The class of problems in both NP and coNP.
Contains graph isomorphism under the assumption 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.)
Equals PNP ∩ coNP [Bra79]. In particular, if a problem in NP ∩ coNP is NP-hard with Turing reduction, then NP = coNP.
Is not believed to contain complete problems.
Is equal to Low(NP)={L : NPL=NP} [Sch83].
The class of decision problems for which both "yes" and "no" answers can be verified by an AM protocol.
If EXP requires exponential time even for AM protocols, then AM ∩ coAM = NP ∩ coNP [GST03].
There exists an oracle relative to which AM ∩ coAM is not contained in PP [Ver95].
The class of problems decidable by an O(f(n))-space Turing machine with three kinds of quantifiers: existential, universal, and randomized.
Contains GAN-SPACE(f(n)).
AUC-SPACE(poly(n)) = SAPTIME = PSPACE [Pap83].
[Con92] shows that AUC-SPACE(log n) has a natural complete problem, and is contained in NP ∩ coNP.
The class of decision problems solvable in polynomial time by a quantum Turing machine, with at most 1/3 probability of error.
One can equivalently define BQP as the class of decision problems solvable by a uniform family of polynomial-size quantum circuits, with at most 1/3 probability of error [Yao93]. Any universal gate set can be used as a basis; however, a technicality is that the transition amplitudes must be efficiently computable, since otherwise one could use them to encode the solutions to hard problems (see [ADH97]).
BQP is often identified as the class of feasible problems for quantum computers.
Contains the factoring and discrete logarithm problems [Sho97], the hidden Legendre symbol problem [DHI02], the Pell's equation and principal ideal problems [Hal02], and some other problems not thought to be in BPP.
Defined in [BV97], where it is also shown that BQP contains BPP and is contained in P with a #P oracle.
BQPBQP = BQP [BV97].
[ADH97] showed that BQP is contained in PP, and [FR98] showed that BQP is contained in AWPP.
There exist oracles relative to which:
If P=BQP relative to a random oracle then BQP=BPP [FR98].
If NP = coNP, then any inconsistent Boolean formula of size n has a proof of inconsistency of size polynomial in n.
If NP does not equal coNP, then P does not equal NP. But the other direction is not known.
See also: NP ∩ coNP.
Every problem in coNP has an IP (interactive proof) system, where moreover the prover can be restricted to BPP#P. If every problem in coNP has an interactive protocol whose rounds are bounded by a polylogarithmic function, then EH collapses to the third level [SS04].
Co-NP is equal to SO-A, the second-order queries where the second-order quantifiers are only universals.
The class of problems A such that ΣkPA = ΣkP; that is, adding A as an oracle does not increase the power of the kth level of the polynomial hierarchy.
L1P = NP ∩ coNP.
For all k, Lk is contained in Lk+1 and in NP.
Defined in [Sch83].
See also HkP.
Together with NP/poly ∩ coNP/poly, has the same relation to NP ∩ coNP as P/poly has to P. A language in (NP ∩ coNP)/poly is defined by a single language in NP ∩ coNP which is then modified by advice. A language in NP/poly ∩ coNP/poly comes from two possibly different languages in NP and coNP which become the same with good advice.
There is an oracle relative to which NP/poly ∩ coNP/poly, indeed NP/1 ∩ coNP/1, is not contained in (NP ∩ coNP)/poly [FFK+93]. Recently they improved this to NP/1 ∩ coNP [FF..].
If NP is contained in (NP ∩ coNP)/poly, then PH collapses to S2PNP ∩ coNP [CCH+01].
The class of functions computable in polynomial time with a shared, untrusted witness for each input size. The input-oblivious version of NP.
L is in ONP if there exists a polynomial time verifier V taking an input and a witness, so that:
Defined in [FSW09], where it was shown NP has size nk circuits for some constant k if and only if ONP/1 has size nj circuits for some constant j.
ONP is contained in P/poly and NP [FSW09].
ONP = NP iff NP is in P/poly [FSW09].
If NE is not E then ONP is not P [GM15].
See also YP for an input oblivious analogue of NP ∩ coNP.
The class of decision problems for which there exists a polynomial-time machine M such that:
Defined in a recent post of the blog Shtetl-Optimized. See there for an explanation of the class's name.
Contains ZPP by the same argument that places BPP in P/poly.
Also contains P with TALLY ∩ NP ∩ coNP oracle.
Is contained in NP ∩ coNP and YPP.
Is equal to ONP ∩ coONP.
Defined as RBQP ∩ coRBQP. Equivalently, the class of problems in NP ∩ coNP such that both positive and negative witnesses are in FBQP.
For example, the language of square-free numbers is in ZBQP, because factoring is in FBQP and a factorization can be certified in ZPP (indeed in P, by [AKS02]).
Unlike EQP or ZQP, ZBQP would generalize ZPP in practice if quantum computers existed, in the sense that it provides proven answers.
Contains ZPP and is contained in RBQP and ZQP. Also, ZBQPZBQP = ZBQP. Defined here to clarify EQP and ZQP.