https://complexityzoo.net/api.php?action=feedcontributions&user=Soulsand&feedformat=atomComplexity Zoo - User contributions [en]2024-03-29T10:38:43ZUser contributionsMediaWiki 1.35.0https://complexityzoo.net/index.php?title=Complexity_Zoo:A&diff=6508Complexity Zoo:A2016-07-25T06:10:23Z<p>Soulsand: /* AW[t]: Alternating W[t] */ Addition of a result</p>
<hr />
<div>__NOTOC__<br />
{{CZ-Letter-Menu|A}}<br />
<br />
<br />
===== <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]] and to [[#Complexity_Zoo:A#AL|AL]] the alternating logtime hierarchy.<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 />
<br />
===== <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 />
----<br />
<br />
===== <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 />
----<br />
<br />
===== <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 />
----<br />
<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 />
----<br />
===== <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 />
----<br />
===== <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 />
----<br />
<br />
===== <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 />
----<br />
<br />
===== <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 />
----<br />
===== <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 />
----<br />
===== <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 />
----<br />
===== <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 />
----<br />
===== <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 />
----<br />
===== <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 />
----<br />
===== <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 />
----<br />
<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>Soulsandhttps://complexityzoo.net/index.php?title=Complexity_Zoo_References&diff=6507Complexity Zoo References2016-07-25T06:08:51Z<p>Soulsand: /* D */ addition of a reference</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="abjl2014" style="color:maroon">[ABFL2014]</span><br />
S. Aaronson, A. Bouland, J. Fitzsimons, M. Lee<br />
The space "just above" BQP<br />
arXiv:[http://arxiv.org/abs/1412.6507 arxiv.org/abs/1412.6507]<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 />
ECCC [http://eccc.uni-trier.de/eccc-reports/1996/TR96-024/ TR96-024],<br />
DIMACS [http://dimacs.rutgers.edu/TechnicalReports/abstracts/1997/97-40.html TR 97-40].<br />
<br />
<span id="abv95" style="color:maroon">[ABV95]</span><br />
W. Aiello, M. Bellare, and R. Venkatesan.<br />
Knowledge on the average - perfect, statistical, and logarithmic,<br />
<i>Proceedings of ACM STOC'95</i>, 1995.<br />
<br />
<span id="acg99" style="color:maroon">[ACG+99]</span><br />
G. Ausiello, P. Crescenzi, G. Gambosi, V. Kann, A. Marchetti-Spaccamela, and M. Protasi.<br />
<i>Complexity and Approximation: Combinatorial optimization problems and their approximability properties</i>,<br />
Springer-Verlag, 1999.<br />
See also "A compendium of NP optimization problems" (P. Crescenzi and V. Kann, eds.),<br />
[http://www.nada.kth.se/~viggo/wwwcompendium/ http://www.nada.kth.se/~viggo/wwwcompendium/].<br />
<br />
<span id="adh97" style="color:maroon">[ADH97]</span><br />
L. Adleman, J. DeMarrais, and M. Huang.<br />
Quantum computability,<br />
<i>SIAM Journal on Computing</i> 26:1524-1540, 1997.<br />
<br />
<span id="adl78" style="color:maroon">[Adl78]</span><br />
L. Adleman.<br />
Two theorems on random polynomial time.<br />
FOCS 78.<br />
<br />
<span id="afm01" style="color:maroon">[AFM01]</span><br />
L. Antu&ntilde;es, L. Fortnow, and D. van Melkebeek.<br />
Computational depth,<br />
<i>Proceedings of IEEE Complexity'01</i>, pp. 266-273, 2001.<br />
[http://people.cs.uchicago.edu/~fortnow/papers/depth.ps http://people.cs.uchicago.edu/~fortnow/papers/depth.ps]<br />
<br />
<span id="ag00" style="color:maroon">[AG00]</span><br />
C. Alvarez and R. Greenlaw.<br />
A compendium of problems complete for symmetric logarithmic space,<br />
<i>Journal of Computational Complexity</i> 9:73-95, 2000.<br />
ECCC [http://eccc.uni-trier.de/eccc-reports/1996/TR96-039/ TR96-039].<br />
<br />
<span id="ag04" style="color:maroon">[AG04]</span><br />
S. Aaronson and D. Gottesman.<br />
Improved Simulation of Stabilizer Circuits,<br />
<i>Phys. Rev. A</i> 70, 052328, 2004.<br />
[http://arxiv.org/abs/quant-ph/0406196 arXiv:quant-ph/0406196].<br />
<br />
<span id="agh90" style="color:maroon">[AGH90]</span><br />
W. Aiello, S. Goldwasser, and J. H&aring;stad.<br />
On The Power Of Interaction.<br />
Combinatorica 10 (1990), no. 1, 3--25.<br />
<br />
<span id="agk07" style="color:maroon">[AGK07]</span><br />
D. Aharonov, D. Gottesman, and J. Kempe;stad.<br />
The power of quantum systems on a line.<br />
FOCS 2007.<br />
<br />
{{Reference<br />
|tag=Agr01<br />
|authors=M. Agrawal<br />
|title=For completeness, sublogarithmic space is no space<br />
|journal=Information Processing Letters (82), 2001-2002<br />
|srcdetail=iss. 6, 321-325<br />
|link=http://www.cse.iitk.ac.in/~manindra/isomorphism/sublog-completeness.pdf<br />
}}<br />
<br />
{{Reference<br />
|id=Ajt83<br />
|tag=AJT83<br />
|authors=M. Ajtai<br />
|title=Σ-1-1-Formulae on finite structures<br />
|journal=Annals of Pure and Applied Logic (24), 1983<br />
|srcdetail=1-48<br />
}}<br />
<br />
<span id="ah87" style="color:maroon">[AH87]</span><br />
L. Adleman and M. Huang.<br />
Recognizing primes in random polynomial time,<br />
<i>Proceedings of ACM STOC'87</i>, pp. 462-470, 1987.<br />
<br />
<span id="ah87b" style="color:maroon">[AH87b]</span><br />
W. Aiello and J. H&aring;stad.<br />
Perfect zero-knowledge languages can be recognized in two rounds,<br />
<i>Proceedings of IEEE FOCS 1987</i>, pp. 439-448, 1987.<br />
<br />
<span id="aik04" style="color:maroon">[AIK04]</span><br />
B. Applebaum, Y. Ishai, and E. Kushilevitz.<br />
Cryptography in NC<sup>0</sup>,<br />
<i>Proceedings of IEEE FOCS 2004</i>.<br />
[http://www.cs.technion.ac.il/~yuvali/pubs/AIK04.ps http://www.cs.technion.ac.il/~yuvali/pubs/AIK04.ps].<br />
<br />
<span id="aj93" style="color:maroon">[AJ93]</span><br />
C. Alvarez and B. Jenner.<br />
A very hard log-space counting class,<br />
<i>Theoretical Computer Science</i> 107:3-30, 1993.<br />
<br />
<span id="ak02" style="color:maroon">[AK02]</span><br />
V. Arvind and P. Kurur.<br />
Graph isomorphism is in SPP,<br />
ECCC [http://eccc.uni-trier.de/eccc-reports/2002/TR02-037/ TR02-037], 2002.<br />
<br />
<span id="ak06" style="color:maroon">[AK06]</span><br />
S. Aaronson and G. Kuperberg.<br />
Quantum Versus Classical Proofs and Advice.<br />
arXiv:[http://arxiv.org/quant-ph/0604056 quant-ph/0604056], 2006.<br />
<br />
<span id="ak96" style="color:maroon">[AK96]</span><br />
F. Ablayev and M. Karpinski.<br />
On the power of randomized branching programs,<br />
<i>Proceedings of the International Colloquium on Automata, Languages, and Programming (ICALP)</i>, Springer-Verlag 1099, pp. 348-356, 1996.<br />
ECCC [http://eccc.uni-trier.de/eccc-reports/1995/TR95-054/ TR95-054],<br />
DIMACS [http://dimacs.rutgers.edu/TechnicalReports/abstracts/1996/96-46.html TR 96-46].<br />
<br />
<span id="akl79" style="color:maroon">[AKL+79]</span><br />
R. Aleliunas, R. M. Karp, R. J. Lipton, L. Lov&aacute;sz, and C. Rackoff.<br />
Random walks, traversal sequences, and the complexity of maze problems,<br />
<i>Proceedings of IEEE FOCS'79</i>, pp. 218-223, 1979.<br />
<br />
{{Reference<br />
|tag=AKR+03<br />
|authors=E. Allender, M. Koucký, D. Ronneburger, et al<br />
|title=Derandomization and distinguishing complexity<br />
|journal=Proceedings of the 18th Annual IEEE Conference on Computational Complexity<br />
|srcdetail=209-220<br />
}}<br />
<br />
{{Reference<br />
|tag=AKS94<br />
|authors=V. Arvind, J. Köbler and R. Schuler<br />
|title=On helping and interactive proof systems<br />
|journal=Algorithms and Computation: 5th International Symposium<br />
|srcdetail=137-145<br />
}}<br />
<br />
<span id="aks02" style="color:maroon">[AKS02]</span><br />
M. Agrawal, N. Kayal, and N. Saxena.<br />
Primes is in P,<br />
Annals of Mathematics, 160 (2004), 781-793.<br />
[http://www.cse.iitk.ac.in/primality.pdf http://www.cse.iitk.ac.in/primality.pdf].<br />
<br />
<span id="aks95" style="color:maroon">[AKS+95]</span><br />
V. Arvind, J. K&ouml;bler, U. Sch&ouml;ning, and R. Schuler.<br />
If NP has polynomial-size circuits, then MA=AM,<br />
<i>Theoretical Computer Science</i> 137, 1995.<br />
[http://www.informatik.hu-berlin.de/Institut/struktur/algorithmenII/Papers/ma-am.ps.gz http://www.informatik.hu-berlin.de/Institut/struktur/algorithmenII/Papers/ma-am.ps.gz]<br />
<br />
<span id="all96" style="color:maroon">[All96]</span><br />
E. Allender.<br />
Circuit complexity before the dawn of the new millennium,<br />
<i>Proceedings of the 16th Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FST&amp;TCS)</i>, Lecture Notes in Computer Science 1180, pp. 1-18, 1996.<br />
DIMACS [http://dimacs.rutgers.edu/TechnicalReports/abstracts/1997/97-49.html TR 97-49].<br />
<br />
<span id="all99" style="color:maroon">[All99]</span><br />
E. Allender.<br />
The permanent requires large uniform threshold circuits,<br />
<i>Chicago Journal of Theoretical Computer Science</i> 7, 1999.<br />
DIMACS [http://dimacs.rutgers.edu/TechnicalReports/abstracts/1997/97-51.html TR 97-51].<br />
<br />
<span id="alm98" style="color:maroon">[ALM+98]</span><br />
S. Arora, C. Lund, R. Motwani, M. Sudan, and M. Szegedy.<br />
Proof verification and hardness of approximation problems,<br />
<i>Journal of the ACM</i> 45(3):501-555, 1998.<br />
ECCC [http://eccc.uni-trier.de/eccc-reports/1998/TR98-008/ TR98-008].<br />
<br />
{{Reference<br />
|id=am04 |tag=AM04<br />
|title=Visibly Pushdown Languages<br />
|authors=R. Alur and P. Madhusudan<br />
|journal=Proceedings of ACM STOC'04, 2004.<br />
|srcdetail=202-211<br />
}}<br />
<br />
{{Reference<br />
|id=am09 |tag=AM09<br />
|title=Adding Nesting Structure to Words.<br />
|authors=R. Alur and P. Madhusudan<br />
|journal=Journal of the ACM 56(3)<br />
|srcdetail=Article 16, May 2009<br />
}}<br />
<br />
<span id="amb14" style="color:maroon">[Amb14]</span><br />
A. Ambainis.<br />
On physical problems that are slightly more difficult than QMA,<br />
<i>Proceedings of the 2014 IEEE 29th Conference on Computational Complexity</i>, 2014.<br />
arXiv:[http://arxiv.org/abs/1312.4758 quant-ph/1312.4758].<br />
<br />
<span id="amp02" style="color:maroon">[AMP02]</span><br />
F. Ablayev, C. Moore, and C. Pollett.<br />
Quantum and stochastic branching programs of bounded width,<br />
<i>Proceedings of the International Colloquium on Automata, Languages, and Programming (ICALP)</i>, 2002.<br />
arXiv:[http://arxiv.org/abs/quant-ph/0201139 quant-ph/0201139],<br />
ECCC [http://eccc.uni-trier.de/eccc-reports/2002/TR02-013/ TR02-013].<br />
<br />
<span id="ams06" style="color:maroon">[AMS06]</span><br />
N. Alon, D. Moshkovitz, and S. Safra. <br />
Algorithmic construction of sets for k-restrictions, <br />
<i>ACM Transactions on Algorithms (TALG)</i> 2(2): 153–177, 2006.<br />
[http://dx.doi.org/10.1145/1150334.1150336 doi:10.1145/1150334.1150336]<br />
<br />
<span id="an02" style="color:maroon">[AN02]</span><br />
D. Aharonov and T. Naveh.<br />
Quantum NP - a survey,<br />
arXiv:[http://arxiv.org/abs/quant-ph/0210077 quant-ph/0210077].<br />
<br />
<span id="ap95" style="color:maroon">[AP95]</span><br />
G. Ausiello and M. Protasi<br />
Local search, reducibility, and approximability of NP optimization problems,<br />
<i>Information Processing Letters</i> 54:73-79, 1995.<br />
<br />
<span id="ar01" style="color:maroon">[AR01]</span><br />
M. Alekhnovich and A. A. Razborov.<br />
Resolution is not automatizable unless W[P] is tractable,<br />
<i>Proceedings of IEEE FOCS'01</i>, pp. 210-219, 2001.<br />
<br />
<span id="ar03" style="color:maroon">[AR03]</span><br />
D. Aharonov and O. Regev.<br />
A lattice problem in quantum NP,<br />
arXiv:[http://arxiv.org/abs/quant-ph/0307220 quant-ph/0307220].<br />
<br />
<span id="ar88" style="color:maroon">[AR88]</span><br />
E. Allender and R. Rubinstein.<br />
P-printable sets,<br />
<i>SIAM Journal on Computing</i> 17(6):1193-1202, 1988.<br />
<br />
<span id="ar16" style="color:maroon">[AR16]</span><br />
B. Applebaum and P. Raykov.<br />
From Private Simultaneous Messages to Zero-Information Arthur-Merlin Protocols and Back,<br />
<i>Proceedings of TCC(A2)</i>, pp. 65-82, 2016.<br />
<br />
<span id="aro96" style="color:maroon">[Aro96]</span><br />
S. Arora.<br />
Polynomial time approximation scheme for Euclidean TSP and other geometric problems,<br />
<i>Proceedings of IEEE FOCS'96</i>, pp. 2-11, 1996.<br />
[http://www.cs.princeton.edu/~arora/pubs/tsp1.ps http://www.cs.princeton.edu/~arora/pubs/tsp1.ps]<br />
<br />
<span id="arz99" style="color:maroon">[ARZ99]</span><br />
E. Allender, K. Reinhardt, and S. Zhou.<br />
Isolation, matching, and counting: uniform and nonuniform upper bounds,<br />
<i>Journal of Computer and System Sciences</i> 59:164-181, 1999.<br />
[http://www.cs.rutgers.edu/pub/allender/matching.pdf http://www.cs.rutgers.edu/pub/allender/matching.pdf].<br />
<br />
<span id="as94" style="color:maroon">[AS94]</span><br />
E. Allender and M. Strauss.<br />
Measure on small complexity classes with applications for BPP,<br />
<i>Proceedings of IEEE FOCS'94</i>, pp. 807-818, 1994.<br />
ECCC [http://eccc.uni-trier.de/eccc-reports/1994/TR94-004/ TR94-004],<br />
DIMACS [http://dimacs.rutgers.edu/TechnicalReports/abstracts/1994/94-18.html TR 94-18].<br />
<br />
<span id="as98" style="color:maroon">[AS98]</span><br />
S. Arora and M. Safra.<br />
Probabilistic checking of proofs: a new characterization of NP,<br />
<i>Journal of the ACM</i> 45(1):70-122, 1998.<br />
[http://www.cs.princeton.edu/~arora/pubs/as.ps http://www.cs.princeton.edu/~arora/pubs/as.ps].<br />
<br />
<span id="asv00" style="color:maroon">[ASV00]</span><br />
A. Ambainis, L. Schulman, and U. Vazirani.<br />
Quantum computing with highly mixed states,<br />
<i>Proceedings of ACM STOC'2000</i>, pp. 705-714, 2000.<br />
arXiv:[http://arxiv.org/abs/quant-ph/0003136 quant-ph/0003136].<br />
<br />
<span id="atw00" style="color:maroon">[ATW+00]</span><br />
R. Armoni, A. Ta-Shma, A. Wigderson, and S. Zhou.<br />
An O(log(n)<sup>4/3</sup>) algorithm for (s,t) connectivity in undirected graphs,<br />
<i>Journal of the ACM</i> 47(2):294-311, 2000.<br />
[http://whiteboard.cs.tau.ac.il/~amnon/Papers/ATWZ.jacm00.pdf http://whiteboard.cs.tau.ac.il/~amnon/Papers/ATWZ.jacm00.pdf]<br />
<br />
{{Reference<br />
|tag=AV04<br />
|title=Abelian permutation group problems and logspace counting classes<br />
|authors=V. Arvind and T. C. Vijayaraghavan<br />
|journal=Proceedings of the 19th IEEE Conference on Computational Complexity<br />
|srcdetails=204-214, 2004<br />
}}<br />
<br />
<span id="aw90" style="color:maroon">[AW90]</span><br />
E. Allender and K. W. Wagner.<br />
Counting hierarchies: polynomial time and constant depth circuits,<br />
<i>Bulletin of the EATCS</i> 40, February 1990.<br />
[http://people.cs.uchicago.edu/~fortnow/beatcs/column40.ps http://people.cs.uchicago.edu/~fortnow/beatcs/column40.ps].<br />
<br />
===== B =====<br />
<br />
<span id="babai85" style="color:maroon">[Bab85]</span><br />
L. Babai.<br />
Trading Group Theory for Randomness.<br />
In <i>17th STOC</i>, pages 421--429, 1985.<br />
<br />
<span id="bab87" style="color:maroon">[Bab87]</span><br />
L. Babai.<br />
Random oracles separate PSPACE from the polynomial-time hierarchy.<br />
Information Processing Letters, 26 (1987) 51-53.<br />
<br />
<span id="bar02" style="color:maroon">[Bar02]</span><br />
B. Barak.<br />
A probabilistic-time hierarchy theorem for "slightly non-uniform" algorithms,<br />
<i>Proceedings of RANDOM'2002</i>, 2002.<br />
[http://www.math.weizmann.ac.il/~/boaz/Papers/bptime.ps http://www.math.weizmann.ac.il/~/boaz/Papers/bptime.ps]<br />
<br />
<span id="bar89" style="color:maroon">[Bar89]</span><br />
D. A. M. Barrington.<br />
Bounded-width polynomial-size branching programs can recognize exactly those languages in NC<sub>1</sub>,<br />
<i>Journal of Computer and System Sciences</i> 38:150-164, 1989.<br />
<br />
<span id="baz95" style="color:maroon">[Baz95]</span><br />
C. Bazgan.<br />
Approximation de probl&egrave;mes d'optimisation et de fonctions totales de NP,<br />
PhD thesis, INRIA, Orsay, France, 1998.<br />
[http://l1.lamsade.dauphine.fr/~bazgan/Papers/these.ps http://l1.lamsade.dauphine.fr/~bazgan/Papers/these.ps]<br />
<br />
<span id="bb12" style="color:maroon">[BB12]</span><br />
M. Bläser and B. Manthey.<br />
Smoothed Complexity Theory,<br />
<i>Proceedings of the 37th Int. Symp. on Mathematical Foundations of Computer Science</i>, 2012.<br />
ArXiv: [http://arxiv.org/pdf/1202.1936.pdf 1202.1936].<br />
<br />
<span id="bb92" style="color:maroon">[BB92]</span><br />
A. Berthiaume and G. Brassard.<br />
The quantum challenge to structural complexity theory.<br />
Proceedings of Structure in Complexity Theory, 1992, 132--137.<br />
<br />
<span id="bbb97" style="color:maroon">[BBB+97]</span><br />
C. H. Bennett, E. Bernstein, G. Brassard, and U. Vazirani.<br />
Strengths and weaknesses of quantum computing,<br />
<i>SIAM Journal on Computing</i>, 26(5):1510-1523, 1997.<br />
arXiv:[http://arxiv.org/abs/quant-ph/9701001 quant-ph/9701001].<br />
<br />
<span id="bbf98" style="color:maroon">[BBF98]</span><br />
R. Beigel, H. Buhrman, and L. Fortnow.<br />
NP might not be as easy as detecting unique solutions,<br />
<i>Proceedings of ACM STOC'98</i>, pp. 203-208, 1998.<br />
[http://people.cs.uchicago.edu/~fortnow/papers/newiso.ps http://people.cs.uchicago.edu/~fortnow/papers/newiso.ps].<br />
<br />
<span id="bbr94" style="color:maroon">[BBR94]</span><br />
D. A. M. Barrington, R. Beigel, and S. Rudich.<br />
Representing Boolean functions as polynomials modulo composite integers,<br />
<i>Computational Complexity</i>, 4:367-382, 1994.<br />
[http://www.cis.temple.edu/~beigel/papers/bbr-mods-cc.html http://www.cis.temple.edu/~beigel/papers/bbr-mods-cc.html].<br />
<br />
<span id="bbs86" style="color:maroon">[BBS86]</span><br />
J. Balc&aacute;zar, R. Book, and U. Sch&ouml;ning.<br />
Sparse sets, lowness, and highness,<br />
<i>SIAM Journal on Computing</i> 15:739-747, 1986.<br />
<br />
<span id="bce95" style="color:maroon">[BCE+95]</span><br />
P. Beame, S. Cook, J. Edmonds, R. Impagliazzo, and T. Pitassi.<br />
The relative complexity of NP search problems,<br />
<i>Proceedings of ACM STOC'95</i>, pp. 303-314, 1995.<br />
[http://www.cs.washington.edu/homes/beame/search.ps http://www.cs.washington.edu/homes/beame/search.ps]<br />
<br />
<span id="bch86" style="color:maroon">[BCH86]</span><br />
P. Beame, S. Cook, and J. Hoover.<br />
Log depth circuits for division and related problems,<br />
<i>SIAM Journal on Computing</i> 15:994-1003, 1986<br />
[http://www.cs.washington.edu/homes/beame/papers/division.ps http://www.cs.washington.edu/homes/beame/papers/division.ps].<br />
<br />
<span id="bcg92" style="color:maroon">[BCG+92]</span><br />
S. Ben-David, B. Chor, O. Goldreich, and M. Luby.<br />
On the theory of average case complexity,<br />
<i>Journal of Computer and System Sciences</i> 44(2):193-219, 1992.<br />
[http://www.cs.technion.ac.il/~shai/aver.pdf http://www.cs.technion.ac.il/~shai/aver.pdf]<br />
<br />
<span id="bcs97" style="color:maroon">[BCS+97]</span><br />
L. Blum, F. Cucker, M. Shub, and S. Smale.<br />
<i>Complexity and Real Computation</i>,<br />
Springer-Verlag, 1997.<br />
<br />
<span id="bcd89" style="color:maroon">[BCD+89]</span><br />
A. Borodin, S. A. Cook, P. W. Dymond, W. L. Ruzzo, and M. L. Tompa.<br />
Two applications of inductive counting for complementation problems,<br />
<i>SIAM Journal on Computing</i> 18:559-578, 1989.<br />
<br />
<span id="bcp83" style="color:maroon">[BCP83]</span><br />
A. Borodin, S. A. Cook, and N. Pippenger.<br />
Parallel computations for well-endowed rings and space-bounded probabilistic machines,<br />
<i>Information and Control</i> 58:113-136, 1983.<br />
<br />
<span id="bd99" style="color:maroon">[BD99]</span><br />
H. Buhrman and W. van Dam.<br />
Bounded quantum query complexity,<br />
<i>Proceedings of IEEE Complexity'99</i>, pp. 149-156, 1999.<br />
arXiv:[http://arxiv.org/abs/quant-ph/9903035 quant-ph/9903035].<br />
<br />
<span id="bdg88" style="color:maroon">[BDG88]</span><br />
J. L. Balc&aacute;zar, J. D&iacute;az, and J. Gabarr&oacute;<br />
Structural complexity 1<br />
<br />
<span id="bdh92" style="color:maroon">[BDH+92]</span><br />
G. Buntrock, C. Damm, U. Hertrampf, and Ch. Meinel.<br />
Structure and importance of logspace-MOD-classes,<br />
<i>Mathematical Systems Theory</i> 25:223-237, 1992.<br />
[http://www.num.math.uni-goettingen.de/damm/papers/BDHM92.ps.gz http://www.num.math.uni-goettingen.de/damm/papers/BDHM92.ps.gz].<br />
<br />
<span id="bei89" style="color:maroon">[Bei89]</span><br />
R. Beigel.<br />
On the relativized power of additional accepting paths,<br />
<i>Proceedings of IEEE Complexity'89</i>, pp. 216-224, 1989.<br />
[http://www.cis.temple.edu/~beigel/papers/ukp-structures.PS.gz http://www.cis.temple.edu/~beigel/papers/ukp-structures.PS.gz].<br />
<br />
<span id="bei94" style="color:maroon">[Bei94]</span><br />
R. Beigel.<br />
Perceptrons, PP, and the polynomial hierarchy,<br />
<i>Computational Complexity</i> 4:339-349, 1994.<br />
[http://www.cis.temple.edu/~beigel/papers/delta2p-cc.PS.gz http://www.cis.temple.edu/~beigel/papers/delta2p-cc.PS.gz].<br />
<br />
<span id="ber80" style="color:maroon">[Ber80]</span><br />
L. Berman.<br />
The complexity of logical theories,<br />
<i>Theoretical Computer Science</i> 11:71-78, 1980.<br />
<br />
<span id="bf92" style="color:maroon">[BF92]</span><br />
R. Beigel and J. Feigenbaum.<br />
On Being Incoherent Without Being Very Hard.<br />
Comput. Complexity 2 (1992), no. 1, 1--17<br />
[http://www.cis.temple.edu/~beigel/papers/bf-coherent-cc.html http://www.cis.temple.edu/~beigel/papers/bf-coherent-cc.html]<br />
<br />
<span id="bf99" style="color:maroon">[BF99]</span><br />
H. Buhrman and L. Fortnow.<br />
One-sided versus two-sided randomness,<br />
<i>Proceedings of the 16th Symposium on Theoretical Aspects of Computer Science (STACS)</i>, pp. 100-109, 1999.<br />
[http://people.cs.uchicago.edu/~fortnow/papers/rpvsbpp.ps http://people.cs.uchicago.edu/~fortnow/papers/rpvsbpp.ps].<br />
<br />
{{Reference<br />
|tag=BF03<br />
|authors=R. Beigel<br />
|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 />
<br />
<span id="bfl91" style="color:maroon">[BFL91]</span><br />
L. Babai, L. Fortnow, and C. Lund.<br />
Nondeterministic exponential time has two-prover interactive protocols,<br />
<i>Computational Complexity</i> 1:3-40, 1991.<br />
[http://people.cs.uchicago.edu/~fortnow/papers/mip2.ps http://people.cs.uchicago.edu/~fortnow/papers/mip2.ps].<br />
<br />
<span id="bfm88" style="color:maroon">[BFM88]</span><br />
M. Blum, P. Feldman, and S. Micali. <br />
Non-interactive zero-knowledge proofs and their applications,<br />
<i>Proceedings of the 20th STOC, ACM</i>, 1988.<br />
<br />
<span id="bfs86" style="color:maroon">[BFS86]</span><br />
L. Babai, P. Frankl, and J. Simon.<br />
Complexity classes in communication complexity theory,<br />
<i>Proceedings of IEEE FOCS'86</i>, pp. 337-347, 1986.<br />
<br />
<span id="bft98" style="color:maroon">[BFT98]</span><br />
H. Buhrman, L. Fortnow, and T. Thierauf.<br />
Nonrelativizing separations,<br />
<i>Proceedings of IEEE Complexity'98</i>, pp. 8-12, 1998.<br />
[http://people.cs.uchicago.edu/~fortnow/papers/nonrel.ps http://people.cs.uchicago.edu/~fortnow/papers/nonrel.ps].<br />
<br />
<span id="bgs75" style="color:maroon">[BGS75]</span><br />
T. Baker, J. Gill, and R. Solovay.<br />
Relativizations of the P=?NP question,<br />
<i>SIAM Journal on Computing</i> 4:431-442, 1975.<br />
<br />
<span id="bh77" style="color:maroon">[BH77]</span><br />
L. Berman and J. Hartmanis.<br />
On isomorphism and density of NP and other complete sets,<br />
<i>SIAM Journal on Computing</i> 6:305-322, 1977.<br />
<br />
<span id="bg03" style="color:maroon">[BG03]</span><br />
M. Ben-Or and D. Gutfreund.<br />
Trading help for interaction in statistical zero-knowledge proofs,<br />
<i>J. Cryptology</i> 16 (2003), no. 2, 95--116.<br />
[http://www.cs.huji.ac.il/~danig/pubs/help_interaction.ps http://www.cs.huji.ac.il/~danig/pubs/help_interaction.ps]<br />
<br />
<span id="bg69" style="color:maroon">[BG69]</span><br />
R. Book and S. Greibach.<br />
Quasi-realtime languages,<br />
<i>Proceedings of ACM STOC</i> pp. 15-18, 1969.<br />
http://portal.acm.org/citation.cfm?id=800169.805416<br />
<br />
<span id="bg81" style="color:maroon">[BG81]</span><br />
C. H. Bennett and J. Gill.<br />
Relative to a random oracle A, P<sup>A</sup> != NP<sup>A</sup> != coNP<sup>A</sup> with probability 1,<br />
<i>SIAM Journal on Computing</i>, 10(1):96-113, 1981.<br />
DOI:[http://dx.doi.org/10.1137/0210008 10.1137/0210008]<br />
<br />
<span id="bg92" style="color:maroon">[BG92]</span><br />
R. Beigel and J. Gill.<br />
Counting classes: thresholds, parity, mods, and fewness,<br />
<i>Theoretical Computer Science</i> 103(1):3-23, 1992.<br />
[http://www.cis.temple.edu/~beigel/papers/bg-mods-tcs.PS.gz http://www.cis.temple.edu/~beigel/papers/bg-mods-tcs.PS.gz].<br />
<br />
<span id="bg98" style="color:maroon">[BG98]</span><br />
R. Beigel and J. Goldsmith.<br />
Downward separation fails catastrophically for limited nondeterminism classes,<br />
<i>SIAM Journal on Computing</i> 17(5):1420-1429, 1998.<br />
[http://www.cis.temple.edu/~beigel/papers/bg-beta-draft.PS.gz http://www.cis.temple.edu/~beigel/papers/bg-beta-draft.PS.gz].<br />
<br />
<span id="bg94" style="color:maroon">[BG94]</span><br />
M. Bellare and S. Goldwasser.<br />
The complexity of decision versus search,<br />
<i>SIAM Journal on Computing</i> 23(1):91-119, 1994.<br />
[http://www.cs.ucsd.edu/users/mihir/papers/compip.pdf http://www.cs.ucsd.edu/users/mihir/papers/compip.pdf]<br />
<br />
<span id="bgg90" style="color:maroon">[BGG+90]</span><br />
M. Ben-Or, O. Goldreich, S. Goldwasser, J. H&aring;stad, J. Kilian, S. Micali, and P. Rogaway.<br />
Everything provable is provable in zero-knowledge,<br />
<i>Advances in Cryptology: CRYPTO'88</i> (S. Goldwasser, ed.), Lecture Notes in Computer Science 403, Springer-Verlag, pp. 37-56, 1990.<br />
<br />
<span id="bgk88" style="color:maroon">[BGK+88]</span><br />
M. Ben-Or, S. Goldwasser, J. Kilian, and A. Wigderson.<br />
Multi-prover interactive proofs: how to remove intractability,<br />
<i>Proceedings of ACM STOC'88</i>, pp. 113-131, 1988.<br />
<br />
<span id="bg82" style="color:maroon">[BG82]</span><br />
A. Blass and Y. Gurevich.<br />
On the unique satisfiability problem,<br />
<i>Information and Control</i> 55(1-3):80-88, 1982.<br />
<br />
<span id="bgm02" style="color:maroon">[BGM02]</span><br />
E. B&ouml;hler, C. Gla&szlig;er, and D. Meister.<br />
Error-bounded probabilistic computations between MA and AM,<br />
Mathematical foundations of computer science 2003, 249--258.<br />
[http://haegar.informatik.uni-wuerzburg.de/users/glasser/publications/sbp-ma-am-tr.pdf http://haegar.informatik.uni-wuerzburg.de/users/glasser/publications/sbp-ma-am-tr.pdf]<br />
<br />
<span id="bgr93" style="color:maroon">[BGR93]</span> <br />
B. von Braunmühl, R. Gengler, and R. Rettinger.<br />
The alternation hierarchy for sublogarithmic space is infinite,<br />
Computational Complexity, v.3 n.3, p.207-230, July 1993 <br />
[doi>10.1007/BF01271368]<br />
[http://portal.acm.org/citation.cfm?id=218886]<br />
<br />
<span id="bh91" style="color:maroon">[BH91]</span><br />
S. R. Buss and L. Hay.<br />
On truth-table reducibility to SAT,<br />
<i>Information and Computation</i> 91(1):86-102, 1991.<br />
<br />
<span id="bh97" style="color:maroon">[BH97]</span><br />
C. Berg and J. H&aring;stad.<br />
On the BPP hierarchy problem,<br />
Technical Report TRITA-NA-9702, Royal Institute of Technology, Sweden, 1997.<br />
[ftp://ftp.nada.kth.se/pub/documents/Theory/Christer-Berg/bpp.ps ftp://ftp.nada.kth.se/pub/documents/Theory/Christer-Berg/bpp.ps].<br />
<br />
{{Reference<br />
|tag=BH08<br />
|title=NP-Hard sets are exponentially eense unless NP is contained in coNP/poly<br />
|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 />
}}<br />
<br />
<span id="bhr00" style="color:maroon">[BHR00]</span><br />
B. Borchert, L. Hemaspaandra, and J. Rothe.<br />
Restrictive Acceptance Suffices for Equivalence Problems.<br />
LMS J. Comput. Math. 3 (2000), 86--95<br />
arXiv:[http://arxiv.org/cs.CC/9907041 cs.CC/9907041].<br />
<br />
<span id="bhw89" style="color:maroon">[BHW89]</span><br />
R. Beigel, L. Hemachandra, and G. Wechsung.<br />
On the power of probabilistic polynomial time,<br />
<i>Proceedings of IEEE Complexity'89</i>, pp. 225-230, 1989.<br />
<br />
<span id="bhz87" style="color:maroon">[BHZ87]</span><br />
R. B. Boppana, J. H&aring;stad, and S. Zachos.<br />
Does co-NP have short interactive proofs?,<br />
<i>Information Processing Letters</i> 25:127-132, 1987.<br />
<br />
<span id="bk89" style="color:maroon">[BK89]</span><br />
M. Blum and S. Kannan.<br />
Designing programs that check their work,<br />
<i>Proceedings of ACM STOC'89</i>, 1989.<br />
<br />
<span id="bkl00" style="color:maroon">[BKL+00]</span><br />
D. A. M. Barrington, P. Kadau, K.-J. Lange, and P. McKenzie.<br />
On the complexity of some problems on groups input as multiplication tables,<br />
[http://www-fs.informatik.uni-tuebingen.de/~lange/Arbeiten/fologlog/bklm/neu.ps.gz http://www-fs.informatik.uni-tuebingen.de/~lange/Arbeiten/fologlog/bklm/neu.ps.gz]<br />
<i>Proceedings of IEEE Complexity'2000</i>, 2000.<br />
<br />
<span id="bks95" style="color:maroon">[BKS95]</span><br />
R. Beigel, M. Kummer, and F. Stephan.<br />
Approximable sets,<br />
<i>Information and Computation</i> 120(2):304-314, 1995.<br />
[http://www.cis.temple.edu/~beigel/papers/bks-queries2-ic.PS.gz http://www.cis.temple.edu/~beigel/papers/bks-queries2-ic.PS.gz].<br />
<br />
<span id="blm98" style="color:maroon">[BLM+98]</span><br />
D. A. M. Barrington, C.-J. Lu, P. B. Miltersen, and S. Skyum.<br />
Searching constant width mazes captures the AC<sup>0</sup> hierarchy,<br />
<i>Proceedings of the 1998 Symposium of Theoretical Aspects of Computer Science (STACS'98)</i>, 1998.<br />
ECCC [http://eccc.uni-trier.de/eccc-reports/1997/TR97-044/ TR97-044].<br />
<br />
<span id="blm99" style="color:maroon">[BLM+99]</span><br />
D. A. M. Barrington, C.-J. Lu, P. B. Miltersen, and S. Skyum.<br />
On monotone planar circuits,<br />
<i>Proceedings of IEEE Complexity'99</i>, 1999.<br />
[http://www.brics.dk/~bromille/Papers/mpc.ps http://www.brics.dk/~bromille/Papers/mpc.ps]<br />
<br />
<span id="bls84" style="color:maroon">[BLS84]</span><br />
R. Book, T. Long, and A. Selman.<br />
Quantitative relativizations of complexity classes,<br />
<i>SIAM Journal on Computing</i> 13(3):461-487, 1984.<br />
<br />
<span id="blu67" style="color:maroon">[Blu67]</span><br />
M. Blum. A Machine-Independent Theory of the Complexity of Recursive Functions. <i>J. ACM</i> 14: 322-336, 1967. <br />
<br />
<span id="bm04" style="color:maroon">[BM04]</span><br />
J. Buresh-Oppenheim and T. Morioka.<br />
Relativized NP search problems and propositional proof systems,<br />
<i>Proceedings of IEEE Complexity 2004</i>, pp. 54-67, 2004.<br />
ECCC [http://eccc.uni-trier.de/eccc-reports/2003/TR03-084/ TR03-084].<br />
<br />
<span id="bm88" style="color:maroon">[BM88]</span><br />
L. Babai and S. Moran.<br />
Arthur-Merlin games: a randomized proof system, and a hierarchy of complexity classes,<br />
<i>Journal of Computer and Systems Sciences</i> 36:254-276, 1988.<br />
<br />
<span id="boo72" style="color:maroon">[Boo72]</span><br />
R. Book.<br />
On languages accepted in polynomial time,<br />
<i>SIAM Journal on Computing</i> 1(4):281-287, 1972.<br />
<br />
<span id="boo74" style="color:maroon">[Boo74]</span><br />
R. Book.<br />
Comparing complexity classes,<br />
<i>Journal of Computer and System Sciences</i> 3(9):213-229, 1974.<br />
<br />
<span id="boo94" style="color:maroon">[Boo94]</span><br />
R. Book.<br />
On collapsing the polynomial-time hierarchy,<br />
<i>Information Processing Letters</i> 52(5):235-237, 1994.<br />
<br />
<span id="bor77" style="color:maroon">[Bor77]</span><br />
A. Borodin.<br />
On relating time and space to size and depth,<br />
<i>SIAM Journal on Computing</i> 6:733-744, 1977.<br />
<br />
<span id="bra77" style="color:maroon">[Bra77]</span><br />
F.-J. Brandenburg.<br />
On one-way auxiliary pushdown automata,<br />
<i>Proceedings of the Third GI-Conference on Theoretical Computer Science</i>, Springer LNCS vol. 48, pp. 132-144, 1977.<br />
<br />
<span id="bra79" style="color:maroon">[Bra79]</span><br />
G. Brassard.<br />
A note on the complexity of cryptography<br />
<i>IEEE Transactions on Information Theory</i>, 25(2):232-233, 1979.<br />
<br />
<span id="bra06" style="color:maroon">[Bra06]</span><br />
S. Bravyi.<br />
Efficient algorithm for a quantum analogue of 2-SAT,<br />
arXiv:[http://arxiv.org/abs/quant-ph/0602108v1 quant-ph/0602108v1], 2006.<br />
<br />
<span id="brs91" style="color:maroon">[BRS91]</span><br />
R. Beigel, N. Reingold, and D. A. Spielman.<br />
PP is closed under intersection,<br />
<i>Proceedings of ACM STOC'91</i>, pp. 1-9, 1991.<br />
[http://www.cis.temple.edu/~beigel/papers/brs-pp-jcss.PS.gz http://www.cis.temple.edu/~beigel/papers/brs-pp-jcss.PS.gz].<br />
<br />
<span id="bru90" style="color:maroon">[Bru90]</span><br />
J. Bruck.<br />
Harmonic analysis of polynomial threshold functions,<br />
SIAM J. Discrete Math., 3 (1990) 168-177.<br />
<br />
<span id="bs00" style="color:maroon">[BS00]</span><br />
B. Borchert and F. Stephan.<br />
Looking for an Analogue of Rice's Theorem in Circuit Complexity Theory.<br />
MLQ Math. Log. Q. 46 (2000), no. 4, 489--504<br />
[http://math.uni-heidelberg.de/logic/bb/papers/Rice.ps http://math.uni-heidelberg.de/logic/bb/papers/Rice.ps]<br />
<br />
<span id="bs90" style="color:maroon">[BS90]</span><br />
J. Bruck and R. Smolensky.<br />
Polynomial threshold functions, AC<sup>0</sup> functions and spectral norms,<br />
<i>Proceedings of IEEE FOCS'90</i>, pp. 632-641, 1990.<br />
<br />
<span id="bs90b" style="color:maroon">[BS90b]</span><br />
R. B. Boppana and M. Sipser. The complexity of finite functions. <br />
chapter in <i>Handbook of Theoretical Computer Science</i>, Volume A (J. van Leeuwen, editor), Elsevier, 1990.<br />
<br />
<span id="bsf02" style="color:maroon">[BSF02]</span><br />
A. Ben-Hur, H. T. Siegelmann, and S. Fishman.<br />
A theory of complexity for continuous time systems,<br />
<i>Journal of Complexity</i> 18(1):51-86, 2002.<br />
[http://cmgm.stanford.edu/~asab/Papers/dds2.ps.gz http://cmgm.stanford.edu/~asab/Papers/dds2.ps.gz]<br />
<br />
<span id="bt04" style="color:maroon">[BT04]</span><br />
H. Buhrman and L. Torenvliet.<br />
Separating complexity classes using structural properties,<br />
<i>Proceedings of IEEE Complexity 2004</i>, pp. 130-138, 2004.<br />
[http://staff.science.uva.nl/~leen/PAPERS/superrobustsets.pdf http://staff.science.uva.nl/~leen/PAPERS/superrobustsets.pdf]<br />
<br />
{{Reference<br />
|tag=BT06<br />
|authors=A. Bogdanov and L. Trevisan<br />
|title=Average-Case Complexity<br />
|journal=ECCC Report TR06-073<br />
|srcdetail=Revision 01, accepted on Fri Sep 29 22:13:11 2006<br />
}}<br />
<br />
<span id="bt88" style="color:maroon">[BT88]</span><br />
D. A. M. Barrington and D. Th&eacute;rien.<br />
Finite monoids and the fine structure of NC<sup>1</sup>,<br />
<i>Journal of the ACM</i> 35(4):941-952, 1988.<br />
<br />
<span id="bur00" style="color:maroon">[Bur00]</span><br />
P. B&uuml;rgisser.<br />
<i>Completeness and Reduction in Algebraic Complexity Theory</i>,<br />
Springer Series in Algorithms and Computation in Mathematics, Volume 7, 2000.<br />
<br />
{{Reference<br />
|tag=Buss93<br />
|authors = S. Buss<br />
|title = Algorithms for Boolean formula evaluation and for tree-contraction<br />
|journal=Proof Theory, Complexity, and Arithmetic, P. Clote and J. Krajicek (eds) <br />
|srcdetail=Oxford University Press, 1993, pp. 95-115<br />
|link=http://math.ucsd.edu/~sbuss/ResearchWeb/Boolean3/index.html<br />
}}<br />
<br />
<span id="bv97" style="color:maroon">[BV97]</span><br />
E. Bernstein and U. Vazirani.<br />
Quantum complexity theory,<br />
<i>SIAM Journal on Computing</i>, 26(5):1411-1473, 1997.<br />
[http://www.cs.berkeley.edu/~vazirani/bv.ps http://www.cs.berkeley.edu/~vazirani/bv.ps]<br />
<br />
<span id="bvw98" style="color:maroon">[BVW98]</span><br />
R. Book, H. Vollmer, and K. W. Wagner.<br />
Probabilistic type-2 operators and "almost"-classes,<br />
<i>Computational Complexity</i> 7(3):265-289, 1998.<br />
<br />
<span id="bvw07" style="color:maroon">[BVW07]</span><br />
H. Burhman, N. Vereshchajin, R. de Wolf.<br />
On computation and communication with small bias.<br />
<i>Proceedings of IEEE Conference on Computational Complexity 2007</i> 24-32, 2007.<br />
<br />
<span id="bw03" style="color:maroon">[BW03]</span><br />
H. Buhrman and R. de Wolf.<br />
Quantum Zero-Error Algorithms Cannot be Composed,<br />
Information Processing Letters, 87(2):79-84, 2003.<br />
arXiv:[http://arxiv.org/abs/quant-ph/0211029 quant-ph/0211029].<br />
<br />
===== C =====<br />
<br />
<span id="cai01" style="color:maroon">[Cai01]</span><br />
J.-Y. Cai.<br />
S<sub>2</sub>P is contained in ZPP<sup>NP</sup>,<br />
<i>Proceedings of IEEE FOCS'2001</i>, pp. 620-629, 2001.<br />
<br />
<span id="cai86" style="color:maroon">[Cai86]</span><br />
J.-Y. Cai.<br />
With probability one, a random oracle separates PSPACE from the polynomial hierarchy,<br />
<i>Proceedings of ACM STOC'86</i>, pp. 21-29, 1986.<br />
<br />
<span id="cai87" style="color:maroon">[Cai87]</span><br />
J. Cai.<br />
Probability one separation of the Boolean hieararchy,<br />
Lecture Notes in Computer Science, vol 247, p148-158, 1987.<br />
<br />
<span id="can96" style="color:maroon">[Can96]</span><br />
R. Canetti.<br />
More on BPP and the polynomial-time hierarchy,<br />
<i>Information Processing Letters</i> 57:237-241, 1996.<br />
<br />
<span id="cc93" style="color:maroon">[CC93]</span><br />
L. Cai and J. Chen.<br />
On fixed-parameter tractability and approximability of NP-hard optimization problems,<br />
<i>Proceedings of ISTCS'93 - Israel Symposium on Theory of Computing and Systems</i>, pp. 118-126, 1993.<br />
<br />
<span id="cc97" style="color:maroon">[CC97]</span><br />
L. Cai and J. Chen.<br />
On fixed-parameter tractability and approximability of NP optimization problems,<br />
<i>Journal of Computer and System Sciences</i> 54(3):465-474, 1997.<br />
<br />
<span id="ccd03" style="color:maroon">[CCD+03]</span><br />
A. M. Childs, R. Cleve, E. Deotto, E. Farhi, S. Gutmann, and D. A. Spielman.<br />
Exponential algorithmic speedup by quantum walk,<br />
<i>Proceedings of ACM Symposium on Theory of Computing</i>, pp. 59-68, 2003.<br />
arXiv:[http://arxiv.org/abs/quant-ph/0209131 quant-ph/0209131].<br />
<br />
<span id="ccg94" style="color:maroon">[CCG+94]</span><br />
R. Chang, B. Chor, O. Goldreich, J. Hartmanis, J. H&aring;stad, D. Ranjan, and P. Rohatgi.<br />
The random oracle hypothesis is false,<br />
<i>Journal of Computer and System Sciences</i> 49(1):24-39, 1994.<br />
<br />
<span id="cch01" style="color:maroon">[CCH+01]</span><br />
J.-Y. Cai, V. Chakaravarthy, L. Hemaspaandra, and M. Ogihara.<br />
Some Karp-Lipton-type theorems based on S<sub>2</sub>,<br />
University of Rochester Computer Science Technical Report TR-759, November 2001.<br />
<br />
<span id="cd05" style="color:maroon">[CD05]</span><br />
X. Chen and X. Deng<br />
3-NASH is PPAD-Complete, online: [http://eccc.uni-trier.de/eccc-reports/2005/TR05-134/Paper.pdf http://eccc.uni-trier.de/eccc-reports/2005/TR05-134/Paper.pdf], nov. 2005.<br />
<br />
<span id="cdl01" style="color:maroon">[CDL01]</span><br />
A. Chiu, G. Davida, and B. Litow.<br />
Division in logspace-uniform NC<sub>1</sub>,<br />
<i>Theoretical Informatics and Applications</i> 35(3):259, 2001.<br />
<br />
<span id="cf91" style="color:maroon">[CF91]</span><br />
J.-Y. Cai and M. Furst.<br />
PSPACE survives constant-width bottlenecks,<br />
<i>International Journal of Foundations of Computer Science</i> 2(1):67-76, 1991.<br />
<br />
{{Reference<br />
|id=cf07 |tag=CP07<br />
|title=On parameterized path and chordless path problems<br />
|authors=Y. Chen and J. Flum<br />
|journal=Proceedings of the IEEE Conference on Computational Complexity 2007<br />
|srcdetail=250-263<br />
}}<br />
<br />
{{Reference<br />
|tag=CFL83<br />
|title=Unbounded fan-in circuits and associative functions<br />
|authors=A. Chandra, S. Fortune, R. Lipton<br />
|journal=Proceedings of the fifteenth annual ACM symposium on Theory of computing<br />
|srcdetail=52-60, 1983<br />
}}<br />
<br />
<span id="cfl93" style="color:maroon">[CFL+93]</span><br />
A. Condon, J. Feigenbaum, C. Lund, and P. Shor.<br />
Probabilistically checkable debate systems and approximation algorithms for PSPACE-hard functions (extended abstract),<br />
<i>Proceedings of ACM STOC'93</i>, pp. 305-314, 1993.<br />
<br />
<span id="cgh88" style="color:maroon">[CGH+88]</span><br />
J.-Y. Cai, T. Gundermann, J. Hartmanis, L. Hemachandra, V. Sewelson, K. Wagner, and G. Wechsung.<br />
The Boolean hierarchy I: structural properties,<br />
<i>SIAM Journal on Computing</i> 17:1232-1252, 1988.<br />
Part II: applications in 18:95-111, 1989.<br />
<br />
<span id="cgr04" style="color:maroon">[CGR+04]</span><br />
M. Crasmaru, C. Gla&szlig;er, K. W. Regan, and S. Sengupta.<br />
A protocol for serializing unique strategies,<br />
submitted, 2004.<br />
[http://www.cse.buffalo.edu/faculty/regan/papers/ps/CGRS03.ps http://www.cse.buffalo.edu/faculty/regan/papers/ps/CGRS03.ps].<br />
<br />
<span id="ch89" style="color:maroon">[CH89]</span><br />
J.-Y. Cai and L. A. Hemachandra.<br />
On the power of parity polynomial time,<br />
<i>Proceedings of the Symposium on Theoretical Aspects of Computer Science (STACS)</i>, Lecture Notes in Computer Science 349, pp. 229-240, Springer, 1989.<br />
<br />
<span id="cht04" style="color:maroon">[CHT+04]</span><br />
R. Cleve, P. H&oslash;yer, B. Toner, and J. Watrous.<br />
Consequences and limits of nonlocal strategies,<br />
<i>Proceedings of IEEE Complexity</i>, pp. 236-249, 2004.<br />
arXiv:[http://arxiv.org/abs/quant-ph/0404076 quant-ph/0404076].<br />
<br />
<span id="chu41" style="color:maroon">[Chu41]</span><br />
A. Church.<br />
The calculi of lambda-conversion,<br />
<i>Annals of Mathematical Studies</i> 6, Princeton Univ. Press, 1941.<br />
<br />
{{Reference<br />
|tag=CIK+03<br />
|title=The complexity of Unique <math>k</math>-SAT: An isolation lemma for <math>k</math>-CNFs.<br />
|authors=C. Calabro, R. Impagliazzo, V. Kabanets, et al<br />
|journal=Proceedings of the IEEE Conference on Computational Complexity 2003<br />
|srcdetail=135-141<br />
}}<br />
<br />
<span id="ckk95" style="color:maroon">[CKK+95]</span><br />
F. Cucker, M. Karpinski, P. Koiran, T. Lickteig, and K. Werther.<br />
On real Turing machines that toss coins,<br />
<i>Proceedings of ACM STOC'95</i>, pp. 335-342, 1995.<br />
<br />
<span id="cks81" style="color:maroon">[CKS81]</span><br />
A. K. Chandra, D. C. Kozen, and L. J. Stockmeyer.<br />
Alternation,<br />
<i>Journal of the ACM</i> 28:114-133, 1981.<br />
<br />
<span id="cks99" style="color:maroon">[CKS+99]</span><br />
P. Crescenzi, V. Kann, R. Silvestri, and L. Trevisan.<br />
Structure in approximation classes,<br />
<i>SIAM Journal on Computing</i> 28:1759-1782, 1999.<br />
ECCC [http://eccc.uni-trier.de/eccc-reports/1996/TR96-066/ TR96-066].<br />
<br />
<span id="cksu05" style="color:maroon">[CKSU05]</span><br />
H. Cohn, R. Kleinberg, B. Szegedy, and C. Umans. Group-theoretic Algorithms for Matrix Multiplication. <i>Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS)</i> 379-388, 2005.<br />
<br />
<span id="cm01" style="color:maroon">[CM01]</span><br />
M. Cryan and P. B. Miltersen.<br />
On pseudorandom generators in NC<sup>0</sup>,<br />
<i>Proceedings of the 26th International Symposium on Mathematical Foundations of Computer Science (MFCS)</i>, pp. 272-284, 2001.<br />
<br />
{{Reference<br />
|tag=CMTV98<br />
|title=Nondeterministic NC1 computation<br />
|authors=H. Caussinus, P. McKenzie, D. Th&eacute;rien, and H. Vollmer<br />
|journal=Journal of Computer and System Sciences<br />
|srcdetail=200-212, 1998<br />
}}<br />
<br />
<span id="cmi00" style="color:maroon">[CMI00]</span><br />
Clay Mathematics Institute.<br />
The P versus NP problem (a millennium prize problem), with official problem description by S. Cook,<br />
[http://www.claymath.org/prizeproblems/pvsnp.htm http://www.claymath.org/prizeproblems/pvsnp.htm], 2000.<br />
<br />
<span id="cns99" style="color:maroon">[CNS99]</span><br />
J.-Y. Cai, A. P. Nerurkar, and D. Sivakumar.<br />
Hardness and hierarchy theorems for probabilistic quasi-polynomial time,<br />
<i>Proceedings of ACM STOC'99</i>, pp. 726-735, 1999.<br />
<br />
<span id="cob64" style="color:maroon">[Cob64]</span><br />
A. Cobham.<br />
The intrinsic computational difficulty of functions,<br />
<i>Proceedings of the 1964 Congress on Logic, Mathematics and the Methodology of Science</i>, pp. 24-30, 1964.<br />
<br />
<span id="cob66" style="color:maroon">[Cob66]</span><br />
A. Cobham.<br />
The recognition problem for the set of perfect squares,<br />
<i>Proceedings of the 7th Symposium on Switching and Automata Theory</i>, pp. 78-87, 1966.<br />
<br />
<span id="con73" style="color:maroon">[Con73]</span><br />
R. Constable.<br />
Type 2 computational complexity,<br />
<i>Proceedings of ACM STOC'73</i>, pp. 108-121, 1973.<br />
<br />
<span id="con92" style="color:maroon">[Con92]</span><br />
A. Condon.<br />
The complexity of stochastic games,<br />
<i>Information and Computation</i> 96(2):203-224, 1992.<br />
<br />
<span id="coo71" style="color:maroon">[Coo71]</span><br />
S. A. Cook.<br />
The complexity of theorem-proving procedures,<br />
<i>Proceedings of ACM STOC'71</i>, pp. 151-158, 1971.<br />
<br />
<span id="coo71b" style="color:maroon">[Coo71b]</span><br />
S. A. Cook.<br />
Characterizations of pushdown machines in terms of time-bounded computers,<br />
<i>Journal of the ACM</i> 18(1):4-18, 1971.<br />
<br />
<span id="coo79" style="color:maroon">[Coo79]</span><br />
S. A. Cook.<br />
Deterministic CFL's are accepted simultaneously in polynomial time and log squared space,<br />
<i>Proceedings of ACM STOC'79</i>, pp. 338-345, 1979.<br />
<br />
<span id="coo85" style="color:maroon">[Coo85]</span><br />
S. A. Cook.<br />
A taxonomy of problems with fast parallel algorithms,<br />
<i>Information and Control</i> 64:2-22, 1985.<br />
<br />
<span id="cp95" style="color:maroon">[CP95]</span><br />
P. Crescenzi and C. Papadimitriou.<br />
Reversible simulation of space-bounded computations,<br />
<i>Theoretical Computer Science</i> 143:159-165, 1995.<br />
<br />
{{Reference<br />
|id=cp07 |tag=CP07<br />
|title=Bounded queries and the NP Machine Hypothesis.<br />
|authors=R. Chang and S. Purini<br />
|journal=Proceedings of the IEEE Conference on Computational Complexity 2007<br />
|srcdetail=52-59<br />
}}<br />
<br />
<span id="cr96" style="color:maroon">[CR96]</span><br />
S. Chaudhuri and J. Radhakrishnan. <br />
Deterministic Restrictions in Circuit Complexity, <br />
<i>Proceedings of ACM STOC 1996</i>, pp. 30-36, 1996.<br />
<br />
{{Reference<br />
|id=cr06 |tag=CR06<br />
|authors=V. Chakaravarthy and S. Roy<br />
|title=Oblivious symmetric alternation<br />
|journal=Proceedings of the 23rd Symposium on Theoretical Aspects of Computer Science (STACS) 2006<br />
|srcdetail=230-241<br />
}}[http://www.cs.wisc.edu/~venkat/o2.ps]<br />
<br />
<span id="cs92" style="color:maroon">[CS92]</span><br />
J. Castro and C. Seara.<br />
Characterizations of some complexity classes between &#920;<sub>2</sub><sup>p</sup> and &#916;<sub>2</sub><sup>p</sup>,<br />
<i>Proceedings of STACS 1992</i>, pp. 305-317, 1992.<br />
<br />
<span id="ct94" style="color:maroon">[CT94]</span><br />
P. Crescenzi and L. Trevisan.<br />
An approximation scheme preserving reducibility and its applications,<br />
<i>Proceedings of 14th Annual Conference on Foundations of Software Technology and Theoretical Computer Computer Science (FSTTCS)</i>, pp. 330-341, Lecture Notes in Computer Science 880, Springer-Verlag, 1994.<br />
<br />
<span id="ct97" style="color:maroon">[CT97]</span><br />
M. Cesati and L. Trevisan.<br />
On the efficiency of polynomial time approximation schemes,<br />
ECCC [http://eccc.uni-trier.de/eccc-reports/1997/TR97-001/ TR97-001], 1997.<br />
<br />
<span id="ct07" style="color:maroon">[CT07]</span><br />
X. Chen and S.-H. Teng.<br />
Paths beyond local search: A nearly tight bound for randomized fixed-point computation.<br />
FOCS 2007.<br />
<br />
<span id="cw82" style="color:maroon">[CW82]</span> D. Coppersmith and S. Winograd. On the Asymptotic Complexity of Matrix Multiplication. <i>SIAM J. Comput.</i> 11(3): 472-492,1982.<br />
<br />
{{Reference<br />
|tag=CW90<br />
|title=Matrix multiplication via arithmetic progressions<br />
|journal=Journal of Symbolic Computation<br />
|srcdetail=9:251–280, 1990<br />
|authors=D. Coppersmith and S. Winograd<br />
}}<br />
<br />
<span id="cw00" style="color:maroon">[CW00]</span><br />
R. Cleve and J. Watrous.<br />
Fast parallel circuits for the quantum Fourier transform,<br />
<i>Proceedings of IEEE Focs'2000</i>, pp. 526-536, 2000.<br />
arXiv:[http://arxiv.org/abs/quant-ph/0006004 quant-ph/0006004].<br />
<br />
===== D =====<br />
<br />
<span id="dam90" style="color:maroon">[Dam90]</span><br />
C. Damm.<br />
Problems complete for L,<br />
<i>Information Processing Letters</i> 36:247-250, 1990.<br />
<br />
<span id="dc89" style="color:maroon">[DC89]</span><br />
P. W. Dymond and S. Cook.<br />
Complexity theory of parallel time and hardware,<br />
<i>Information and Computation</i> 80:205-226, 1989.<br />
<br />
<span id="ddp98" style="color:maroon">[DDP+98]</span><br />
A. De Santis, G. Di Crescenzo, G. Persiano, and M. Yung.<br />
Image density is complete for non-interactive SZK,<br />
<i>Proceedings of the 25th International Colloquium on Automata, Languages, and Programming (ICALP)</i>, Lecture Notes in Computer Science, pp. 784-795, 1998.<br />
(Note: Some results in the paper were later retracted.)<br />
<br />
<span id="dek76" style="color:maroon">[Dek76]</span><br />
M. I. Dekhtyar.<br />
On the relativization of deterministic and nondeterministic complexity classes,<br />
<i>Mathematical Foundations of Computer Science</i>, pp. 255-259, Springer LNCS 45, 1976.<br />
<br />
<span id="df97" style="color:maroon">[DF97]</span><br />
R. G. Downey and M. R. Fellows.<br />
Threshold dominating sets and an improved characterization of W[2],<br />
<i>Theoretical Computer Science</i> 189, 1997.<br />
<br />
<span id="df99" style="color:maroon">[DF99]</span><br />
R. G. Downey and M. R. Fellows.<br />
<i>Parameterized Complexity</i>,<br />
Springer-Verlag Monographs in Computer Science, 1999.<br />
<br />
<span id="dft96" style="color:maroon">[DFT96]</span><br />
R. G. Downey, M. R. Fellows, and U. Taylor.<br />
On the parameteric complexity of relational database queries and a sharper characterization of W[1],<br />
<i>Combinatorics, Complexity, and Logic</i>, Proceedings of DMTCS'96, Springer-Verlag, pp. 194-213, 1996.<br />
<br />
<span id="dft98" style="color:maroon">[DFT96]</span><br />
R. G. Downey, M. R. Fellows, and U. Taylor.<br />
Parameterized circuit complexity and the W<br />
hierarchy.<br />
<i>Theoret. Computer Sci.</i>, 191(1–2):97–115, January 1998.<br />
<br />
<span id="dgp05" style="color:maroon">[DGP05]</span><br />
C. Daskalakis, P. W. Goldberg, and C. H. Papadimitriou<br />
The Complexity of Computing a Nash Equilibrium, online: [http://www.cs.berkeley.edu/~christos/papers/ppad.ps ppad.ps], sep. 2005.<br />
<br />
<span id="dhi02" style="color:maroon">[DHI02]</span><br />
W. van Dam, S. Hallgren, and L. Ip.<br />
Quantum algorithms for hidden shift problems,<br />
arXiv:[http://arxiv.org/abs/quant-ph/0211140 quant-ph/0211140], 2002.<br />
<br />
<span id="dp05" style="color:maroon">[DP05]</span><br />
C. Daskalakis and C. H. Papadimitriou<br />
The Complexity of Computing a Nash Equilibrium, online: [http://www.cs.berkeley.edu/~christos/papers/3players.pdf 3players.pdf], nov. 2005.<br />
<br />
{{Reference-ECCC<br />
|tag=DP08 |year=2008 |date=Feb 28 |eccc-num=TR08-014<br />
|authors=M. David and T. Pitassi<br />
|title=Separating NOF communication complexity classes RP and NP<br />
}}<br />
<br />
<span id="dw86" style="color:maroon">[DW86]</span><br />
E. Dahlhaus and M. K. Warmuth.<br />
Membership for growing context-sensitive grammars is polynomial,<br />
<i>Journal of Computer and System Sciences</i> 33:456-472, 1986.<br />
<br />
===== E =====<br />
<br />
<span id="edm65" style="color:maroon">[Edm65]</span><br />
J. Edmonds.<br />
Paths, trees, and flowers,<br />
<i>Canadian Journal of Mathematics</i> 17(3):449-467, 1965.<br />
<br />
<span id="ey07" style="color:maroon">[EY07]</span><br />
K. Etessami and M. Yannakakis.<br />
On the Complexity of Nash Equilibria and Other Fixed Points.<br />
Unpublished.<br />
<br />
Paths, trees, and flowers,<br />
<i>Canadian Journal of Mathematics</i> 17(3):449-467, 1965.<br />
<br />
===== F =====<br />
<br />
<span id="fag73" style="color:maroon">[Fag73]</span><br />
R. Fagin.<br />
Contributions to the Model Theory of Finite Strucutres,<br />
<i>Ph.D. Thesis (1973), U.C. Berkeley</i><br />
<br />
<span id="fag74" style="color:maroon">[Fag74]</span><br />
R. Fagin.<br />
Generalized first-order spectra and polynomial-time recognizable sets,<br />
<i>Complexity of Computation</i> (R. M. Karp, ed.), SIAM-AMS Proceedings Vol. 7, 1974.<br />
<br />
<span id="fen02" style="color:maroon">[Fen02]</span><br />
S. Fenner.<br />
PP-lowness and a simple definition of AWPP,<br />
<i>Theory Comput. Syst.</i> 36 (2003), no. 2, 199--212.<br />
ECCC [http://eccc.uni-trier.de/eccc-reports/2002/TR02-036/ TR02-036].<br />
<br />
<span id="ff.." style="color:maroon">[FF..]</span><br />
S. Fenner, L. Fortnow,<br />
Unpublished.<br />
<br />
<span id="ffk93" style="color:maroon">[FFK+93]</span><br />
S. Fenner, L. Fortnow, S. Kurtz, and L. Li.<br />
An oracle builder's toolkit,<br />
<i>Proceedings of Structure in Complexity Theory</i>, pages 120-131, 1993.<br />
[http://people.cs.uchicago.edu/~fortnow/papers/obt.ps http://people.cs.uchicago.edu/~fortnow/papers/obt.ps].<br />
<br />
<span id="ffk94" style="color:maroon">[FFK94]</span><br />
S. Fenner, L. Fortnow, and S. Kurtz.<br />
Gap-definable counting classes,<br />
<i>Journal of Computer and System Sciences</i> 48(1):116-148, 1994.<br />
[http://people.cs.uchicago.edu/~fortnow/papers/gaps.ps http://people.cs.uchicago.edu/~fortnow/papers/gaps.ps].<br />
<br />
<span id="fg02" style="color:maroon">[FG02]</span><br />
J. Flum and M. Grohe.<br />
The parameterized complexity of counting problems,<br />
<i>Proceedings of IEEE FOCS'2002</i>, 2002.<br />
<br />
<span id="fgh98" style="color:maroon">[FGH+98]</span><br />
S. Fenner, F. Green, S. Homer, and R. Pruim.<br />
Quantum NP is hard for PH,<br />
<i>Proceedings of the Sixth Italian Conference on Theoretical Computer Science</i>, World-Scientific, pp. 241-252, 1998.<br />
arXiv:[http://arxiv.org/abs/quant-ph/9812056 quant-ph/9812056].<br />
<br />
<span id="fgl91" style="color:maroon">[FGL+91]</span><br />
U. Feige, S. Goldwasser, L. Lov&aacute;sz, S. Safra, and M. Szegedy.<br />
Approximating clique is almost NP-complete,<br />
<i>Proceedings of IEEE FOCS'91</i>, pp. 2-12, 1991.<br />
<br />
<span id="fgmsz89" style="color:maroon">[FGM+89]</span><br />
M. Furer, O. Goldreich, Y. Mansour, M. Sipser, and S. Zachos.<br />
On Completeness and Soundness in Interactive Proof Systems.<br />
In <i>Advances in Computing Research: a research annual</i>,<br />
Vol.~5 (Randomness and Computation, S. Micali, ed.),<br />
pages 429--442, 1989.<br />
<br />
<span id="fie98" style="color:maroon">[Fie98]</span><br />
U. Feige.<br />
A threshold of ln(n) for approximating set cover.<br />
<i>Journal of the ACM (JACM)</i>, 45(4): 634--652, 1998.<br />
[http://dx.doi.org/10.1145/285055.285059 doi:10.1145/285055.285059]<br />
<br />
<span id="fk05" style="color:maroon">[FK05]</span><br />
L. Fortnow and A. Klivans.<br />
NP with small advice,<br />
<i>Proceedings of IEEE Complexity'2005</i>, pp. 228-234, 2005.<br />
[http://people.cs.uchicago.edu/~fortnow/papers/fk.ps http://people.cs.uchicago.edu/~fortnow/papers/fk.ps].<br />
<br />
<span id="fk97" style="color:maroon">[FK97]</span><br />
U. Feige and J. Kilian.<br />
Limited versus polynomial nondeterminism,<br />
<i>Chicago Journal of Theoretical Computer Science</i> Article 1, 1997.<br />
<br />
<span id="fk97b" style="color:maroon">[FK97b]</span><br />
U. Feige and J. Kilian.<br />
Making games short,<br />
<i>Proceedings of ACM STOC'1997</i>, pp. 506-516, 1997.<br />
<br />
<span id="for94" style="color:maroon">[For94]</span><br />
L. Fortnow.<br />
The role of relativization in complexity theory,<br />
<i>Bulletin of the EATCS</i> 52, February 1994.<br />
[http://people.cs.uchicago.edu/~fortnow/papers/relative.ps http://people.cs.uchicago.edu/~fortnow/papers/relative.ps].<br />
<br />
<span id="for02" style="color:maroon">[For02]</span><br />
J. Forster.<br />
A linear lower bound on the unbounded error probabilistic communication complexity,<br />
<i>Journal of Computer and System Sciences</i> 65(4):612-625, 2002.<br />
<br />
<span id="fr74" style="color:maroon">[FR74]</span><br />
M. J. Fischer and M. O. Rabin.<br />
Super-exponential complexity of Presburger arithmetic,<br />
<i>Complexity of Computation</i> (R. M. Karp, ed.), SIAM-AMS Symposium on Applied Mathematics, 1974.<br />
<br />
<span id="fr98" style="color:maroon">[FR98]</span><br />
L. Fortnow and J. D. Rogers.<br />
Complexity limitations on quantum computation,<br />
<i>Proceedings of IEEE Complexity'98</i>, pp. 202-209, 1998.<br />
arXiv:[http://arxiv.org/abs/cs.CC/9811023 cs.CC/9811023].<br />
<br />
<span id="fri57" style="color:maroon">[Fri57]</span><br />
R. M. Friedberg.<br />
Two recursively enumerable sets of incomparable degrees of unsolvability,<br />
<i>Proceedings of the National Academy of Sciences</i>, 43:236-238, 1957.<br />
[http://www.pubmedcentral.nih.gov/picrender.fcgi?artid=528418&amp;blobtype=pdf http://www.pubmedcentral.nih.gov/picrender.fcgi?artid=528418&amp;blobtype=pdf].<br />
<br />
<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 />
<br />
<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 />
<br />
<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 />
<br />
<span id="fur07" style="color:maroon">[Fur07]</span><br />
M. Furer.<br />
Fast Integer Multiplication,<br />
STOC, 2007.<br />
<br />
<span id="fv93" style="color:maroon">[FV93]</span><br />
T. Feder and M. Y. Vardi.<br />
Monotone monadic SNP and constraint satisfaction,<br />
<i>Proceedings of the 25th ACM Symposium on Theory of Computing</i>, pp. 612-622, 1993.<br />
DOI:[http://doi.acm.org/10.1145/167088.167245 10.1145/167088.167245].<br />
<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 />
[http://portal.acm.org/citation.cfm?id=114454&dl=GUIDE&coll=GUIDE&CFID=74222314&CFTOKEN=30698817]<br />
<br />
<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 />
<br />
<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 />
<br />
<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 />
Extended abstract appeared in <i>Proceeedings of the 39th International Colloquium on Automata, Languages, and Programming (ICALP)</i>, pages 387-398, Springer, 2012.<br />
<br />
<span id="gy16" style="color:maroon">[GY16]</span><br />
S. Gharibian, and J. Yirka.<br />
The complexity of estimating local physical quantities, <br />
arXiv:[http://arxiv.org/abs/1606.05626 quant-ph/1606.05626], 2016.<br />
<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 />
<br />
<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 />
arXiv:[http://arxiv.org/abs/1302.0290], 2013.<br />
<br />
{{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 />
<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="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 />
<i>Proceedings of ICALP'16</i>, to appear, 2016.<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="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 />
<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 />
arXiv:[http://arxiv.org/abs/quant-ph/0211179 quant-ph/0211179], 2002.<br />
<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 />
In <i>Proceedings of the 39th ACM Symposium on Theory of Computing (STOC'07)</i>, pages 565-574, 2007.<br />
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 />
arXiv:[http://arxiv.org/abs/1011.2787 arXiv:1011.2787].<br />
<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="gw14" style="color:maroon">[GW14]</span><br />
M. Göös and T. Watson.<br />
Communication Complexity of Set-Disjointness for All Probabilities,<br />
<i>Proceedings of RANDOM'14</i>, 721-736, 2014.<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.cse.psu.edu/~hallgren/pell.pdf http://www.cse.psu.edu/~hallgren/pell.pdf].<br />
<br />
<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 />
Some optimal inapproximability results,<br />
''Journal of the ACM'', 48(4):798-3859, 2001.<br />
[http://www-sunos4.nada.kth.se/~johanh/optimalinap.ps http://www-sunos4.nada.kth.se/~johanh/optimalinap.ps]<br />
<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 />
<br />
<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 />
<br />
<span id="hes01" style="color:maroon">[Hes01]</span><br />
W. Hesse.<br />
Division is in uniform TC<sup>0</sup>,<br />
<i>Proceedings of the International Colloquium on Automata, Languages, and Programming (ICALP)</i>, 2001.<br />
[http://www.cs.umass.edu/~whesse/div.ps http://www.cs.umass.edu/~whesse/div.ps]<br />
<br />
<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 />
<br />
<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 />
arXiv:[http://arxiv.org/abs/cs.CC/9910002 cs.CC/9910002].<br />
<br />
<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 />
<br />
<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 />
<br />
<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 />
<br />
<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="hm13" style="color:maroon">[HM13]</span><br />
A. W. Harrow and A. Montanaro.<br />
Testing product states, quantum Merlin-Arthur games and tensor optimisation,<br />
<i>Journal of the ACM</i> vol. 60 no. 1, 2013<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 />
<br />
<span id="hn06" style="color:maroon">[HN06]</span><br />
D. Harnik and M. Naor.<br />
On the compressibility of NP instances and cryptographic applications.<br />
<i>Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS)</i>, 719-728, 2006.<br />
[http://www.cs.technion.ac.il/~harnik/Compress.pdf http://www.cs.technion.ac.il/~harnik/Compress.pdf]<br />
<br />
<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 />
Springer-Verlag, 2002.<br />
See also [http://www.cs.rochester.edu/u/lane/=companion/ http://www.cs.rochester.edu/u/lane/=companion/].<br />
<br />
<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="hr90" style="color:maroon">[HR90]</span><br />
B. Halstenberg and R. Reischuk.<br />
Relations between communication complexity classes,<br />
<i>Journal of Computer and System Sciences</i> 41(3):402-429, 1990.<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 />
[http://www.thi.uni-hannover.de/forschung/publikationen/daten/he-re-vo99.ps.gz http://www.thi.uni-hannover.de/forschung/publikationen/daten/he-re-vo99.ps.gz]<br />
<br />
<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 />
<br />
<span id="ht03" style="color:maroon">[HT03]</span><br />
C. M. Homan and M. Thakur.<br />
One-way permutations and self-witnessing languages,<br />
<i>Journal of Computer and System Sciences</i> 67 (2003), 608-622.<br />
[https://www.sciencedirect.com/science/article/pii/S0022000003000680].<br />
<br />
<span id="ht06" style="color:maroon">[HT06]</span><br />
L. Hella and J. M. Turull-Torres.<br />
Computing queries with higher-order logics,<br />
<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 />
<br />
<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 />
In search of an easy witness: exponential time vs. probabilistic polynomial time,<br />
<i>Proceedings of IEEE Complexity'2001</i>, 2001.<br />
[http://www.cs.sfu.ca/~kabanets/papers/exp_journal.ps.gz http://www.cs.sfu.ca/~kabanets/papers/exp_journal.ps.gz]<br />
<br />
<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 />
<br />
{{Reference<br />
|tag=IM03<br />
|title=A zero one law for RP<br />
|authors=R. Impagliazzo and P. Moser<br />
|journal=Proceedings of the 18th Conference on Computational Complexity<br />
|srcdetail=48-52. IEEE Computer Society Press, 2003<br />
}}<br />
<br />
{{Reference<br />
|tag=Imp95<br />
|title=A personal view of average-case complexity<br />
|authors=R. Impagliazzo<br />
|journal=Proceedings of the 10th Conference on Structure in Complexity Theory<br />
|srcdetail=134-147. IEEE Computer Society Press, 1995<br />
}}<br />
<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 />
Expressibility and Parallel Complexity<br />
<i>SIAM Journal on Computing</i>, 18:625-638, 1989.<br />
[http://www.cs.umass.edu/~immerman/pub/parallel.pdf]<br />
<br />
<span id="imm98" style="color:maroon">[Imm98]</span><br />
N. Immerman.<br />
<i>Descriptive Complexity</i>,<br />
Springer Graduate Texts in Computer Science, 1998.<br />
<br />
<span id="imp02" style="color:maroon">[Imp02]</span><br />
R. Impagliazzo.<br />
Hardness as randomness: a survey of universal derandomization,<br />
<i>Proceedings of the ICM</i>, Beijing, vol. 3, pp. 649-658, 2002.<br />
arXiv:[http://arxiv.org/abs/cs.CC/0304040 cs.CC/0304040].<br />
<br />
<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 />
[http://www.wisdom.weizmann.ac.il/~naor/PAPERS/subset.ps.gz http://www.wisdom.weizmann.ac.il/~naor/PAPERS/subset.ps.gz]<br />
<br />
<span id="ipz01" style="color:maroon">[IPZ01]</span><br />
R. Impagliazzo, R. Paturi, and F. Zane.<br />
Which problems have strongly exponential complexity,<br />
<i>Journal of Computer and System Sciences</i> 63(4):512-530, 2001.<br />
[http://cm.bell-labs.com/cm/ms/who/francis/papers/focs98-subexp.pdf http://cm.bell-labs.com/cm/ms/who/francis/papers/focs98-subexp.pdf].<br />
<br />
<span id="is91" style="color:maroon">[IS91]</span><br />
R. Impagliazzo and M. Sudan.<br />
Private communication,<br />
cited in [#bg94" style="color:maroon">[BG94], 1991.<br />
<br />
<span id="it89" style="color:maroon">[IT89]</span><br />
R. Impagliazzo and G. Tardos.<br />
Decision versus search problems in super-polynomial time,<br />
in <i>Proceedings of IEEE FOCS 1989</i>, pp. 222-227, 1989.<br />
<br />
<span id="iv12" style="color:maroon">[IV12]</span><br />
T. Ito and T. Vidick.<br />
A multi-prover interactive proof for NEXP sound against entangled provers,<br />
to appear in <i>Proceedings of IEEE FOCS 2012</i><br />
arXiv:[http://arxiv.org/abs/1207.0550 1207.0550].<br />
<br />
<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 />
Approximate counting in bounded arithmetic,<br />
''Journal of Symbolic Logic'' 72(3):959–993, 2007.<br />
<br />
<span id="jer12" style="color:maroon">[Jeř12]</span><br />
E. Jeřábek.<br />
Integer factoring and modular square roots,<br />
http://arxiv.org/abs/1207.5220<br />
<br />
<span id="JJUW09" style="color:maroon">[JJUW09]</span><br />
R. Jain, Z. Ji, S. Upadhyay, and J. Watrous.<br />
QIP = PSPACE,<br />
arXiv:[http://arxiv.org/abs/0907.4737 0907.4737], 2009.<br />
<br />
<span id="jks02" style="color:maroon">[JKS02]</span><br />
J. C. Jackson, A. R. Klivans, and R. A. Servedio.<br />
Learnability beyond AC<sup>0</sup>,<br />
<i>Proceedings of ACM STOC'2002</i>, pp. 776-784, 2002.<br />
<br />
<span id="jl95" style="color:maroon">[JL95]</span><br />
D. W. Juedes and J. H. Lutz.<br />
The complexity and distribution of hard problems,<br />
<i>SIAM Journal on Computing</i> 24(2):279-295, 1995.<br />
[http://www.cs.iastate.edu/~lutz/%3DPAPERS/cdhp.ps http://www.cs.iastate.edu/~lutz/%3DPAPERS/cdhp.ps]<br />
<br />
<span id="joh90" style="color:maroon">[Joh90]</span><br />
D. S. Johnson.<br />
A catalog of complexity classes,<br />
chapter 2 in <i>Handbook of Theoretical Computer Science</i>, Volume A (J. van Leeuwen, editor), Elsevier, 1990.<br />
<br />
<span id="jon98" style="color:maroon">[Jon98]</span><br />
N. D. Jones.<br />
Logspace and Ptime Characteried by Programming Languages,<br />
[ftp://ftp.diku.dk/pub/diku/semantics/papers/D-398.ps.gz]<br />
<br />
<span id="jpy88" style="color:maroon">[JPY88]</span><br />
D. S. Johnson, C. H. Papadimitriou, and M. Yannakakis.<br />
How easy is local search?,<br />
<i>Journal of Computer and System Sciences</i> 37:79-100, 1988.<br />
<br />
<span id="jsv01" style="color:maroon">[JSV01]</span><br />
M. Jerrum, A. Sinclair, and E. Vigoda.<br />
A polynomial-time approximation algorithm for the permanent of a matrix with non-negative entries,<br />
<i>Proceedings of ACM STOC'2001</i>, pp. 712-721, 2001.<br />
ECCC [http://eccc.uni-trier.de/eccc-reports/2000/TR00-079/ TR00-079].<br />
<br />
<span id="jun85" style="color:maroon">[Jun85]</span><br />
H. Jung.<br />
On probabilistic time and space,<br />
<i>Proceedings of 12th International Colloquium on Automata, Languages, and Programming (ICALP)</i>, Lecture Notes in Computer Science, 194:310-317, 1985.<br />
<br />
<span id="jw04" style="color:maroon">[JW04]</span><br />
M. Jerrum and U. Wagner.<br />
<i>Counting, Sampling, and Integrating: Algorithms and Complexity</i>,<br />
Chapter 3 (lecture notes labeled as under construction).<br />
[http://www.dcs.ed.ac.uk/home/mrj/ETHbook/chap3.ps http://www.dcs.ed.ac.uk/home/mrj/ETHbook/chap3.ps].<br />
<br />
<span id="jw09" style="color:maroon">[JW09]</span><br />
R. Jain and J. Watrous.<br />
Parallel approximation of non-interactive zero-sum quantum games,<br />
<i>Proceedings of the 24th Annual IEEE Conference on Computational Complexity</i>, pages 243–253, 2009.<br />
arXiv:[http://arxiv.org/abs/0808.2775 0808.2775 [quant-ph]].<br />
<br />
<span id="jwb03" style="color:maroon">[JWB03]</span><br />
D. Janzing, P. Wocjan, and T. Beth.<br />
Cooling and low energy state preparation for 3-local Hamiltonians are FQMA-complete,<br />
arXiv:[http://arxiv.org/abs/quant-ph/0303186 quant-ph/0303186], 2003.<br />
<br />
<span id="jy88" style="color:maroon">[JY88]</span><br />
D. S. Johnson and M. Yannakakis.<br />
On generating all maximal independent sets,<br />
<i>Information Processing Letters</i> 27(3):119-123, 1988.<br />
<br />
===== K =====<br />
<br />
<span id="kad88" style="color:maroon">[Kad88]</span><br />
J. Kadin.<br />
The polynomial time hierarchy collapses if the Boolean hierarchy collapses,<br />
<i>SIAM Journal on Computing</i> 17:1263-1282, 1988.<br />
<br />
<span id="kan82" style="color:maroon">[Kan82]</span><br />
R. Kannan.<br />
Circuit-size lower bounds and non-reducibility to sparse sets,<br />
<i>Information and Control</i> 55:40-56, 1982.<br />
<br />
<span id="kar72" style="color:maroon">[Kar72]</span><br />
R. M. Karp.<br />
Reducibility among combinatorial problems,<br />
<i>Complexity of Computer Computations</i> (J. W. Thatcher and R. E. Miller, eds.), Plenum Press, 1972.<br />
<br />
<span id="kar86" style="color:maroon">[Kar86]</span><br />
H. Karloff.<br />
A Las Vegas algorithm for maximum matching,<br />
<i>Combinatorica</i> 6:387-392, 1986.<br />
<br />
<span id="kf84" style="color:maroon">[KF84]</span><br />
C. M. R. Kintala and P. Fischer.<br />
Refining nondeterminism in relativized complexity classes,<br />
<i>SIAM Journal on Computing</i> 13:329-337, 1984.<br />
<br />
<span id="kha79" style="color:maroon">[Kha79]</span><br />
L. G. Khachiyan.<br />
A polynomial algorithm for linear programming,<br />
<i>Soviet Math Doklady</i> 20:191-194, 1979.<br />
<br />
<span id="kha93" style="color:maroon">[Kha93]</span><br />
M. Kharitonov.<br />
Cryptographic hardness of distribution-specific learning,<br />
<i>Proceedings of ACM STOC'93</i>, pp. 372-381, 1993.<br />
<br />
<span id="ki02" style="color:maroon">[KI02]</span><br />
V. Kabanets and R. Impagliazzo.<br />
Derandomizing polynomial identity tests means proving circuit lower bounds,<br />
ECCC [http://eccc.uni-trier.de/eccc-reports/2002/TR02-055/ TR02-055], 2002.<br />
<br />
<span id="kkr04" style="color:maroon">[KKR04]</span><br />
J. Kempe, A. Kitaev, and O. Regev.<br />
The Complexity of the Local Hamiltonian Problem,<br />
<i>SIAM Journal of Computing</i>, Vol. 35(5), p. 1070-1097 (2006).<br />
arXiv:[http://arxiv.org/abs/quant-ph/0406180 quant-ph/0406180].<br />
<br />
<span id="kl82" style="color:maroon">[KL82]</span><br />
R. M. Karp and R. J. Lipton.<br />
Turing machines that take advice,<br />
<i>Enseign. Math.</i> 28:191-201, 1982.<br />
<br />
<span id="kla03" style="color:maroon">[Kla03]</span><br />
H. Klauck.<br />
Rectangle size bounds and threshold covers in communication complexity,<br />
<i>Proceedings of IEEE CCC'03</i>, pp. 118-134, 2003.<br />
<br />
<span id="kla07" style="color:maroon">[Kla07]</span><br />
H. Klauck.<br />
Lower bounds for quantum communication complexity,<br />
<i>SIAM Journal on Computing</i> 37(1):20-46, 2007.<br />
<br />
<span id="kla11" style="color:maroon">[Kla11]</span><br />
H. Klauck.<br />
On Arthur Merlin games in communication complexity,<br />
<i>Proceedings of IEEE CCC'11</i>, pp. 189-199, 2011.<br />
<br />
<span id="kle71" style="color:maroon">[Kle71]</span><br />
S. C. Kleene.<br />
<i>Introduction to Metamathematics</i>,<br />
Elsevier, 1971.<br />
<br />
<span id="km02" style="color:maroon">[KM02]</span><br />
H. Kobayashi and K. Matsumoto.<br />
Quantum multi-prover interactive proof systems with limited prior entanglement,<br />
<i>Proceedings of ISAAC'2002</i>, pp. 115-127, 2002.<br />
arXiv:[http://arxiv.org/abs/cs.CC/0102013 cs.CC/0102013].<br />
<br />
<span id="km92" style="color:maroon">[KM92]</span><br />
D. Koller and N. Megiddo.<br />
On the Complexity of Two-person Zero-sum Games in Extensive Form,<br />
Games and Economic Behavior 4, 528-552, 1992.<br />
[http://theory.stanford.edu/~megiddo/pdf/recall.pdf http://theory.stanford.edu/~megiddo/pdf/recall.pdf]<br />
<br />
<span id="km99" style="color:maroon">[KM99]</span><br />
A. Klivans and D. van Melkebeek.<br />
Graph nonisomorphism has subexponential size proofs unless the polynomial hierarchy collapses,<br />
in <i>Proceedings of ACM STOC'99</i>, pp. 659-667, 1999.<br />
<br />
<span id="kms99" style="color:maroon">[KMS+99]</span><br />
S. Khanna, R. Motwani, M. Sudan, and U. Vazirani.<br />
On syntactic versus computational views of approximability,<br />
<i>SIAM Journal on Computing</i> 28:164-191, 1999.<br />
ECCC [http://eccc.uni-trier.de/eccc-reports/1995/TR95-023/ TR95-023].<br />
<br />
<span id="kmsy14" style="color:maroon">[KMSY14]</span><br />
G. Kol, S. Moran, A. Shpilka, and A. Yehudayoff.<br />
Approximate nonnegative rank is equivalent to the smooth rectangle bound,<br />
<i>Proceedings of the ICALP'14</i>, pp. 701-712, 2014.<br />
<br />
<span id="kmy01" style="color:maroon">[KMY01]</span><br />
H. Kobayashi, K. Matsumoto, and T. Yamakami.<br />
Quantum certificate verification: single versus multiple quantum certificates,<br />
arXiv:[http://arxiv.org/abs/quant-ph/0110006 quant-ph/0110006], 2001.<br />
<br />
<span id="ko82" style="color:maroon">[Ko82]</span><br />
K. Ko.<br />
Some observations on the probabilistic algorithms and NP-hard problems,<br />
<i>Information Processing Letters</i> 14(1):39-43, 1982.<br />
<br />
<span id="ko85" style="color:maroon">[Ko85]</span><br />
K. Ko.<br />
On some natural complete operators,<br />
<i>Theoretical Computer Science</i> 37(1):1-30, 1985.<br />
<br />
<span id="kob02" style="color:maroon">[Kob02]</span><br />
H. Kobayashi.<br />
Non-interactive quantum statistical and perfect zero-knowledge,<br />
arXiv:[http://arxiv.org/abs/quant-ph/0207158 quant-ph/0207158], 2002.<br />
<br />
<span id="kob89" style="color:maroon">[Kob89]</span><br />
J. K&ouml;bler.<br />
<i>Strukturelle Komplexit&auml;t von Anzahlproblemen</i>,<br />
PhD thesis, Universit&auml;t Stuttgart, 1989.<br />
<br />
<span id="koi96" style="color:maroon">[Koi96]</span><br />
P. Koiran.<br />
Hilbert's Nullstellensatz is in the polynomial hierarchy,<br />
<i>Journal of Complexity</i> 12(4):273-286, 1996,<br />
DIMACS [http://dimacs.rutgers.edu/TechnicalReports/abstracts/1996/96-27.html TR 96-27].<br />
<br />
<span id="koz92" style="color:maroon">[Koz92]</span><br />
D. C. Kozen.<br />
On the Myhill-Nerode theorem for trees,<br />
<i>Bulletin of the EATCS</i> 47:170-173, 1992.<br />
<br />
<span id="koz97" style="color:maroon">[Koz97]</span><br />
D. C. Kozen.<br />
<i>Automata and Computability</i>,<br />
Springer-Verlag, 1997.<br />
<br />
<span id="kp89" style="color:maroon">[KP89]</span><br />
J. Krajicek and P. Pudlak.<br />
Propositional Proof Systems, the Consistency of First Order Theories and the Complexity of Computations,<br />
<i>J. Symb. Log.</i>, 54:1063-79, 1989.<br />
<br />
<span id="kr03" style="color:maroon">[KR03]</span><br />
J. Kempe and O. Regev.<br />
3-Local Hamiltonian is QMA-complete,<br />
<i>Quantum Inf. Comput.</i>, 3(3):258-264, 2003.<br />
arXiv:[http://arxiv.org/abs/quant-ph/0302079 quant-ph/0302079].<br />
<br />
<span id="kra.." style="color:maroon">[Kra..]</span><br />
H. Krawczyk.<br />
Unpublished.<br />
<br />
<span id="krc00" style="color:maroon">[KRC00]</span><br />
V. Kabanets, C. Rackoff, and S. A. Cook.<br />
Efficiently approximable real-valued functions,<br />
ECCC [http://eccc.uni-trier.de/eccc-reports/2000/TR00-034/ TR00-034], 2000.<br />
<br />
<span id="kre88" style="color:maroon">[Kre88]</span><br />
M. Krentel.<br />
The complexity of optimization problems,<br />
<i>Journal of Computer and System Sciences</i> 36:490-509, 1988.<br />
<br />
<span id="krs90" style="color:maroon">[KRS90]</span><br />
C. P. Kruskal, L. Rudolph, and M. Snir.<br />
A complexity theory of efficient parallel algorithms,<br />
<i>Theoretical Computer Science</i> 71:95-132, 1990.<br />
<br />
{{Reference<br />
|tag=KS05<br />
|authors=N. Kayal and N. Saxena<br />
|title=On the ring isomorphism and automorphism problems<br />
|journal=Proceedings of the 20th Annual IEEE Conference on Computational Complexity<br />
|srcdetail=2-12, 2005}}<br />
<br />
<span id="kst89" style="color:maroon">[KST+89]</span><br />
J. K&ouml;bler, U. Sch&ouml;ning, and J. Tor&aacute;n.<br />
On counting and approximation,<br />
<i>Acta Informatica</i> 26:363-379, 1989.<br />
<br />
<span id="kst89b" style="color:maroon">[KST+89b]</span><br />
J. K&ouml;bler, U. Sch&ouml;ning, S. Toda, and J. Tor&aacute;n.<br />
Turing machines with few accepting computations and low sets for PP,<br />
<i>Proceedings of IEEE Complexity'89</i>, pp. 208-215, 1989.<br />
[http://www.informatik.hu-berlin.de/Forschung_Lehre/algorithmenII/Papers/few.ps.gz http://www.informatik.hu-berlin.de/Forschung_Lehre/algorithmenII/Papers/few.ps.gz]<br />
<br />
<span id="kst92" style="color:maroon">[KST92]</span><br />
J. K&ouml;bler, U. Sch&ouml;ning, and J. Tor&aacute;n.<br />
Graph isomorphism is low for PP,<br />
<i>Computational Complexity</i> 2:301-330, 1992.<br />
<br />
<span id="kst93" style="color:maroon">[KST93]</span><br />
J. K&ouml;bler, U. Sch&ouml;ning, and J. Tor&aacute;n.<br />
<i>The Graph Isomorphism Problem: Its Structural Complexity</i>,<br />
Birkh&auml;user, 1993.<br />
<br />
<span id="ksv02" style="color:maroon">[KSV02]</span><br />
A. Kitaev, A. Shen, and M. N. Vyalyi.<br />
<i>Classical and Quantum Computation</i>,<br />
American Mathematical Society, 2002.<br />
<br />
<span id="kt94" style="color:maroon">[KT94]</span><br />
P. G. Koliatis and M. N. Thakur.<br />
Logical definability of NP optimization problems,<br />
<i>Information and Computation</i> 115:321-353, 1994.<br />
<br />
<span id="kt96" style="color:maroon">[KT96]</span><br />
J. K&ouml;bler and S. Toda.<br />
On the power of generalized MOD-classes,<br />
<i>Mathematical Systems Theory</i> 29(1):33-46, 1996.<br />
[ftp://theorie.informatik.uni-ulm.de/pub/papers/ti/mod.ps.gz ftp://theorie.informatik.uni-ulm.de/pub/papers/ti/mod.ps.gz]<br />
<br />
<span id="kup09" style="color:maroon">[Kup09]</span><br />
G. Kuperberg.<br />
How hard is it to approximate the Jones polynomial?, 2009.<br />
arXiv:[http://arxiv.org/abs/0908.0512 quant-ph/0908.0512v1].<br />
<br />
<span id="kur64" style="color:maroon">[Kur64]</span><br />
S. Y. Kuroda.<br />
Classes of languages and linear-bounded automata,<br />
<i>Information and Control</i> 7:207-233, 1964.<br />
<br />
<span id="kur83" style="color:maroon">[Kur83]</span><br />
S. Kurtz.<br />
On the random oracle hypothesis,<br />
<i>Information and Control</i> 57:40-47, 1983.<br />
<br />
<span id="kur85" style="color:maroon">[Kur85]</span><br />
S. Kurtz.<br />
On Relativized Exponential and Probabilistic Complexity Classes,<br />
<i>Information and Control</i> 71:231-243, 1985.<br />
<br />
<span id="kuw86" style="color:maroon">[KUW86]</span><br />
R. Karp, E. Upfal, and A. Wigderson.<br />
Constructing a perfect matching is in random NC,<br />
<i>Combinatorica</i> 6:35-48, 1986.<br />
<br />
<span id="kv88" style="color:maroon">[KV88]</span><br />
M. Karpinski and R. Verbeek.<br />
Randomness, provability, and the separation of Monte Carlo time and space,<br />
<i>Lecture Notes in Computer Science</i> 270, pp. 189-207, Springer, 1988.<br />
<br />
<span id="kv94" style="color:maroon">[KV94]</span><br />
M. Kearns and L. Valiant.<br />
Cryptographic limitations on learning Boolean formulae and finite automata,<br />
<i>Journal of the ACM</i> 41(1):67-95, 1994.<br />
[http://www.cis.upenn.edu/~mkearns/papers/crypto.pdf http://www.cis.upenn.edu/~mkearns/papers/crypto.pdf]<br />
<br />
<span id="kv96" style="color:maroon">[KV96]</span><br />
M. Karpinski and R. Verbeek.<br />
On Randomized Versus Deterministic Computation,<br />
Theoret. Comput. Sci. 154 (1996), no. 1, 23--39.<br />
ECCC [http://eccc.uni-trier.de/eccc-reports/1995/TR95-021/ TR95-021].<br />
<br />
<span id="kw.." style="color:maroon">[KW..]</span><br />
A. Kitaev and J. Watrous.<br />
Unpublished.<br />
<br />
<span id="kw00" style="color:maroon">[KW00]</span><br />
A. Kitaev and J. Watrous.<br />
Parallelization, amplification, and exponential time simulation of quantum interactive proof systems,<br />
<i>Proceedings of ACM STOC'2000</i>, pp. 608-617, 2000.<br />
[https://cs.uwaterloo.ca/~watrous/Papers/QuantumInteractiveProofs.pdf https://cs.uwaterloo.ca/~watrous/Papers/QuantumInteractiveProofs.pdf]<br />
<br />
<span id="kw88" style="color:maroon">[KW88]</span><br />
M. Karchmer and A. Wigderson.<br />
Monotone circuits for connectivity require superlogarithmic depth,<br />
<i>Proceedings of ACM STOC'88</i>, pp. 539-550, 1988.<br />
<br />
<span id="kw93" style="color:maroon">[KW93]</span><br />
M. Karchmer and A. Wigderson.<br />
On span programs,<br />
<i>Proceedings of IEEE Complexity'93</i>, pp. 102-111, 1993.<br />
<br />
<span id="kw98" style="color:maroon">[KW98]</span><br />
J. K&ouml;bler and O. Watanabe.<br />
New collapse consequences of NP having small circuits,<br />
<i>SIAM Journal on Computing</i> 28(1):311-324, 1998.<br />
[http://www.informatik.hu-berlin.de/forschung/gebiete/algorithmenII/Publikationen/Papers/collapse.ps.gz http://www.informatik.hu-berlin.de/forschung/gebiete/algorithmenII/Publikationen/Papers/collapse.ps.gz]<br />
<br />
<span id="kw15" style="color:maroon">[KW15]</span><br />
J. Kwisthout.<br />
Tree-width and the computational complexity of MAP approximations in Bayesian networks,<br />
<i>Journal of Artificial Intelligence Research</i> 53:699-720, 2015.<br />
[http://www.socsci.ru.nl/johank/tree-width_PP.pdf http://www.socsci.ru.nl/johank/tree-width_PP.pdf]<br />
<br />
===== L =====<br />
<br />
<span id="lad75" style="color:maroon">[Lad75]</span><br />
R. Ladner.<br />
On the structure of polynomial time reducibility,<br />
<i>Journal of the ACM</i> 22:155-171, 1975.<br />
<br />
<span id="lau83" style="color:maroon">[Lau83]</span><br />
C. Lautemann.<br />
BPP and the polynomial time hierarchy,<br />
<i>Information Processing Letters</i> 17:215-218, 1983.<br />
<br />
<span id="lee02" style="color:maroon">[Lee02]</span><br />
T. Lee.<br />
Arithmetical definability over finite structures,<br />
Mathematical Logic Quarterly, Vol. 49(4), 2003. <br />
[http://www.lri.fr/~lee/arith.pdf http://www.lri.fr/~lee/arith.pdf].<br />
<br />
<span id="lev73" style="color:maroon">[Lev73]</span><br />
L. A. Levin.<br />
Universal search problems (in Russian),<br />
<i>Problemy Peredachi Informatsii</i> 9(3):265-266, 1973.<br />
<br />
<span id="lev86" style="color:maroon">[Lev86]</span><br />
L. A. Levin.<br />
Average case complete problems,<br />
<i>SIAM Journal on Computing</i>, 15:285-286, 1986.<br />
<br />
<span id="lfk90" style="color:maroon">[LFK+90]</span><br />
C. Lund, L. Fortnow, H. Karloff, and N. Nisan.<br />
Algebraic methods for interactive proofs,<br />
<i>Proceedings of IEEE FOCS'90</i>, pp. 1-10, 1990.<br />
[http://people.cs.uchicago.edu/~fortnow/papers/ip.ps http://people.cs.uchicago.edu/~fortnow/papers/ip.ps]<br />
<br />
<span id="li93" style="color:maroon">[Li93]</span><br />
L. Li.<br />
<i>On the Counting Functions</i>,<br />
PhD thesis, University of Chicago, 1993.<br />
<br />
{{Reference<br />
|tag=LiRe93<br />
|authors=M. Liskiewicz, R. Reischuk<br />
|title=The complexity world below logarithmic space<br />
|journal=Proceedings of the Structure in Complexity Theory Conference<br />
|srcdetail=1993, 64-78<br />
}}<br />
<br />
<span id="li11" style="color:maroon">[Li11]</span><br />
Y. D. Li.<br />
BQP and PPAD,<br />
<i>Electronic Colloquium on Computational Complexity</i> TR11-103, 2011.<br />
<br />
<span id="ll76" style="color:maroon">[LL76]</span><br />
R. Ladner and N. A. Lynch.<br />
Relativization of questions about log space computability,<br />
<i>Mathematical Systems Theory</i> 10:19-32, 1976.<br />
<br />
<span id="lmn93" style="color:maroon">[LMN93]</span><br />
N. Linial, Y. Mansour, and N. Nisan.<br />
Constant depth circuits, Fourier transform, and learnability,<br />
<i>Journal of the ACM</i> 40(3):607-620, 1993.<br />
<br />
<span id="lmt97" style="color:maroon">[LMT97]</span><br />
K. Lange, P. McKenzie, and A. Tapp.<br />
Reversible space equals deterministic space (extended abstract),<br />
<i>Proceedings of IEEE FOCS'97</i>, pp. 45-50, 1997.<br />
<br />
<span id="lp82" style="color:maroon">[LP82]</span><br />
H. R. Lewis and C. H. Papadimitriou.<br />
Symmetric space-bounded computation,<br />
<i>Theoretical Computer Science</i> 19:161-187, 1982.<br />
<br />
<span id="ls74" style="color:maroon">[LS74]</span><br />
E. A. Lamagna and J. E. Savage<br />
Combinational complexity of some monotone functions,<br />
<i>FOCS</i> 140-44, 1974.<br />
<br />
<span id="lut91" style="color:maroon">[Lut91]</span><br />
J. H. Lutz.<br />
An upward measure separation theorem,<br />
<i>Theoretical Computer Science</i> 81:127-135, 1991.<br />
<br />
{{Reference<br />
|tag=Lut93<br />
|authors=J. H. Lutz<br />
|title=The quantitative structure of exponential time<br />
|journal=Proc. 8th Structure in Complexity Theory Conference<br />
|srcdetail=(IEEE Comput. Soc. Press, 1993) 158-175<br />
}}[http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.53.1845&rep=rep1&type=pdf]<br />
<br />
<span id="lv97" style="color:maroon">[LV97]</span><br />
M. Li and P. Vit&aacute;nyi.<br />
<i>An Introduction to Kolmogorov Complexity and Its Applications</i> (second edition),<br />
Springer, 1997.<br />
<br />
===== M =====<br />
<br />
<span id="m08" style="color:maroon">[M08]</span><br />
L. Malka.<br />
How to achieve perfect simulation, and a complete problem for non-interactive perfect zero-knowledge. <i>IACR 5th Theory of Cryptography Conference (TCC)</i>, 2008.<br />
[http://www.cs.uvic.ca/~liorma/publications/NIZK9.pdf].<br />
<br />
<span id="mah82" style="color:maroon">[Mah82]</span><br />
S. R. Mahaney.<br />
Sparse complete sets for NP: Solution of a conjecture by Berman and Hartmanis,<br />
<i>Journal of Computer and System Sciences</i> 25:130-143, 1982.<br />
<br />
<span id="may94" style="color:maroon">[May94]</span><br />
E. Mayordomo.<br />
Almost every set in exponential time is P-bi-immune,<br />
<i>Theoretical Computer Science</i> 136(2):487-506, 1994.<br />
<br />
<span id="may94b" style="color:maroon">[May94b]</span><br />
E. Mayordomo.<br />
<i>Contributions to the study of resource-bounded measure</i>,<br />
PhD thesis, Universitat Politecnica de Catalunya, 1994.<br />
<br />
<span id="ms89" style="color:maroon">[MS89]</span><br />
E. W. Mayr and A. Subramanian. <br />
The complexity of circuit value and network stability, Proceedings of the Fourth Annual Conference on <i>Structure in Complexity Theory</i>, pp.114-123, 1989.<br />
<br />
<span id="mc00" style="color:maroon">[MC00]</span><br />
C. Moore and J. P. Crutchfield.<br />
Quantum automata and quantum grammars,<br />
<i>Theoretical Computer Science</i> 237:275-306, 2000.<br />
<br />
{{Reference<br />
|tag=Mel00<br />
|authors=D. Melkebeek<br />
|title=The zero-one law holds for BPP<br />
|journal=Theoretical Computer Science<br />
|srcdetail=Volume 244, Issues 1-2, 6 August 2000, Pages 283-288<br />
}}<br />
<br />
<span id="mes99" style="color:maroon">[Mes99]</span><br />
J. Messner.<br />
On optimal algorithms and optimal proof systems,<br />
<i>Lecture Notes in Computer Science</i> 1563:541-550, 1999.<br />
<br />
<span id="mil76" style="color:maroon">[Mil76]</span><br />
G. Miller.<br />
Riemann's hypothesis and tests for primality,<br />
<i>Journal of Computer and System Sciences</i>, 13:300-317, 1976.<br />
<br />
<span id="mil92" style="color:maroon">[Mil92]</span><br />
P. B. Miltersen.<br />
Circuit depth relative to a random oracle,<br />
<i>Information Processing Letters</i> 42(6):295-298, 1992.<br />
<br />
<span id="mn02" style="color:maroon">[MN02]</span><br />
C. Moore and M. Nilsson.<br />
Parallel quantum computation and quantum codes,<br />
<i>SIAM Journal on Computing</i> 31(3):799-815, 2002.<br />
arXiv:[http://arxiv.org/abs/quant-ph/9808027 quant-ph/9808027].<br />
<br />
<span id="mon80" style="color:maroon">[Mon80]</span><br />
B. Monien.<br />
On a subclass of pseudopolynomial problems,<br />
<i>Mathematical Foundations of Computer Science (MFCS'80)</i>, Springer LNCS 88, pp. 414-425, 1980.<br />
<br />
<span id="moo99" style="color:maroon">[Moo99]</span><br />
C. Moore.<br />
Quantum circuits: fanout, parity, and counting,<br />
ECCC [http://eccc.uni-trier.de/eccc-reports/1999/TR99-032/ TR99-032].<br />
<br />
<span id="mor01" style="color:maroon">[Mor01]</span><br />
T. Morioka.<br />
Classification of search problems and their definability in bounded arithmetic,<br />
master's thesis, University of Toronto, 2001.<br />
ECCC [http://eccc.uni-trier.de/eccc-reports/2001/TR01-082/ TR01-082].<br />
<br />
<span id="mp75" style="color:maroon">[MP75]</span><br />
D. E. Muller and F. P. Preparata.<br />
Bounds to complexities of networks for sorting and for switching,<br />
<i>Journal of the ACM</i> 22:195-201, 1975.<br />
<br />
<span id="mp91" style="color:maroon">[MP91]</span><br />
N. Megiddo and C. H. Papadimitriou.<br />
On total functions, existence theorems, and computational complexity,<br />
<i>Theoretical Computer Science</i> 81(2):317-324, 1991.<br />
<br />
<span id="mr95" style="color:maroon">[MR95]</span><br />
R. Motwani and P. Raghavan.<br />
<i>Randomized Algorithms</i>,<br />
Cambridge University Press, 1995.<br />
<br />
<span id="ms02" style="color:maroon">[MS02]</span><br />
K. Mulmuley and M. Sohoni.<br />
Geometric complexity theory I: An approach to the P vs. NP and related problems,<br />
<i>SIAM Journal on Computing</i> 31(2):496-526, 2002.<br />
<br />
<span id="muc56" style="color:maroon">[Muc56]</span><br />
A. A. Muchnik.<br />
On the unsolvability of the problem of reducibility in the theory of algorithms,<br />
<i>Doklady Akademii Nauk SSSR</i> 108:194-197, 1956.<br />
<br />
<span id="mv99" style="color:maroon">[MV99]</span><br />
P. B. Miltersen and N. V. Vinodchandran.<br />
Derandomizing Arthur-Merlin games using hitting sets,<br />
<i>Proceedings of IEEE FOCS'99</i>, pp. 71-80, 1999.<br />
<br />
<span id="mvv87" style="color:maroon">[MVV87]</span><br />
K. Mulmuley, U. V. Vazirani, and V. V. Vazirani.<br />
Matching is as easy as matrix inversion,<br />
<i>Proceedings of ACM STOC'87</i>, pp. 345-354, 1987.<br />
<br />
<span id="mvw99" style="color:maroon">[MVW99]</span><br />
P. B. Miltersen, N. V. Vinodchandran, and O. Watanabe.<br />
Super-polynomial versus half-exponential circuit size in the exponential hierarchy,<br />
<i>Proceedings of the 5th Annual Conference on Computing and Combinatorics (COCOON'99)</i>, pp. 210-220, Lecture Notes in Computer Science 1627, Springer-Verlag, 1999.<br />
<br />
<span id="mw05" style="color:maroon">[MW05]</span><br />
C. Marriott and J. Watrous.<br />
Quantum Arthur-Merlin Games,<br />
<i>Computational Complexity</i>, 14(2):122-152, 2005.<br />
arXiv:[http://arxiv.org/abs/cs/0506068 cs/0506068].<br />
<br />
===== N =====<br />
<br />
<span id="nc00" style="color:maroon">[NC00]</span><br />
M. Nielsen and I. Chuang.<br />
<i>Quantum Computation and Quantum Information</i>,<br />
Cambridge University Press, 2000.<br />
<br />
<span id="nhk00" style="color:maroon">[NHK00]</span><br />
M. Nakanishi, K. Hamaguchi, and T. Kashiwabara.<br />
Ordered quantum branching programs are more powerful than ordered probabilistic branching programs under a bounded-width restriction,<br />
<i>Proceedings of COCOON'2000 (Computing and Combinatorics)</i>, Springer LNCS 1858, pp. 467-476, 2000.<br />
<br />
<span id="nie02" style="color:maroon">[Nie02]</span><br />
G. Niemann and J. R. Woinowski.<br />
The Growing Context-Sensitive Languages Are the Acyclic Context-Sensitive Languages,<br />
<i>Developments in Language Theory</i>. LNCS 2295, pp. 197-205.<br />
</span><br />
<br />
<span id="nis02" style="color:maroon">[Nis02]</span><br />
T. Nishino.<br />
Mathematical models of quantum computation,<br />
New Gen. Comput. 20 (2002), no 4, 317-337.<br />
<br />
<span id="nis92" style="color:maroon">[Nis92]</span><br />
N. Nisan.<br />
RL is contained in SC,<br />
<i>Proceedings of ACM STOC'92</i>, pp. 619-623, 1992.<br />
<br />
<span id="nr97" style="color:maroon">[NR97]</span><br />
M. Naor and O. Reingold.<br />
Number-theoretic constructions of efficient pseudorandom functions,<br />
<i>Proceedings of IEEE FOCS'97</i>, pp. 458-467, 1997.<br />
<br />
<span id="nr98" style="color:maroon">[NR98]</span><br />
R. Niedermeier and P. Rossmanith.<br />
Unambiguous computations and locally definable acceptance types,<br />
<i>Theoretical Computer Science</i> 194:137-161, 1998.<br />
<br />
<span id="nrr01" style="color:maroon">[NRR01]</span><br />
M. Naor, O. Reingold, and A. Rosen.<br />
Pseudo-random functions and factoring,<br />
ECCC [http://eccc.uni-trier.de/eccc-reports/2001/TR01-064/ TR01-064].<br />
<br />
<span id="ns05" style="color:maroon">[NS05]</span><br />
A. Nickelsen and B. Schelm.<br />
Average-case computations - comparing AvgP, HP, and Nearly-P,<br />
<i>Proceedings of IEEE Complexity'2005</i>, pp. 235-242, 2005.<br />
[http://www.thi.uni-hannover.de/forschung/publikationen/daten/ni-sc05.pdf http://www.thi.uni-hannover.de/forschung/publikationen/daten/ni-sc05.pdf].<br />
<br />
<span id="nsw92" style="color:maroon">[NSW92]</span><br />
N. Nisan, E. Szemer&eacute;di, and A. Wigderson.<br />
Undirected connectivity in O(log<sup>1.5</sup>n) space,<br />
<i>Proceedings of IEEE FOCS'92</i>, pp. 24-29, 1992.<br />
<br />
<span id="nt95" style="color:maroon">[NT95]</span><br />
N. Nisan and A. Ta-Shma.<br />
Symmetric logspace is closed under complement,<br />
<i>Proceedings of ACM STOC'95</i>, pp. 140-146, 1995.<br />
ECCC [http://eccc.uni-trier.de/eccc-reports/1994/TR94-003/ TR94-003].<br />
<br />
<span id="nw94" style="color:maroon">[NW94]</span><br />
N. Nisan and A. Wigderson.<br />
Hardness versus randomness,<br />
<i>Journal of Computer and System Sciences</i> 49:149-167, 1994.<br />
<br />
<span id="ny03" style="color:maroon">[NY03]</span><br />
H. Nishimura and T. Yamakami.<br />
Polynomial time quantum computation with advice,<br />
arXiv:[http://arxiv.org/abs/quant-ph/0305100 quant-ph/0305100],<br />
ECCC [http://eccc.uni-trier.de/eccc-reports/2003/TR03-059/ TR03-059], 2003.<br />
<br />
<span id="ny03b" style="color:maroon">[NY03b]</span><br />
H. Nishimura and T. Yamakami.<br />
An algorithmic argument [http://www.cheatcodesforsim3.com/ for] query complexity lower bounds of advised quantum computation,<br />
arXiv:[http://arxiv.org/abs/quant-ph/0312003 quant-ph/0312003], 2003.<br />
<br />
===== O =====<br />
<br />
<span id="ogi94" style="color:maroon">[Ogi94]</span><br />
M. Ogihara.<br />
On serializable languages,<br />
<i>International Journal of Foundations of Computer Science</i> 5(3-4):303-318, 1994.<br />
<br />
<span id="oh93" style="color:maroon">[OH93]</span><br />
M. Ogihara and L. Hemachandra.<br />
A complexity theory for feasible closure properties,<br />
<i>Journal of Computer and System Sciences</i> 46(3):295-325, 1993.<br />
<br />
<span id="oka96" style="color:maroon">[Oka96]</span><br />
T. Okamoto.<br />
On relationships between statistical zero-knowledge proofs,<br />
<i>Proceedings of ACM STOC'96</i>, 1996.<br />
<br />
<span id="oks94" style="color:maroon">[OKS+94]</span><br />
P. Orponen, K.-I. Ko, U. Sch&ouml;ning, and O. Watanabe.<br />
Instance complexity,<br />
<i>Journal of the ACM</i> 41:96-121, 1994.<br />
<br />
<span id="ost91" style="color:maroon">[Ost91]</span><br />
R. Ostrovsky.<br />
One-way functions, hard on average problems and statistical zero-knowledge proofs,<br />
<i>Proceedings of IEEE Complexity'91</i>, pp. 51-59, 1991.<br />
<br />
<span id="ow93" style="color:maroon">[OW93]</span><br />
R. Ostrovsky and A. Wigderson.<br />
One-way functions are essential for non-trivial zero-knowledge,<br />
<i>Proceedings of the 2nd Israel Symposium on Theory of Computing and Systems (ISTCS-93)</i>, 1993.<br />
<br />
===== P =====<br />
<br />
<span id="pap83" style="color:maroon">[Pap83]</span><br />
C. H. Papadimitriou.<br />
Games against nature,<br />
<i>Proceedings of IEEE FOCS'83</i>, pp. 446-450, 1983.<br />
<br />
<span id="pap90" style="color:maroon">[Pap90]</span><br />
C. H. Papadimitriou.<br />
On graph-theoretic lemmata and complexity classes,<br />
<i>Proceedings of IEEE FOCS'90</i>, pp. 794-801, 1990.<br />
<br />
<span id="pap94" style="color:maroon">[Pap94]</span><br />
C. H. Papadimitriou.<br />
<i>Computational Complexity</i>,<br />
Addison-Wesley, 1994.<br />
<br />
<span id="pap94b" style="color:maroon">[Pap94b]</span><br />
C. H. Papadimitriou.<br />
On the complexity of the parity argument and other inefficient proofs of existence,<br />
<i>Journal of Computer and System Sciences</i> 48(3):498-532, 1994.<br />
<br />
{{Reference<br />
|tag=Per07<br />
|authors=K. Pervyshev<br />
|title=On heuristic time hierarchies<br />
|journal=Proceedings of the 22nd Annual IEEE Conference on Computational Complexity<br />
|srcdetail=347-357, 2007<br />
}}<br />
<br />
<span id="pos44" style="color:maroon">[Pos44]</span><br />
E. L. Post.<br />
Recursively enumerable sets of positive integers and their decision problems,<br />
<i>Bulletin of the American Mathematical Society</i> 50:284-316, 1944.<br />
<br />
<span id="pp00" style="color:maroon">[PP00]</span><br />
S. Parker and M. B. Plenio.<br />
Efficient factorization with a single pure qubit and log N mixed qubits,<br />
<i>Physical Review Letters</i> 85:3049, 2000.<br />
arXiv:[http://arxiv.org/abs/quant-ph/0001066 quant-ph/0001066].<br />
<br />
<span id="pps83" style="color:maroon">[PPS+83]</span><br />
W. J. Paul, N. Pippenger, E. Szemer&eacute;di, and W. T. Trotter.<br />
On determinism versus nondeterminism and related problems,<br />
<i>Proceedings of IEEE FOCS'83</i>, pp. 429-438, 1983.<br />
<br />
<span id="pps14" style="color:maroon">[PPS14]</span><br />
P. Papakonstantinou, D. Scheder, and H. Song.<br />
Overlays and limited memory communication,<br />
<i>Proceedings of IEEE CCC'14</i>, pp. 298-308, 2014.<br />
<br />
<span id="pra74" style="color:maroon">[Pra74]</span><br />
V. R. Pratt.<br />
The power of negative thinking in multiplying Boolean matrices,<br />
<i>STOC '74: Proceedings of the sixth annual ACM Symposium on Theory of Computing</i>, 80-83, 1974.<br />
<br />
<span id="pra75" style="color:maroon">[Pra75]</span><br />
V. R. Pratt.<br />
Every prime has a succinct certificate,<br />
<i>SIAM Journal on Computing</i>, 4:214-220, 1975.<br />
<br />
<span id="ps86" style="color:maroon">[PS86]</span><br />
R. Paturi and J. Simon.<br />
Probabilistic communication complexity,<br />
<i>Journal of Computer and System Sciences</i>, 33(1):106-123, 1986.<br />
<br />
<span id="pv04" style="color:maroon">[PV04]</span><br />
A. Pavan and N. V. Vinodchandran.<br />
[http://ftp.eccc.uni-trier.de/eccc-reports/2004/TR04-053/ TR04-053].<br />
<br />
<span id="py84" style="color:maroon">[PY84]</span><br />
C. H. Papadimitriou and M. Yannakakis.<br />
The complexity of facets (and some facets of complexity),<br />
<i>Journal of Computer and System Sciences</i> 28:244-259, 1984.<br />
<br />
<span id="py88" style="color:maroon">[PY88]</span><br />
C. H. Papadimitriou and M. Yannakakis.<br />
Optimization, approximation, and complexity classes,<br />
<i>Proceedings of ACM STOC'88</i>, pp. 229-234, 1988.<br />
<br />
<span id="py96" style="color:maroon">[PY96]</span><br />
C. H. Papadimitriou and M. Yannakakis.<br />
On limited nondeterminism and the complexity of the VC dimension,<br />
<i>Journal of Computer and System Sciences</i> 53(2):161-170, 1996.<br />
<br />
<span id="pz83" style="color:maroon">[PZ83]</span><br />
C. H. Papadimitriou and S. Zachos.<br />
Two remarks on the power of counting,<br />
<i>Proceedings of the 6th GI Conference in Theoretical Computer Science</i>, Lecture Notes in Computer Science Vol. 145, Springer-Verlag, pp. 269-276, 1983.<br />
<br />
===== R =====<br />
<br />
<span id="ra00" style="color:maroon">[RA00]</span><br />
K. Reinhardt and E. Allender.<br />
Making nondeterminism unambiguous,<br />
<i>SIAM Journal on Computing</i> 29:1118-1131, 2000.<br />
ECCC [http://eccc.uni-trier.de/eccc-reports/1997/TR97-014/ TR97-014],<br />
DIMACS [http://dimacs.rutgers.edu/TechnicalReports/abstracts/1997/97-46.html TR 97-46].<br />
<br />
<span id="rab60" style="color:maroon">[Rab60]</span><br />
M. O. Rabin.<br />
Degree of difficulty of computing a function and a partial ordering of recursive sets,<br />
Tech Report No. 2, Hebrew University, 1960.<br />
<br />
<span id="rac82" style="color:maroon">[Rac82]</span><br />
C. Rackoff.<br />
Relativized questions involving probabilistic algorithms,<br />
<i>Journal of the ACM</i> 29(1):261-268, 1982.<br />
<br />
<span id="raz05" style="color:maroon">[Raz05]</span><br />
R. Raz.<br />
Quantum information and the PCP theorem,<br />
to appear in <i>Proc. IEEE FOCS</i>, 2005.<br />
ECCC [http://www.eccc.uni-trier.de/eccc-reports/2005/TR05-038/index.html TR05-038].<br />
<br />
<span id="raz85" style="color:maroon">[Raz85]</span><br />
A. A. Razborov.<br />
Lower bounds on the monotone complexity of some Boolean functions,<br />
<i>Dokl. Akad. Nauk SSSR</i> 281(4):798-801, 1985.<br />
English translation in <i>Soviet Math. Dokl.</i> 31:354-357, 1985.<br />
<br />
<span id="raz85b" style="color:maroon">[Raz85b]</span><br />
A. A. Razborov.<br />
A lower bound on the monotone network complexity of the logical permanent,<br />
<i>Mat. Zametky</i> 37(6):887-900, 1985.<br />
English translation in <i>Russian Mathematical Notes</i> 37:485-493, 1985.<br />
<br />
<span id="raz87" style="color:maroon">[Raz87]</span><br />
A. A. Razborov.<br />
Lower bounds for the size of circuits of bounded depth with basis {&amp;,},<br />
<i>Mathematicheskie Zametki</i> 41(4):598-607, 1987.<br />
English translation in <i>Math. Notes. USSR</i> 41(4):333-338, 1987.<br />
<br />
<span id="raz92" style="color:maroon">[Raz92]</span><br />
A. A. Razborov.<br />
On the distributional complexity of disjointness,<br />
<i>Theoretical Computer Science</i> 106(2):385-390, 1992.<br />
<br />
<span id="raz94" style="color:maroon">[Raz94]</span><br />
A. A. Razborov.<br />
On provably disjoint NP-pairs,<br />
ECCC [http://eccc.uni-trier.de/eccc-reports/1994/TR94-006/ TR94-006], 1994.<br />
<br />
<span id="reg02" style="color:maroon">[Reg02]</span><br />
K. Regan.<br />
Understanding the Mulmuley-Sohoni approach to P vs. NP,<br />
<i>Bulletin of the EATCS</i> 78, October 2002.<br />
[http://people.cs.uchicago.edu/~fortnow/beatcs/column78.pdf http://people.cs.uchicago.edu/~fortnow/beatcs/column78.pdf].<br />
<br />
<span id="rei04" style="color:maroon">[Rei04]</span><br />
O. Reingold.<br />
Undirected st-connectivity in log-space,<br />
ECCC [http://eccc.uni-trier.de/eccc-reports/2004/TR04-094/ TR04-094], 2004.<br />
<br />
<span id="rr95" style="color:maroon">[RR95]</span><br />
K. Regan and J. Royer.<br />
On Closure Properties of Bounded two-Sided Error Complexity Classes,<br />
Math. Systems Theory, 28 (1995) 229-243.<br />
ftp://ftp.cis.syr.edu/users/royer/coinflips.ps<br />
<br />
<span id="rr97" style="color:maroon">[RR97]</span><br />
A. A. Razborov and S. Rudich.<br />
Natural proofs,<br />
<i>Journal of Computer and System Sciences</i> 55(1):24-35, 1997.<br />
ECCC [http://eccc.uni-trier.de/eccc-reports/1994/TR94-010/ TR94-010].<br />
<br />
<span id="rs98" style="color:maroon">[RS98]</span><br />
A. Russell and R. Sundaram.<br />
Symmetric alternation captures BPP,<br />
<i>Computational Complexity</i> 7(2):152-162, 1998.<br />
<br />
<span id="rs10" style="color:maroon">[RS10]</span><br />
A. Razborov and A. Sherstov.<br />
The sign-rank of AC0,<br />
<i>SIAM Journal on Computing</i> 39(5):1833-1855, 2010.<br />
<br />
<span id="rtv05" style="color:maroon">[RTV05]</span><br />
O. Reingold and L. Trevisan and S. Vadhan.<br />
Pseudorandom walks in biregular graphs and the RL vs. L problem,<br />
ECCC [http://eccc.uni-trier.de/eccc-reports/2005/TR05-022/ TR05-022], 2004.<br />
<br />
<span id="rub88" style="color:maroon">[Rub88]</span><br />
R. Rubinstein.<br />
<i>Structural Complexity Classes of Sparse Sets: Intractability, Data Compression, and Printability</i>,<br />
PhD Thesis, Northeastern University (Boston, MA), 1988.<br />
<br />
<span id="rud97" style="color:maroon">[Rud97]</span><br />
S. Rudich.<br />
Super-bits, demi-bits, and NP/qpoly-natural proofs,<br />
<i>RANDOM: International Workshop on Randomization and Approximation Techniques in Computer Science</i>, Lecture Notes in Computer Science, Springer-Verlag, 1997.<br />
<br />
<span id="rus85" style="color:maroon">[Rus85]</span><br />
D. A. Russo.<br />
Structural Properties of Complexity Classes.<br />
PhD thesis, UC Santa Barbara, 1985.<br />
<br />
<span id="ruv12" style="color:maroon">[RUV12]</span><br />
B. W. Reichardt, F. Unger, and U. Vazirani.<br />
A classical leash for a quantum system: Command of quantum systems via rigidity of CHSH games,<br />
<i>Nature</i> 496:456–460, 2013.<br />
<br />
<span id="ruz81" style="color:maroon">[Ruz81]</span><br />
W. L. Ruzzo.<br />
On uniform circuit complexity,<br />
<i>Journal of Computer and System Sciences</i> 22(3):365-383, 1971.<br />
<br />
<span id="rv97" style="color:maroon">[RV97]</span><br />
K. Regan and H. Vollmer.<br />
"[http://dx.doi.org/10.1016/S0304-3975(96)00288-5 Gap-languages and log-time complexity classes]",<br />
''Theoretical Computer Science'' 188(1–2):101–116, 1997.<br />
<br />
<span id="rw01" style="color:maroon">[RW01]</span><br />
S. Reith and K. Wagner.<br />
On Boolean lowness and Boolean highness,<br />
<i>Theoretical Computer Science</i> 261(2):305-321, 2001.<br />
<br />
===== S =====<br />
<br />
{{Reference<br />
|tag=San07<br />
|title=Circuit lower bounds for Merlin-Arthur classes<br />
|journal=Electronic Colloquium on Computational Complexity<br />
|authors=R. Santhanam<br />
|srcdetail=Report TR07-005<br />
}}<br />
<br />
<span id="sav70" style="color:maroon">[Sav70]</span><br />
W. Savitch.<br />
Relationships between nondeterministic and deterministic tape complexities,<br />
<i>Journal of Computer and System Sciences</i> 4(2):177-192, 1970.<br />
<br />
<span id="sch02a" style="color:maroon">[Sch02a]</span><br />
M. Schaefer and C. Umans.<br />
Completeness in the Polynomial-Time Hierarchy: A Compendium,<br />
<i>Sigact News</i> September, 2002.<br />
<br />
<span id="sch02b" style="color:maroon">[Sch02b]</span><br />
M. Schaefer and C. Umans.<br />
Completeness in the Polynomial-Time Hierarchy: Part II,<br />
<i>Sigact News</i> December, 2002.<br />
<br />
<span id="sch03" style="color:maroon">[Sch03]</span><br />
P. Schnoebelen.<br />
Oracle circuits for branching-time model checking,<br />
<i>Proceedings of ICALP 2003</i>, pp. 790-801, 2003.<br />
<br />
<span id="sch78" style="color:maroon">[Sch78]</span><br />
C. P. Schnorr.<br />
Satisfiability Is Quasilinear Complete in NQL,<br />
<i>Journal of the ACM</i> 25(1):136-145, 1978.<br />
<br />
<span id="sch83" style="color:maroon">[Sch83]</span><br />
U. Sch&ouml;ning.<br />
A low and a high hierarchy within NP,<br />
<i>Journal of Computer and System Sciences</i> 27:14-28, 1983.<br />
<br />
<span id="sch86" style="color:maroon">[Sch86]</span><br />
U. Sch&ouml;ning.<br />
Complete Sets and Closeness to Complexity Classes,<br />
<i>Mathematical Systems Theory</i> 19:29-41, 1986.<br />
DOI:[http://dx.doi.org/10.1007/BF01704904 10.1007/BF01704904]<br />
<br />
<span id="sel79" style="color:maroon">[Sel79]</span><br />
A. Selman.<br />
P-selective sets, tally languages, and the behavior of polynomial time reducibilities in NP,<br />
<i>Mathematical Systems Theory</i> 13(1):55-65, 1979.<br />
<br />
<span id="sch2015" style="color:maroon">[Sch2015]</span><br />
M. Schwarz.<br />
An exponential time upper bound for quantum Merlin Arthur games with unentangled provers,<br />
arXiv:[http://arxiv.org/abs/1510.08447 1510.08447], 2015.<br />
<br />
<span id="ses05" style="color:maroon">[SES05]</span><br />
E. Allender, S. Datta, and S. Roy.<br />
The directed planar reachability problem,<br />
<i>Proceedings of FSTTCS</i>, #1373 in Computer Science<br />
<br />
<span id="sf98" style="color:maroon">[SF98]</span><br />
H. T. Siegelmann and S. Fishman.<br />
Analog computation with dynamical systems,<br />
Physica 120D, p. 214, 1998.<br />
<br />
<span id="sfm78" style="color:maroon">[SFM78]</span><br />
J. Seiferas, M. Fischer, and A. Meyer.<br />
Separating nondeterministic time complexity classes,<br />
<i>Journal of the ACM</i> 25:146-167, 1978.<br />
<br />
<span id="sha90" style="color:maroon">[Sha90]</span><br />
A. Shamir.<br />
IP=PSPACE,<br />
<i>Proceedings of IEEE FOCS'90</i>, pp. 11-15, 1990.<br />
<br />
<span id="she59" style="color:maroon">[She59]</span><br />
J. C. Shepherdson.<br />
The reduction of two-way automata to one-way automata,<br />
<i>IBM Journal of Research and Development</i>, 3:198-200, 1959.<br />
<br />
<span id="she08" style="color:maroon">[She08]</span><br />
A. A. Sherstov.<br />
Separating AC<sup>0</sup> from depth-2 majority circuits,<br />
<i>Computational Complexity</i>, 17(2):149-178, 2008.<br />
[http://www.cs.utexas.edu/~sherstov/publications/pdf/cc08hsmat.pdf http://www.cs.utexas.edu/~sherstov/publications/pdf/cc08hsmat.pdf]<br />
<br />
<span id="shi03" style="color:maroon">[Shi03]</span><br />
Y. Shi.<br />
Quantum and classical tradeoffs,<br />
arXiv:[http://arxiv.org/abs/quant-ph/0312213 quant-ph/0312213],<br />
ECCC [http://eccc.uni-trier.de/eccc-reports/2004/TR04-023/ TR04-023], 2003.<br />
<br />
<span id="sho97" style="color:maroon">[Sho97]</span><br />
P. Shor.<br />
Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer,<br />
<i>SIAM Journal on Computing</i> 26(5):1484-1509, 1997.<br />
arXiv:[http://arxiv.org/abs/quant-ph/9508027 quant-ph/9508027].<br />
<br />
<span id="sho99" style="color:maroon">[Sho99]</span><br />
R. A. Shore.<br />
The recursively enumerable degrees,<br />
<i>Handbook of Recursion Theory</i> (E. Griffor, ed.), pp. 169-197, North-Holland, Amsterdam, 1999.<br />
<br />
<span id="sip82" style="color:maroon">[Sip82]</span><br />
M. Sipser.<br />
On relativization and the existence of complete sets,<br />
<i>Proceedings of ICALP'82</i>, Springer-Verlag Lecture Notes in Computer Science volume 140, pp. 523-531, 1982.<br />
<br />
<span id="sip92" style="color:maroon">[Sip92]</span><br />
M. Sipser.<br />
The history and status of the P versus NP question,<br />
<i>Proceedings of ACM STOC'92</i>, pp. 603-618, 1992.<br />
<br />
<span id="sm02" style="color:maroon">[SM02]</span><br />
L. J. Stockmeyer and A. R. Meyer.<br />
Cosmological lower bound on the circuit complexity of a small problem in logic,<br />
<i>Journal of the ACM</i> 49(6):753-784, 2002.<br />
[http://theory.lcs.mit.edu/~meyer/stock-circuit-jacm.pdf http://theory.lcs.mit.edu/~meyer/stock-circuit-jacm.pdf].<br />
<br />
{{Reference<br />
|tag=SM03<br />
|authors=R. Santhanam and D. van Melkebeek<br />
|title=Holographic proofs and derandomization<br />
|journal=Proceedings of the 18th Annual IEEE Conference on Computational Complexity<br />
|srcdetail=269-283<br />
}}<br />
<br />
<span id="smo87" style="color:maroon">[Smo87]</span><br />
R. Smolensky.<br />
Algebraic methods in the theory of lower bounds for Boolean circuit complexity,<br />
<i>Proceedings of ACM STOC'87</i>, pp. 77-82, 1987.<br />
<br />
<span id="sp98" style="color:maroon">[SP98]</span><br />
U. Sch&ouml;ning and R. Pruim.<br />
<i>Gems of Theoretical Computer Science</i>,<br />
Springer-Verlag, 1998.<br />
<br />
<span id="spa02" style="color:maroon">[Spa02]</span><br />
R. &#352;palek.<br />
Quantum circuits with unbounded fan-out,<br />
arXiv:[http://arxiv.org/abs/quant-ph/0208043 quant-ph/0208043], 2002.<br />
<br />
<span id="ss04" style="color:maroon">[SS04]</span><br />
A. Selman and S. Sengupta.<br />
Polylogarithmic-round interactive proofs for coNP collapse the exponential hierarchy,<br />
<i>Proceedings of IEEE Complexity 2004</i>, pp. 82-90, 2004.<br />
ECCC [http://eccc.uni-trier.de/eccc-reports/2004/TR04-007/ TR04-007].<br />
<br />
<span id="ss77" style="color:maroon">[SS77]</span><br />
R. Solovay and V. Strassen.<br />
A fast Monte-Carlo test for primality,<br />
<i>SIAM Journal on Computing</i>, 6:84-86, 1977.<br />
<br />
<span id="sto76" style="color:maroon">[Sto76]</span><br />
L. J. Stockmeyer.<br />
The polynomial hierarchy,<br />
<i>Theoretical Computer Science</i> 3:1-22, 1976.<br />
<br />
<span id="sto85" style="color:maroon">[Sto85]</span><br />
L. J. Stockmeyer.<br />
On approximation algorithms for #P,<br />
<i>SIAM Journal on Computing</i> 14:849-861, 1985.<br />
<br />
<span id="stt05" style="color:maroon">[STT05]</span><br />
H. Spakowski, M. Thakur, and R. Tripathi.<br />
Quantum and Classical Complexity Classes: Separations, Collapses, and Closure Properties,<br />
Inform. and Comput. 200 (2005), no. 1, 1--34.<br />
[http://web.umr.edu/~thakurk/publications/quantum-j.pdf http://web.umr.edu/~thakurk/publications/quantum-j.pdf]<br />
<br />
<span id="su05" style="color:maroon">[SU05]</span><br />
R. Shaltiel and C. Umans.<br />
Pseudorandomness for approximate counting and sampling,<br />
<i>Proceedings of IEEE Complexity'2005</i>, pp. 212-226, 2005.<br />
[http://www.cs.haifa.ac.il/~ronen/online_papers/derand-ccc-final.ps http://www.cs.haifa.ac.il/~ronen/online_papers/derand-ccc-final.ps].<br />
<br />
<span id="sub94" style="color:maroon">[Sub94]</span><br />
A. Subramanian. <br />
A New Approach to Stable Matching Problems, <br />
<i>SIAM Journal on Computing</i> 23(4), 671-701, 1994. <br />
<br />
<span id="sud78" style="color:maroon">[Sud78]</span><br />
I. Sudborough.<br />
On the tape complexity of deterministic context-free languages,<br />
<i>Journal of the ACM</i> 25(3):405-414, 1978.<br />
<br />
<span id="sv97" style="color:maroon">[SV97]</span><br />
A. Sahai and S. Vadhan.<br />
A complete promise problem for statistical zero-knowledge,<br />
<i>Proceedings of IEEE FOCS'97</i>.<br />
[http://www.eecs.harvard.edu/~salil/papers/complete-abs.html http://www.eecs.harvard.edu/~salil/papers/complete-abs.html].<br />
<br />
<span id="sz95" style="color:maroon">[SZ95]</span><br />
M. Saks and S. Zhou.<br />
RSPACE(s) is contained in DSPACE(s<sup>3/2</sup>),<br />
<i>Proceedings of IEEE FOCS'95</i>, pp. 344-353, 1995.<br />
<br />
<span id="sze87" style="color:maroon">[Sze87]</span><br />
R. Szelepcs&eacute;nyi.<br />
The method of forcing for nondeterministic automata,<br />
<i>Bulletin of the EATCS</i> 33:96-100, 1987.<br />
<br />
{{Reference<br />
|id="szep94" |tag=Szep94<br />
|title=Turing Machines With Sublogarithmic Space<br />
|journal=Lecture Notes in Computer Science<br />
|authors=A. Szepietowski<br />
|srcdetail=volume 843<br />
}}<br />
<br />
===== T =====<br />
<br />
<span id="tan07" style="color:maroon">[Tan07]</span><br />
T. Tantau.<br />
Logspace Optimization Problems and Their Approximability Properties,<br />
<i>Theory of Computing Systems</i>, 41:327-350, 2007.<br />
<br />
<span id="tar88" style="color:maroon">[Tar88]</span><br />
E. Tardos.<br />
The gap between monotone and non-monotone circuit complexity is exponential,<br />
<i>Combinatorica</i>, 8:141-142, 1988.<br />
<br />
<span id="tar89" style="color:maroon">[Tar89]</span><br />
G. Tardos.<br />
Query complexity, or why is it difficult to separate NP<sup>A</sup> intersect coNP<sup>A</sup> from P<sup>A</sup> by random oracles A,<br />
<i>Combinatorica</i>, 9:385-392, 1989.<br />
<br />
<span id="tha98" style="color:maroon">[Tha98]</span><br />
J. S. Thathachar.<br />
On Separating the Read-k-Times Branching Program Hierarchy,<br />
<i>Proceedings of the 30th ACM Symposium on Theory of Computing</i>, pp. 653-662, 1998.<br />
ECCC [http://eccc.hpi-web.de/eccc-reports/1998/TR98-002/ TR98-02], DOI:[http://doi.acm.org/10.1145/276698.276881 10.1145/276698.276881].<br />
<br />
<span id="tod89" style="color:maroon">[Tod89]</span><br />
S. Toda.<br />
On the computational power of PP and P,<br />
<i>Proceedings of IEEE FOCS'89</i>, pp. 514-519, 1989.<br />
<br />
<span id="tor00" style="color:maroon">[Tor00]</span><br />
J. Tor&aacute;n.<br />
On the hardness of graph isomorphism,<br />
<i>Proceedings of IEEE FOCS'2000</i>, pp. 180-186, 2000.<br />
<br />
<span id="tor88" style="color:maroon">[Tor88]</span><br />
J. Tor&aacute;n.<br />
Structural Properties of the Counting Hierarchies,<br />
Ph.D Theis, 1988.<br />
<br />
<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 />
<i>Proceedings of the International Conference on Fundamentals of Computation Theory</i>, LNCS volume 199, Springer-Verlag, pp. 485-493.<br />
<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 />
P, NP, and mathematics--a computational complexity perspective, 2006 mimeo.<br />
[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>Soulsand