Equals DTIME(2O(n)).
Does not equal NP [Boo72] or PSPACE [Boo74] relative to any oracle. However, there is an oracle relative to which E is contained in NP (see ZPP), and an oracle relative to PSPACE is contained in E (by equating the former with P).
There exists a problem that is complete for E under polynomial-time Turing reductions but not polynomial-time truth-table reductions [Wat87].
Problems hard for BPP under Turing reductions have measure 1 in E [AS94].
It follows that, if the problems complete for E under Turing reductions do not have measure 1 in E, then BPP does not equal EXP.
[IT89] gave an oracle relative to which E = NE but still there is an exponential-time binary predicate whose corresponding search problem is not in E.
[BF03] gave a proof that if E = NE, then no sparse set is collapsing, where they defined a set to be collapsing if and if for all such that and are Turing reducible to each other, and are many-to-one reducible to each other.
Contrast with EXP.
Same as AC0[m], but now the constant-depth circuit can contain MOD m gates for any m.
Contained in TC0.
Indeed, can be simulated by depth-3 threshold circuits of quasipolynomial size [Yao90].
According to [All96], there is no good evidence for the existence of cryptographically secure functions in ACC0.
There is no non-uniform ACC0 circuits of polynomial size for NTIMES[2n] and no ACC0 circuit of size 2nO(1) for ENP (The class E with an NP oracle). These are the only two known nontrivial lower bounds against ACC0 and were recently discovered by [Wil11].
Has the same relation to E as AvgP does to P.
Has the same relation to E as BPP does to P.
EE = BPE if and only if EXP = BPP [IKW01].
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.
Has roughly the same relationship to E as PH does to P.
More formally, EH is defined as the union of E, NE, NENP, NE with Σ2P oracle, and so on.
See [Har87] for more information.
If coNP is contained in AM[polylog], then EH collapses to S2-EXP•PNP [SS04] and indeed AMEXP [PV04].
On the other hand, coNE is contained in NE/poly, so perhaps it wouldn't be so surprising if NE collapses.
There exists an oracle relative to which EH does not contain SEH [Hem89]. EH and SEH are incomparable for all anyone knows.
Equals DSPACE(2O(n)).
If E = ESPACE then P = BPP [HY84].
Indeed if E has nonzero measure in ESPACE then P = BPP [Lut91].
ESPACE is not contained in P/poly [Kan82].
Is not contained in BQP/mpoly [NY03].
See also: EXPSPACE.
Equals the union of DTIME(2p(n)) over all polynomials p.
If EXP is in P/poly then EXP = MA [BFL91].
Problems complete for EXP under many-one reductions have measure 0 in EXP [May94], [JL95].
There exist oracles relative to which
[BT04] show the following rather striking result: let A be many-one complete for EXP, and let S be any set in P of subexponential density. Then A-S is Turing-complete for EXP.
[SM03] show that if EXP has circuits of polynomial size, then P can be simulated in MAPOLYLOG such that no deterministic polynomial-time adversary can generate a list of inputs for a P problem that includes one which fails to be simulated. As a result, EXP ⊆ MA if EXP has circuits of polynomial size.
[SU05] show that EXP NP/poly implies EXP P||NP/poly.
In descriptive complexity EXPTIME can be defined as SO() which is also SO(LFP)
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 .
Same as MA, except now Arthur is E instead of polynomial-time.
If MAE = NEE then MA = NEXP ∩ coNEXP [IKW01].
Nondeterministic exponential time with linear exponent (i.e. NTIME(2O(n))).
Contained in NEXP.
The class of decision problems such that whether the answer on input x agrees with the answer on input x-1 (that is, the lexicographic predecessor of x) is solvable in polynomial time. The Turing machine has to decide agreement or disagreement without access to the answer for x-1.
Is contained in E, NT*, and ⊕P. Defined in [GHJ+91] to study ⊕P-complete problems. They show that P, NT, NT*, and ⊕P are either all equal or strictly nested. In particular, they differ with probability 1 relative to a random oracle.
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 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 languages L in UP such that the mapping from an input x to the unique witness for x is a permutation of L.
Contains P.
Defined in [HT03], where it was also shown that the closure of PermUP under polynomial-time one-to-one reductions is UP.
On the other hand, they show that if PermUP = UP then E = UE.
See also: SelfNP.
Equals DTIME(nO(log n)).
Has the same relationship to QP that E does to EXP.
The class of languages L in NP such that the union, over all x in L, of the set of valid witnesses for x equals L itself.
Defined in [HT03], where it was shown that the closure of SelfNP under polynomial-time many-one reductions is NP.
They also show that if SelfNP = NP, then E = NE; and that SAT is contained in SelfNP.
See also: PermUP.
Has the same relation to E as UP does to P.
Same as ZPP, but with 2O(n)-time instead of polynomial-time.
ZPE = EE if and only if ZPP = EXP [IKW01].
