QNC]]<sup>1</sup>, but for the exact rather than bounded-error case.<br />
<br />
In contrast to [[Complexity Zoo:N#nc1|NC<sup>1</sup>]], it is not clear how to simulate QNC<sup>1</sup> on a quantum computer in which one qubit is initialized to a pure state, and the remaining qubits are in the maximally mixed state [[zooref#asv00|[ASV00]]].<br />
<br />
See also [[zooref#mn02|[MN02]]].<br />
<br />
----<br />
===== <span id="qp" style="color:red">QP</span>: Quasipolynomial-Time =====<br />
Equals [[Complexity Zoo:D#dtime|DTIME]](2<sup>polylog(n)</sup>).<br />
<br />
----<br />
===== <span id="qplin" style="color:red">QPLIN</span>: Linear Quasipolynomial-Time =====<br />
Equals [[Complexity Zoo:D#dtime|DTIME]](n<sup>O(log n)</sup>).<br />
<br />
Has the same relationship to [[#qp|QP]] that [[Complexity Zoo:E#e|E]] does to [[Complexity Zoo:E#exp|EXP]].<br />
<br />
----<br />
===== <span id="qpspace" style="color:red">QPSPACE</span>: Quasipolynomial-Space =====<br />
Equals [[Complexity Zoo:D#dspace|DSPACE]](2<sup>polylog(n)</sup>).<br />
<br />
According to [[zooref#bg94|[BG94]]], Beigel and Feigenbaum and (independently) Krawczyk showed that QPSPACE is not contained in [[Complexity Zoo:C#check|Check]].<br />
<br />
----<br />
===== <span id="qrg" style="color:red">QRG</span>: Quantum Refereed Games =====<br />
Same as [[Complexity Zoo:R#rg|RG]], except that now the verifier is a [[Complexity Zoo:B#bqp|BQP]] machine, and can exchange polynomially many quantum messages with the competing provers.<br />
<br />
The two provers are computationally unbounded, but must obey the laws of quantum mechanics. Also, we can assume without loss of generality that the provers share no entanglement, because adversaries gain no advantage by sharing information.<br />
<br />
Defined in [[zooref#gut05|[Gut05]]], where it was also shown that QRG is contained in [[Complexity Zoo:N#nexp|NEXP]] &#8745; [[Complexity Zoo:C#conexp|coNEXP]].<br />
<br />
QRG trivially contains [[Complexity Zoo:R#rg|RG]] (and hence [[Complexity Zoo:E#exp|EXP]]), as well as [[Complexity Zoo:S#sqg|SQG]].<br />
<br />
QRG is contained in [[Complexity Zoo:E#exp|EXP]] [[zooref#gw07|[GW07]]]. Hence QRG = [[Complexity Zoo:R#rg|RG]] = [[Complexity Zoo:E#exp|EXP]] and finding optimal strategies for zero-sum quantum games is no harder than finding optimal strategies for zero-sum classical games.<br />
<br />
----<br />
===== <span id="qrgk" style="color:red">QRG(<i>k</i>)</span>: <i>k</i>-turn Quantum Refereed Games =====<br />
Same as [[Complexity Zoo:R#qrg|QRG]], except that now the verifier exchanges exactly <i>k</i> messages with each prover where <i>k</i> is a polynomial-bounded function of the input length. Messages are exchanged in parallel. QRG(<i>k</i>) is the quantum version of [[Complexity Zoo:R#rgk|RG(<i>k</i>)]]. By definition, QRG(poly) = [[Complexity Zoo:Q#qrg|QRG]]. See also [[Complexity Zoo:Q#qrg1|QRG(1)]] and [[Complexity Zoo:Q#qrg2|QRG(2)]].<br />
<br />
QRG(<i>k</i>) trivially contains [[Complexity Zoo:R#rgk|RG(k)]] for each <i>k</i> (and hence [[Complexity Zoo:P#pspace|PSPACE]] when <math> k \geq 2</math>). QRG(4) trivially contains [[Complexity Zoo:S#sqg|SQG]].<br />
<br />
QRG(<i>k</i>) is trivially contained in [[Complexity Zoo:Q#qrg|QRG]] for each <i>k</i> (and hence [[Complexity Zoo:E#exp|EXP]]).<br />
<br />
Other than these trivial bounds, very little is known of QRG(<i>k</i>) for intermediate values of <i>k</i>. For example, does QRG(<i>k</i>) = [[Complexity Zoo:R#rgk|RG(<i>k</i>)]] for each <i>k</i>?<br />
<br />
----<br />
===== <span id="qrg2" style="color:red">QRG(2)</span>: Two-turn (one-round) Quantum Refereed Games =====<br />
Same as [[Complexity Zoo:Q#qrg|QRG]], except that now the verifier can exchange only two messages with each prover. Messages are exchanged in parallel, so the verifier cannot process the answer from one prover before preparing the question for the other. QRG(2) is the quantum version of [[Complexity Zoo:R#rg2|RG(2)]]. See also [[Complexity Zoo:Q#qrgk|QRG(<i>k</i>)]].<br />
<br />
QRG(2) trivially contains [[Complexity Zoo:R#rg2|RG(2)]] (and hence [[Complexity Zoo:P#pspace|PSPACE]]).<br />
<br />
QRG(2) is trivially contained in [[Complexity Zoo:S#sqg|SQG]] (and hence [[Complexity Zoo:P#pspace|PSPACE]]). Hence QRG(2) = [[Complexity Zoo:R#rg2|RG(2)]] = [[Complexity Zoo:P#pspace|PSPACE]] and finding optimal strategies for two-turn zero-sum quantum games is no harder than finding optimal strategies for two-turn zero-sum classical games.<br />
<br />
----<br />
===== <span id="qrg1" style="color:red">QRG(1)</span>: One-turn Quantum Refereed Games =====<br />
The class of problems for which there exists a [[Complexity Zoo:B#bqp|BQP]] machine M such that:<br />
<ul><br />
<li>If the answer is "yes," then there exists a quantum state &rho; such that for all quantum states &sigma;, M(&rho;,&sigma;) accepts with probability at least 2/3.</li><br />
<li>If the answer is "no," then there exists a &sigma; such that for all &rho;, M(&rho;,&sigma;) rejects with probability at least 2/3.</li><br />
</ul><br />
In other words, it's the same as [[Complexity Zoo:Q#qrgk|QRG(<i>k</i>)]] for <math>k=1</math>, the class of problems that admit quantum interactive proofs with competing provers in which there's no communication from the verifier back to the provers. QRG(1) is the quantum version of [[Complexity Zoo:R#rg1|RG(1)]].<br />
<br />
Defined in [[zooref#jw09|[JW09]]], where it was shown that QRG(1) is contained in [[Complexity Zoo:S#pspace|PSPACE]] .<br />
<br />
QRG(1) trivially contains [[#qma|QMA]] (and indeed [[Complexity Zoo:P#p|P]]<sup>[[#qma|QMA]]</sup>).<br />
<br />
QRG(1) is trivially contained in [[Complexity Zoo:Q#qrg2|QRG(2)]] (and hence [[Complexity Zoo:P#pspace|PSPACE]]).<br />
<br />
----<br />
<br />
===== <span id="qszk" style="color:red">QSZK</span>: Quantum Statistical Zero-Knowledge =====<br />
A quantum analog of [[Complexity Zoo:S#szk|SZK]] (or more precisely [[Complexity Zoo:H#hvszk|HVSZK]]).<br />
<br />
Arthur is a [[Complexity Zoo:B#bqp|BQP]] (i.e. quantum) verifier who can exchange quantum messages with Merlin. So Arthur and Merlin's states may become entangled during the course of the protocol.<br />
<br />
Arthur's "view" of his interaction with Merlin is taken to be the sequence of mixed states he has, over all steps of the protocol. The zero-knowledge requirement is that each of these states must have trace distance at most (say) 1/10 from a state that Arthur could prepare himself (in [[#bqp|BQP]]), without help from Merlin. Arthur is assumed to be an honest verifier.<br />
<br />
Defined in [[zooref#wat02|[Wat02]]], where the following was also shown:<br />
<ul><br />
<li>QSZK is contained in [[Complexity Zoo:P#pspace|PSPACE]].</li><br />
<li>QSZK is closed under complement.</li><br />
<li>Any protocol can be parallelized to consist of two messages, so that QSZK is in [[#qip2|QIP[2]]].</li><br />
<li>One can assume without loss of generality that protocols are public-coin, as for [[Complexity Zoo:S#szk|SZK]].</li><br />
<li>QSZK has a natural complete promise problem, called <i>Quantum State Distinguishability</i> (QSD). We are given quantum circuits Q<sub>0</sub> and Q<sub>1</sub>. Let &#961;<sub>0</sub> and &#961;<sub>1</sub> be the mixed states they produce respectively, when run on the all-0 state (and when non-output qubits are traced out). We are promised that the trace distance between &#961;<sub>0</sub> and &#961;<sub>1</sub> is either at most &#945; or at least &#946;, where &#945; and &#946; are constants in [0,1] satisfying &#945; &lt; &#946;<sup>2</sup>. The problem is to decide which of these is the case.</li><br />
</ul></div>Vprussohttps://complexityzoo.net/index.php?title=Complexity_Zoo&diff=6260Complexity Zoo2013-08-02T18:52:31Z<p>Vprusso: </p>
<hr />
<div>__NOTOC__<br />
<br />
==Introduction==<br />
<br />
Welcome to the '''Complexity Zoo'''... There are now 495 classes and counting!<br />
[[Image:zoo.gif|thumb|right|200px|what's your problem?]]<br />
<br />
{{CZ-Menu-Content}}<br />
<br />
This information was originally moved from http://www.complexityzoo.com/ in August 2005, and is currently under the watchful eyes of its original creators:<br />
<br />
'''Zookeeper''': [http://www.scottaaronson.com/ Scott Aaronson]<br><br />
'''Co-Zookeeper''': Charles Fu<br><br />
'''Veterinarian''': [http://www.math.ucdavis.edu/~greg/ Greg Kuperberg]<br><br />
'''Tour Guide''': [https://sites.google.com/site/cgranade/ Christopher Granade]<br />
<br />
In 2012, this content was moved again to the University of Waterloo <br />
<br />
'''Zoo Conservationist''': [https://cs.uwaterloo.ca/~vrusso/ Vincent Russo]<br><br />
<br />
Errors? Omissions? Misattributions? Your favorite class not here? Then please contribute to the zoo as you see fit by [[Special:UserLogin | signing up]] and clicking on the edit links. Please include references, or better yet links to papers if available.<br />
<br />
To create a new class, click on the edit link of the class before or after the one that you want to add and copy the format of that class. (The classes are alphabetized by their tag names.) Then add the class to the table of contents and increment the total number of classes. After this, you can use the side edit links to edit the individual sections. For more on using the wiki language, see our [[Help:Contents | simple wiki help page]].<br />
<br />
If you would like to contribute but feel unable to make the updates yourself, email the zookeeper at scott at scottaaronson.com.<br />
<br />
==See Also==<br />
<br />
''Introductory Resources''<br />
* [[Zoo Intro|Introductory Essay]]: New visitors may want to stop here and see what the Zoo is all about.<br />
* [[Petting Zoo]]: A more gentle version of the Zoo with fewer classes, meant for new initiates in complexity. (If you're looking for where the Most Important Classes went, look in the Petting Zoo.)<br />
<br />
<br />
''Other Collections and Resources''<br />
* [[Complexity Garden]]: Problems of interest in complexity theory and some notes about important inclusions.<br />
* [[Zoo Exhibit|Special Exhibit]]: A collection of classes of quantum states and probability distributions.<br />
* [http://www.math.ucdavis.edu/~greg/zoology/intro.html Complexity Zoology]: A computer-assisted survey maintained by the [http://www.math.ucdavis.edu/~greg/ Greg Kuperberg], including [http://www.math.ucdavis.edu/~greg/zoology/diagram.xml active] and [http://www.math.ucdavis.edu/~greg/zoology/diagram.pdf static] inclusion diagrams.<br />
* [http://satoshihada.wordpress.com/complexity-zoo-for-ipad/ Complexity Zoo for iPad (and iPhone)]: An iOS viewer for Complexity Zoo.<br />
<br />
<br />
''Appendices''<br />
*[[Zoo Glossary|Glossary]]: Definitions of some complexity theoretic terms.<br />
*[[Zoo References|References]]: Bibliography for the Zoo.<br />
*[[Zoo Pronunciation|Pronunciation Guide]]: A resource for those who insist on communicating verbally about complexity.<br />
*[[Zoo Conventions|Conventions and Notation]]: Common notational conventions used here at the Zoo.<br />
*[[Zoo Operators|Operators]]: A (very short) list of operators which act upon classes.<br />
*[[Zoo Acknowledgments|Acknowledgments]]: Where the Zookeeper and friends acknowledge those who have helped out with the Zoo.<br />
*[[Meta:Complexity Zoo Contributor's Guide|Complexity Zoo Contributor's Guide]]: A guide on how to get started helping out with the Zoo.<br />
<br />
<br />
''NB:'' Longtime Zoo watchers may recall Chris Bourke's LaTeX version of the Zoo and Chad Brewbaker's graphical inclusion diagram. These references are obsolete until further notice.<br />
<br />
<!-- Moved Most Important Classes to Petting Zoo --><br />
<br />
== All Classes ==<br />
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{{CZ-Categories}}</div>Vprussohttps://complexityzoo.net/index.php?title=Complexity_Zoo&diff=6259Complexity Zoo2013-08-02T18:52:11Z<p>Vprusso: </p>
<hr />
<div>__NOTOC__<br />
<br />
==Introduction==<br />
<br />
Welcome to the '''Complexity Zoo'''... There are now 495 classes and counting!<br />
[[Image:zoo.gif|thumb|right|200px|what's your problem?]]<br />
<br />
{{CZ-Menu-Content}}<br />
<br />
This information was originally moved from http://www.complexityzoo.com/ in August 2005, and is currently under the watchful eyes of its original creators:<br />
<br />
'''Zookeeper''': [http://www.scottaaronson.com/ Scott Aaronson]<br><br />
'''Co-Zookeeper''': Charles Fu<br><br />
'''Veterinarian''': [http://www.math.ucdavis.edu/~greg/ Greg Kuperberg]<br><br />
'''Tour Guide''': [https://sites.google.com/site/cgranade/ Christopher Granade]<br />
<br />
In 2012, this content was moved again to the University of Waterloo <br />
<br />
'''Zoo Conservationist'''[https://cs.uwaterloo.ca/~vrusso/ Vincent Russo]<br><br />
<br />
Errors? Omissions? Misattributions? Your favorite class not here? Then please contribute to the zoo as you see fit by [[Special:UserLogin | signing up]] and clicking on the edit links. Please include references, or better yet links to papers if available.<br />
<br />
To create a new class, click on the edit link of the class before or after the one that you want to add and copy the format of that class. (The classes are alphabetized by their tag names.) Then add the class to the table of contents and increment the total number of classes. After this, you can use the side edit links to edit the individual sections. For more on using the wiki language, see our [[Help:Contents | simple wiki help page]].<br />
<br />
If you would like to contribute but feel unable to make the updates yourself, email the zookeeper at scott at scottaaronson.com.<br />
<br />
==See Also==<br />
<br />
''Introductory Resources''<br />
* [[Zoo Intro|Introductory Essay]]: New visitors may want to stop here and see what the Zoo is all about.<br />
* [[Petting Zoo]]: A more gentle version of the Zoo with fewer classes, meant for new initiates in complexity. (If you're looking for where the Most Important Classes went, look in the Petting Zoo.)<br />
<br />
<br />
''Other Collections and Resources''<br />
* [[Complexity Garden]]: Problems of interest in complexity theory and some notes about important inclusions.<br />
* [[Zoo Exhibit|Special Exhibit]]: A collection of classes of quantum states and probability distributions.<br />
* [http://www.math.ucdavis.edu/~greg/zoology/intro.html Complexity Zoology]: A computer-assisted survey maintained by the [http://www.math.ucdavis.edu/~greg/ Greg Kuperberg], including [http://www.math.ucdavis.edu/~greg/zoology/diagram.xml active] and [http://www.math.ucdavis.edu/~greg/zoology/diagram.pdf static] inclusion diagrams.<br />
* [http://satoshihada.wordpress.com/complexity-zoo-for-ipad/ Complexity Zoo for iPad (and iPhone)]: An iOS viewer for Complexity Zoo.<br />
<br />
<br />
''Appendices''<br />
*[[Zoo Glossary|Glossary]]: Definitions of some complexity theoretic terms.<br />
*[[Zoo References|References]]: Bibliography for the Zoo.<br />
*[[Zoo Pronunciation|Pronunciation Guide]]: A resource for those who insist on communicating verbally about complexity.<br />
*[[Zoo Conventions|Conventions and Notation]]: Common notational conventions used here at the Zoo.<br />
*[[Zoo Operators|Operators]]: A (very short) list of operators which act upon classes.<br />
*[[Zoo Acknowledgments|Acknowledgments]]: Where the Zookeeper and friends acknowledge those who have helped out with the Zoo.<br />
*[[Meta:Complexity Zoo Contributor's Guide|Complexity Zoo Contributor's Guide]]: A guide on how to get started helping out with the Zoo.<br />
<br />
<br />
''NB:'' Longtime Zoo watchers may recall Chris Bourke's LaTeX version of the Zoo and Chad Brewbaker's graphical inclusion diagram. These references are obsolete until further notice.<br />
<br />
<!-- Moved Most Important Classes to Petting Zoo --><br />
<br />
== All Classes ==<br />
{{CZ-Menu-Content}}<br />
<br />
{{CZ-Letter-Section|Symbols}}<br />
{{CZ-Letter-Section|A}}<br />
{{CZ-Letter-Section|B}}<br />
{{CZ-Letter-Section|C}}<br />
{{CZ-Letter-Section|D}}<br />
{{CZ-Letter-Section|E}}<br />
{{CZ-Letter-Section|F}}<br />
{{CZ-Letter-Section|G}}<br />
{{CZ-Letter-Section|H}}<br />
{{CZ-Letter-Section|I}}<br />
<!--{{CZ-Letter-Section|J}}<br />
{{CZ-Letter-Section|K}}--><br />
{{CZ-Letter-Section|L}}<br />
{{CZ-Letter-Section|M}}<br />
{{CZ-Letter-Section|N}}<br />
{{CZ-Letter-Section|O}}<br />
{{CZ-Letter-Section|P}}<br />
{{CZ-Letter-Section|Q}}<br />
{{CZ-Letter-Section|R}}<br />
{{CZ-Letter-Section|S}}<br />
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{{CZ-Letter-Section|U}}<br />
{{CZ-Letter-Section|V}}<br />
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{{CZ-Letter-Section|X}}<br />
{{CZ-Letter-Section|Y}}<br />
{{CZ-Letter-Section|Z}}<br />
<br />
<br />
{{CZ-Categories}}</div>Vprussohttps://complexityzoo.net/index.php?title=Complexity_Zoo_References&diff=5345Complexity Zoo References2013-03-05T15:32:34Z<p>Vprusso: /* K */</p>
<hr />
<div>__NOTOC__<br />
<br />
{{Simple-Alpha-Menu|{{CZ-Navbar}}<br />
----<br />
}}<br />
<br />
<br />
<!-- don't delete blank lines above this.. they're there for spacing reasons --><br />
<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 />
arXiv:[http://arxiv.org/abs/quant-ph/0510230 quant-ph/0510230].<br />
<br />
<span id="ak06" style="color:maroon">[AK06]</span><br />
S. Aaronson and G. Kuperberg.<br />
Quantum versus classical proofs and advice,<br />
submitted, 2006.<br />
arXiv:[http://arxiv.org/abs/quant-ph/0604056 quant-ph/0604056].<br />
<br />
<span id="ab00" style="color:maroon">[AB00]</span><br />
E. Allender and D. A. M. Barrington.<br />
Uniform Circuits for Division: Consequences and Problems.<br />
J. Comput. System Sci. 65 (2002), no. 4, 695--716.<br />
ECCC [http://eccc.uni-trier.de/eccc-reports/2000/TR00-65/ TR00-65], 2000.<br />
<br />
{{Reference<br />
|id=abd08 |tag=ABD+08<br />
|authors=S. Aaronson, S. Beigi, A. Drucker, et al<br />
|title=The power of unentanglement<br />
|journal=Electronic Colloquium on Computational Complexity<br />
|srcdetail=ECCC Report TR08-051, accepted on May 02, 2008<br />
|link=[http://eccc.hpi-web.de/eccc-reports/2008/TR08-051/index.html http://eccc.hpi-web.de/eccc-reports/2008/TR08-051/index.html]<br />
}}<br />
<br />
<span id="abf94" style="color:maroon">[ABF+94]</span><br />
J. Aspnes, R. Beigel, M. L. Furst, and S. Rudich.<br />
The expressive power of voting polynomials,<br />
<i>Combinatorica</i> 14(2):135-148, 1994.<br />
[http://www.cs.yale.edu/~aspnes/stoc91voting.ps http://www.cs.yale.edu/~aspnes/stoc91voting.ps]<br />
<br />
<span id="abk02" style="color:maroon">[ABK+02]</span><br />
E. Allender, H. Buhrman, M. Kouck&yacute;, D. van Melkebeek, and D. Ronneburger.<br />
Power from random strings,<br />
<i>Proceedings of IEEE FOCS'2002</i>, pp. 669-678, 2002.<br />
ECCC [http://eccc.uni-trier.de/eccc-reports/2002/TR02-028/ TR02-028].<br />
<br />
<span id="abl98" style="color:maroon">[ABL98]</span><br />
A. Ambainis, D. M. Barrington, and H. L&ecirc;Thanh.<br />
On counting AC<sup>0</sup> circuits with negative constants,<br />
<i>Proceedings of MFCS (Mathematical Foundations of Computer Science)</i>, pp. 419-427, 1998.<br />
ECCC [http://eccc.uni-trier.de/eccc-reports/1998/TR98-020/ TR98-020].<br />
<br />
<span id="abo99" style="color:maroon">[ABO99]</span><br />
E. Allender, R. Beals, and M. Ogihara.<br />
The complexity of matrix rank and feasible systems of linear equations,<br />
<i>Computational Complexity</i> 8(2):99-126, 1999.<br />
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<br />
===== B =====<br />
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===== D =====<br />
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===== E =====<br />
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===== F =====<br />
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===== G =====<br />
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<br />
<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 />
<br />
<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="gra92" style="color:maroon">[Grä92]</span><br />
E. Grädel<br />
Capturing complexity classes b fragments of second order logic<br />
<i>Information and computaiton</i> 119 (1995), 129-135<br />
<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="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 />
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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 />
<br />
<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 />
<br />
<span id="gv02" style="color:maroon">[GV02]</span><br />
M. de Graaf and P. Valiant.<br />
Comparing EQP and MOD<sub>p^k</sub>P using polynomial degree lower bounds,<br />
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<br />
<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 />
arXiv:[http://arxiv.org/abs/cs.CC/0412102 cs.CC/0412102].<br />
<br />
<span id="gw07" style="color:maroon">[GW07]</span><br />
G. Gutoski and J. Watrous.<br />
Toward a general theory of quantum games,<br />
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arXiv:[http://arxiv.org/abs/quant-ph/0611234 quant-ph/0611234].<br />
<br />
<span id="gw10" style="color:maroon">[GW10]</span><br />
G. Gutoski and X. Wu.<br />
Short quantum games characterize PSPACE,<br />
2010.<br />
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<br />
<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 />
ECCC [http://eccc.uni-trier.de/eccc-reports/1995/TR95-049/ TR95-049].<br />
<br />
<span id="gz97" style="color:maroon">[GZ97]</span><br />
O. Goldreich and D. Zuckerman.<br />
Another proof that BPP subseteq PH (and more),<br />
ECCC [http://eccc.uni-trier.de/eccc-reports/1997/TR97-045/ TR97-045].<br />
<br />
===== H =====<br />
<br />
<span id="hal02" style="color:maroon">[Hal02]</span><br />
S. Hallgren.<br />
Polynomial-time quantum algorithms for Pell's equation and the principal ideal problem,<br />
<i>Proceedings of ACM STOC'2002</i>, 2002.<br />
[http://www.cs.caltech.edu/~hallgren/pell.pdf http://www.cs.caltech.edu/~hallgren/pell.pdf].<br />
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<span id="har78" style="color:maroon">[Har78]</span><br />
J. Hartmanis.<br />
<i>Feasible Computations and Provable Complexity Properties</i>,<br />
SIAM, 1978.<br />
<br />
<span id="har87" style="color:maroon">[Har87]</span><br />
J. Hartmanis.<br />
The collapsing hierarchies,<br />
<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 />
<br />
<span id="has87" style="color:maroon">[Has87]</span><br />
J. H&aring;stad.<br />
<i>Computational Limitations for Small-Depth Circuits</i>,<br />
MIT Press, 1987.<br />
<br />
<span id="has88" style="color:maroon">[Has88]</span><br />
J. H&aring;stad.<br />
Oneway permutations in NC<sup>0</sup>,<br />
<i>Information Processing Letters</i> 26:153-155, 1988.<br />
<br />
<span id="has90" style="color:maroon">[Has90]</span><br />
J. H&aring;stad. Tensor rank is NP-complete, ''J. Algorithms'', 11(4):644-654, 1990.<br />
<br />
<span id="has01" style="color:maroon">[Has01]</span><br />
J. H&aring;stad.<br />
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''Journal of the ACM'', 48(4):798-3859, 2001.<br />
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<br />
<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 />
[http://external.nj.nec.com/homepages/fortnow/beatcs/column47.ps http://external.nj.nec.com/homepages/fortnow/beatcs/column47.ps].<br />
<br />
<span id="hck88" style="color:maroon">[HCK+88]</span><br />
J. Hartmanis, R. Chang, J. Kadin, and S. G. Mitchell.<br />
Some observations about relativization of space bounded computations,<br />
<i>Bulletin of the EATCS</i> 35, June 1988.<br />
[http://external.nj.nec.com/homepages/fortnow/beatcs/column35.ps http://external.nj.nec.com/homepages/fortnow/beatcs/column35.ps].<br />
<br />
<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 />
<br />
<span id="hel84b" style="color:maroon">[Hel84b]</span><br />
H. Heller.<br />
On Relativized Polynomial and Exponential Computations,<br />
<i>SIAM Journal on Computing</i> 13(4):717-725, 1984.<br />
<br />
<span id="hel86" style="color:maroon">[Hel86]</span><br />
H. Heller.<br />
On Relativized Exponential and Probabilistic Complexity Classes,<br />
Inform. and Control 71 (1986), no. 3, 231--243<br />
<br />
<span id="hem89" style="color:maroon">[Hem89]</span><br />
L. Hemachandra.<br />
The strong exponential hierarchy collapses,<br />
<i>Journal of Computer and System Sciences</i> 39(3):299-322, 1989.<br />
<br />
<span id="her90" style="color:maroon">[Her90]</span><br />
U. Hertrampf.<br />
Relations among MOD-classes,<br />
<i>Theoretical Computer Science</i> 74:325-328, 1990.<br />
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<span id="her97" style="color:maroon">[Her97]</span><br />
U. Hertrampf.<br />
Acceptance by transformation monoids (with an application to local self-reductions),<br />
<i>Proceedings of IEEE Complexity'97</i>, pp. 213-224, 1997.<br />
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<span id="hes01" style="color:maroon">[Hes01]</span><br />
W. Hesse.<br />
Division is in uniform TC<sup>0</sup>,<br />
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[http://www.cs.umass.edu/~whesse/div.ps http://www.cs.umass.edu/~whesse/div.ps]<br />
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<span id="hh76" style="color:maroon">[HH76]</span><br />
J. Hartmanis and J. Hopcroft.<br />
Independence results in computer science,<br />
<i>ACM SIGACT News</i> 8(4):13-24, 1976.<br />
<br />
<span id="hh86" style="color:maroon">[HH86]</span><br />
J. Hartmanis and L. Hemachandra.<br />
Complexity classes without machines: on complete languages for UP,<br />
<i>Proceedings of ICALP'86</i>, Springer-Verlag Lecture Notes in Computer Science volume 226, pp. 123-135, 1986.<br />
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<span id="hhh98" style="color:maroon">[HHH98]</span><br />
E. Hemaspaandra, L. Hemaspaandra, and H. Hempel.<br />
What's up with downward collapse: using the easy-hard technique to link Boolean and polynomial hierarchy collapses,<br />
<i>SIGACT News</i> 29(3):10-22, 1998.<br />
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<span id="hhk05" style="color:maroon">[HHK+05]</span><br />
L. Hemaspaandra, C. Homan, S. Kosub, and K. Wagner.<br />
The complexity of computing the size of an interval,<br />
Technical Report TR-856, Department of Computer Science, University of<br />
Rochester, 2005. This is an expanded version of [[#hkw01|HKW01]]<br />
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<span id="hhn95" style="color:maroon">[HHN+95]</span><br />
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 />
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<span id="hhr97" style="color:maroon">[HHR97]</span><br />
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 />
<i>Proceedings of ICALP'97</i>, Springer-Verlag Lecture Notes in Computer Science, 1997.<br />
arXiv:[http://arxiv.org/abs/cs.CC/9907036 cs.CC/9907036].<br />
<br />
<span id="hht97" style="color:maroon">[HHT97]</span><br />
Y. Han, L. Hemaspaandra, and T. Thierauf.<br />
Threshold computation and cryptographic security,<br />
<i>SIAM Journal on Computing</i> 26(1):59-78, 1997.<br />
<br />
<span id="hi02" style="color:maroon">[HI02]</span><br />
W. Hesse and N. Immerman.<br />
Complete problems for dynamic complexity classes,<br />
<i>Proceedings of Logic in Computer Science (LICS)</i>, 2002.<br />
[http://www.cs.umass.edu/~immerman/pub/completeLICS.pdf http://www.cs.umass.edu/~immerman/pub/completeLICS.pdf]<br />
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<span id="hjv93" style="color:maroon">[HJV93]</span><br />
L. Hemaspaandra, R. Jain, and N. K. Vereshchagin.<br />
Banishing robust Turing completeness,<br />
<i>International Journal of Foundations of Computer Science</i>, 3-4:245-265, 1993.<br />
<br />
<span id="hkw01" style="color:maroon">[HKW01]</span><br />
L. Hemaspaandra, S. Kosub, and K. Wagner.<br />
The complexity of computing the size of an interval,<br />
<i>Proceedings of ICALP'01</i>, Springer-Verlag Lecture Notes in Computer Science, 2001.<br />
<br />
<span id="hls65" style="color:maroon">[HLS65]</span><br />
J. Hartmanis, P. L. Lewis II, and R. E. Stearns.<br />
Hierarchies of memory-limited computations,<br />
<i>Proceedings of the 6th Annual IEEE Symposium on Switching Circuit Theory and Logic Design</i>, pp. 179-190, 1965.<br />
<br />
<span id="hmp93" style="color:maroon">[HMP+93]</span><br />
A. Hajnal, W. Maass, P. Pudl&aacute;k, M. Szegedy, and G. Tur&aacute;n.<br />
Threshold circuits of bounded depth,<br />
<i>Journal of Computer and System Sciences</i> 46(2):129-154, 1993.<br />
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<span id="hn06" style="color:maroon">[HN06]</span><br />
D. Harnik and M. Naor.<br />
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[http://www.cs.technion.ac.il/~harnik/Compress.pdf http://www.cs.technion.ac.il/~harnik/Compress.pdf]<br />
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<span id="hno96" style="color:maroon">[HNO+96]</span><br />
L. Hemaspaandra, A. Naik, M. Ogihara, and A. Selman.<br />
Computing solutions uniquely collapses the polynomial hierarchy,<br />
<i>SIAM Journal on Computing</i> 25(4):697-708, 1996.<br />
ECCC [http://eccc.uni-trier.de/eccc-reports/1996/TR96-027/ TR96-027].<br />
<br />
<span id="ho02" style="color:maroon">[HO02]</span><br />
L. Hemaspaandra and M. Ogihara.<br />
<i>The Complexity Theory Companion</i>,<br />
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See also [http://www.cs.rochester.edu/u/lane/=companion/ http://www.cs.rochester.edu/u/lane/=companion/].<br />
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<span id="hpv77" style="color:maroon">[HPV77]</span><br />
J. Hopcroft, W. Paul, and L. Valiant.<br />
On time versus space,<br />
<i>Journal of the ACM</i> 24(2):332-337, 1977.<br />
<br />
<span id="hrv00" style="color:maroon">[HRV00]</span><br />
U. Hertrampf, S. Reith, and H. Vollmer.<br />
A note on closure properties of logspace MOD classes,<br />
<i>Information Processing Letters</i> 75(3):91-93, 2000.<br />
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<span id="hs65" style="color:maroon">[HS65]</span><br />
J. Hartmanis and R. E. Stearns.<br />
On the computational complexity of algorithms,<br />
<i>Transactions of the AMS</i> 117:285-305, 1965.<br />
<br />
<span id="hs92" style="color:maroon">[HS92]</span><br />
S. Homer and A. L. Selman.<br />
Oracles for structural properties: the isomorphism problem and public-key cryptography,<br />
<i>Journal of Computer and System Sciences</i> 44(2):287-301, 1992.<br />
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<span id="ht06" style="color:maroon">[HT06]</span><br />
Lauri Hella , José María Turull-Torres<br />
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<i>Theorical. Comput. Sci.</i> 355 (2006), 197--214.<br />
[http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6V1G-4J614M7-6&_user=1516330&_coverDate=04%2F11%2F2006&_rdoc=1&_fmt=high&_orig=search&_sort=d&_docanchor=&view=c&_searchStrId=1404146870&_rerunOrigin=google&_acct=C000053443&_version=1&_urlVersion=0&_userid=1516330&md5=6d794cde4e4a89dfa74f13967cdacb08].<br />
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<span id="hy84" style="color:maroon">[HY84]</span><br />
J. Hartmanis and Y. Yesha.<br />
Computation times of NP sets of different densities,<br />
<i>Theoretical Computer Science</i> 34:17-32, 1984.<br />
<br />
===== I =====<br />
<br />
<span id="iba72" style="color:maroon">[Iba72]</span><br />
O. Ibarra.<br />
A note concerning nondeterministic tape complexities,<br />
<i>Journal of the ACM</i> 4:608-612, 1972.<br />
<br />
<span id="ikw01" style="color:maroon">[IKW01]</span><br />
R. Impagliazzo, V. Kabanets, and A. Wigderson.<br />
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<span id="il90" style="color:maroon">[IL90]</span><br />
R. Impagliazzo and L. A. Levin.<br />
No better ways to generate hard NP instances than picking uniformly at random,<br />
<i>Proceedings of IEEE FOCS'90</i>, pp. 812-821, 1990.<br />
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{{Reference<br />
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}}<br />
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{{Reference<br />
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<br />
<span id="imm82" style="color:maroon">[Imm82]</span><br />
N. Immerman.<br />
Relational queries computable in in polynomial time.<br />
<i>14th ACM STOC Symp. (1987), 86-104</i><br />
<br />
<br />
<span id="imm83" style="color:maroon">[Imm83]</span><br />
N. Immerman.<br />
Languages That Capture Complexity Classes<br />
<i>15th ACM STOC Symp. (1983), 347-354</i><br />
[http://www.cs.umass.edu/~immerman/pub/capture.pdf]<br />
<br />
<span id="imm88" style="color:maroon">[Imm88]</span><br />
N. Immerman.<br />
Nondeterministic space is closed under complement,<br />
<i>SIAM Journal on Computing</i>, 17:935-938, 1988.<br />
<br />
<span id="imm89" style="color:maroon">[Imm89]</span><br />
N. Immerman.<br />
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<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>Vprusso