Contains languages whose complements are in Π2P.
Along with Π2P, comprises the second level of PH, the polynomial hierarchy.
Note that this is equal to NP with a coNP oracle.
[Uma98] has shown that the following problems are complete for Σ2P:
The problem of deciding if a perfect graph is 2-clique-colorable (defined in [FMF16]) has been shown to be complete for Σ2P.
For any fixed k, there is a problem in Σ2P ∩ Π2P that cannot be solved by circuits of size nk [Kan82].
The class of decision problems for which a "yes" answer can be verified by an Arthur-Merlin protocol, as follows.
Arthur, a 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
Surprisingly, it turns out that such a system is just as powerful as a private-coin one, in which Arthur does not need to send his random coins to Merlin [GS86]. So, Arthur never needs to hide information from Merlin.
Furthermore, define AM[k] similarly to AM, except that Arthur and Merlin have k rounds of interaction. Then for all constant k>2, AM[k] = AM[2] = AM [BM88]. Also, the result of [GS86] can then be stated as follows: IP[k] is contained in AM[k+2] for every k (constant or non-constant).
AM contains graph nonisomorphism.
Contains NP, BPP, and SZK, and is contained in NP/poly.AM is also contained in Π2P and this proof relativizes so the containment holds relative to any oracle.
If AM contains coNP then PH collapses to Σ2P ∩ Π2P [BHZ87].
There exists an oracle relative to which AM is not contained in PP [Ver92].
AM = NP under a strong derandomization assumption: namely that some language in NE ∩ coNE requires nondeterministic circuits of size 2Ω(n) ([MV99], improving [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.)
The class of decision problems solvable by an NP machine such that
(Here all computation paths have the same length.)
Often identified as the class of feasible problems for a computer with access to a genuine random-number source.
Defined in [Gil77].
Contained in Σ2P ∩ Π2P [Lau83], and indeed in ZPPNP [GZ97].
If BPP contains NP, then RP = NP [Ko82,Gil77] and PH is contained in BPP [Zac88].
If any problem in E requires circuits of size 2Ω(n), then BPP = P [IW97] (in other words, BPP can be derandomized).
Contained in O2P since problems requiring exponential sized circuits can be verified in O2E [GLV24] [Li23] which can be used to derandomize [IW97].
Indeed, any proof that BPP = P requires showing either that NEXP is not in P/poly, or else that #P requires superpolynomial-size arithmetic circuits [KI02].
BPP is not known to contain complete languages. [Sip82], [HH86] give oracles relative to which BPP has no complete languages.
There exist oracles relative to which P = RP but still P is not equal to BPP [BF99].
In contrast to the case of P, it is unknown whether BPP collapses to BPTIME(nc) for some fixed constant c. However, [Bar02] and [FS04] have shown hierarchy theorems for BPP with a small amount of advice.
A zero-one law exists stating that BPP has p-measure zero unless BPP = EXP [Mel00].
Equals Almost-P.
See also: BPPpath.
Same as BPP, except that now the computation paths need not all have the same length.
Defined in [HHT97], where the following was also shown:
There exists an oracle relative to which BPPpath is not contained in Σ2P [BGM02].
An alternate characterization of BPPpath uses the idea of post-selection. That is, BPPpath is the class of languages for which there exists a pair of polynomial-time Turing machines and such that the following conditions hold for all :
We say that is the post-selector. Intuitively, this characterization allows a BPP machine to require that its random bits have some special but easily verifiable property. This characterization makes the inclusion NP ⊆ BPPpath nearly trivial.
See Also: PostBQP (quantum analogue).
A bounded-error quantum generalization of Σ2P in which the first proof is classical, the second proof quantum, and the verifier is a quantum circuit.
Defined in [GK14].
The following problems are complete for cq-Σ2 (the first two are, in addition, strongly hard to approximate for cq-Σ2) [GK14]:QUANTUM SUCCINCT SET COVER, QUANTUM IRREDUNDANT, cq-Σ2LH (a quantum generalization of the canonical Σ2P-complete problem ΣiSAT). The first of these roughly asks: Given a frustrated local Hamiltonian, what is the maximum number of local interaction terms which can be discarded before the resulting Hamiltonian is no longer frustrated?
The class of decision problems solvable by a Merlin-Arthur protocol, which goes as follows. Merlin, who has unbounded computational resources, sends Arthur a polynomial-size purported proof that the answer to the problem is "yes." Arthur must verify the proof in BPP (i.e. probabilistic polynomial-time), so that
Defined in [Bab85].
An alternative definition requires that if the answer is "yes," then there exists a proof such that Arthur accepts with certainty. However, the definitions with one-sided and two-sided error can be shown to be equivalent (see [FGM+89]).
Contains NP and BPP (in fact also ∃BPP), and is contained in AM and in QMA.
There exists an oracle relative to which BQP is not in MA [Wat00].
Equals NP under a derandomization assumption: if E requires exponentially-sized circuits, then PromiseBPP = PromiseP, implying that MA = NP [IW97].
Shown in [San07] that MA/1 (that is, MA with 1 bit of advice) does not have circuits of size for any . In the same paper, the result was used to show that MA/1 cannot be solved on more than a fraction of inputs having length by any circuit of size . Finally, it was shown that MA does not have arithmetic circuits of size .
The class of dashed hopes and idle dreams.
More formally: an "NP machine" is a nondeterministic polynomial-time Turing machine.
Then NP is the class of decision problems solvable by an NP machine such that
Equivalently, NP is the class of decision problems such that, if the answer is "yes," then there is a proof of this fact, of length polynomial in the size of the input, that can be verified in P (i.e. by a deterministic polynomial-time algorithm). On the other hand, if the answer is "no," then the algorithm must declare invalid any purported proof that the answer is "yes."
For example, the SAT problem is to decide whether a given Boolean formula has any satisfying truth assignments. SAT is in NP, since a "yes" answer can be proved by just exhibiting a satisfying assignment.
A decision problem is NP-complete if (1) it is in NP, and (2) any problem in NP can be reduced to it (under some notion of reduction). The class of NP-complete problems is sometimes called NPC.
That NP-complete problems exist is immediate from the definition. The seminal result of Cook [Coo71], Karp [Kar72], and Levin [Lev73] is that many natural problems (that have nothing to do with Turing machines) are NP-complete.
The first such problem to be shown NP-complete was SAT [Coo71]. Other classic NP-complete problems include:
For many, many more NP-complete problems, see [GJ79].
There exists an oracle relative to which P and NP are unequal [BGS75]. Indeed, P and NP are unequal relative to a random oracle with probability 1 [BG81] (see [AFM01] for a novel take on this result). Though random oracle results are not always indicative about the unrelativized case [CCG+94].
There even exists an oracle relative to which the P versus NP problem is outside the usual axioms of set theory [HH76].
If we restrict to monotone classes, mP is strictly contained in mNP [Raz85].
Perhaps the most important insight anyone has had into P versus NP is to be found in [RR97]. There the authors show that no 'natural proof' can separate P from NP (or more precisely, place NP outside of P/poly), unless secure pseudorandom generators do not exist. A proof is 'natural' if it satisfies two conditions called constructivity and largeness; essentially all lower bound techniques known to date satisfy these conditions. To obtain unnatural proof techniques, some people suspect we need to relate P versus NP to heavy-duty 'traditional' mathematics, for instance algebraic geometry. See [MS02] (and the survey article [Reg02]) for a development of this point of view.
For more on P versus NP (circa 1992) see [Sip92]. For an opinion poll, see [Gas02]. P vs NP is a "Millenium Prize" problem [CMI00].
If P equals NP, then NP equals its complement coNP. Whether NP equals coNP is also open. NP and coNP can be extended to the polynomial hierarchy PH.
The set of decision problems in NP, but not in P or NPC, is sometimes called NPI. If P does not equal NP then NPI is nonempty [Lad75].
Probabilistic generalizations of NP include MA and AM. If NP is in coAM (or BPP) then PH collapses to Σ2P [BHZ87].
PH also collapses to Σ2P if NP is in P/poly [KL82].
There exist oracles relative to which NP is not in BQP [BBB+97].
An alternate characterization is NP = PCP(log n, O(1)) [ALM+98].
Also, [Fag74] showed that NP is precisely the class of decision problems reducible to a graph-theoretic property expressible in second-order existential logic. This leads to the subclass SNP.
It is known that if any NP-complete language is sparse (contains no more than a polynomial number of strings of length ), then P = NP. [BH08] improved this result, showing that if any language in NP has an NP-hard set of subexponential density, then coNP is contained in NP/poly and thus, by [Yap82], PH collapses to the third level.
NP is equal to SO-E, the second-order queries where the second-order quantifiers are only existantials.
The class of decision problems solvable by a family of polynomial-size Boolean circuits. The family can be nonuniform; that is, there could be a completely different circuit for each input length.
Equivalently, P/poly is the class of decision problems solvable by a polynomial-time Turing machine that receives a trusted 'advice string,' that depends only on the size n of the input, and that itself has size upper-bounded by a polynomial in n.
Contains BPP by the progenitor of derandomization arguments [Adl78] [KL82]. By extension, BPP/poly, BPP/mpoly, and BPP/rpoly all equal P/poly. (By contrast, there is an oracle relative to which BPP/log does not equal BPP/mlog, while BPP/mlog and BPP/rlog are not equal relative to any oracle.)
[KL82] showed that, if P/poly contains NP, then PH collapses to the second level, Σ2P.
They also showed:
It was later shown that, if NP is contained in P/poly, then PH collapses to ZPPNP [KW98] and indeed to O2P [CR06] (which is unconditionally included in P/poly). This seems close to optimal, since there exists an oracle relative to which the collapse cannot be improved to Δ2P [Wil85].
If NP is not contained in P/poly, then P does not equal NP. Much of the effort toward separating P from NP is based on this observation. However, a 'natural proof' as defined by [RR97] cannot be used to show NP is outside P/poly, if there is any pseudorandom generator in P/poly that has hardness 2Ω(n^ε) for some ε>0.
If NP is contained in P/poly, then MA = AM [AKS+95]
The monotone version of P/poly is mP/poly.
P/poly has measure 0 in E with Σ2P oracle [May94b].
Strictly contains IC[log,poly] and P/log.
The complexity class of P with untrusted advice depending only on input size is ONP.
The class of problems for which there exists a polynomial-time predicate P(x,y,z) such that for all x, if the answer on input x is "yes," then
Defined in [Can96], where it was also observed that Φ2P = S2P.
Complement of Σ2P.
Along with Σ2P, comprises the second level of PH, the polynomial hierarchy. For any fixed k, there is a problem in Π2P ∩ Σ2P that cannot be solved by circuits of size nk [Kan82].
The class of decision problems for which there is a polynomial-time predicate P such that, on input x,
Note that this differs from Σ2P in that the quantifiers in the second condition are reversed.
Less formally, S2P is the class of one-round games in which a prover and a disprover submit simultaneous moves to a deterministic, polynomial-time referee. In Σ2P, the prover moves first.
Defined in [RS98], where it was also shown that S2P contains MA and Δ2P. Defined independently in [Can96].
Contains (and contrast with) O2P. If NP is contained in P/poly then PH = S2P (attributed to Sengupta in [Cai01]), and even is equal to O2P [CR06].
The class of decision problems for which the following holds. There exists a #P function f and an FP function g such that, for all inputs x,
Defined in [BGM02], where the following was also shown:
There exists an oracle relative to which SBP is not closed under intersection [GLM+15].
If SAT can be solved by an NP-machine with sub-exponential number of accepting paths, then SBP = AM [Vol20].
A superclass of VPk in Valiant's algebraic complexity theory, but not quite the analogue of NP.
A problem is in VNPk if there exists a polynomial p with the following properties:
Originated in [Val79b].
If the field k has characteristic greater than 2, then the permanent of an n-by-n matrix of indeterminates is VNPk-complete under a type of reduction called p-projections ([Val79b]; see also [Bur00]).
A central conjecture is that for all k, VPk is not equal to VNPk. Bürgisser [Bur00] shows that if this were false then:
In both cases, PH collapses to Σ2P.
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