The class of function problems of the form "compute f(x)," where f is the number of accepting paths of an NP machine.
The canonical #P-complete problem is #SAT.
Defined in [Val79], where it was also shown that #Perfect Matching (or equivalently, Permanent) is #P-complete. What makes that interesting is that the associated decision problem Perfect Matching is in P.
Any function in #P can be approximated to within a polynomial factor in BPP with NP oracle [Sto85]. Likewise, any problem in #P can be approximated to within a constant factor by a machine in FP||NP running in time [SU05].
Has the same relation to L as #P does to P.
#L is contained in the function class version of DET [AJ93]. In fact, the determinant is GapL-complete (see refs at GapL), where GapL consists of functions that are the difference of two #L functions.
Roughly, the analogue of #P for parameterized complexity. I.e. the class of parameterized counting problems that are fixed-parameter parsimonious reducible to #WSAT.Defined in [FG02], which should be consulted for the full definition. [FG02] also showed that there exist #W[1]-complete problems whose corresponding decision problems are fixed-parameter tractable (i.e. in FPT).
Complete problems for #W[1] include counting the paths or cycles of a given length in a graph and counting the chordless paths of a given length (the latter both under parsimonious and Turing reductions). Complete problems for #W[2] include the maximal chordless path problem. [CF07]
Same as SBP, except that f is a nonnegative-valued GapP function rather than a #P function.
Defined in [Vya03], where the following was also shown:
Kuperberg ([Kup09]) showed that A0PP = SBQP.
The class of decision problems such that for some #P function f(x,0m),
Defined in [GKR+95].
Contains PH and ModPH. Contained in MP.
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.
Equals BPTIME(2O((log n)^k)); that is, the class of problems solvable in quasipolynomial-time on a bounded-error machine.
Defined in [CNS99], where the following was also shown:
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].
The class of #P function problems such that some underlying NP machine witnessing membership in #P has"clustered" accepting paths. That is:
Defined in [HHK+05].
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 subclass of #P counting problems whose answer, y, is approximable in the following sense. There exists a randomized algorithm that, with probability at least 1-δ, approximates y to within an ε multiplicative factor in time polynomial in n (the input size), 1/ε, and log(1/δ).
The permanent of a nonnegative matrix is in FPRAS [JSV01].
Has the same relation to L as GapP does to P. (And therefore, has the same relation to #L as GapP does to #P.)
The determinant is GapL-complete [Vin91] [Dam91] [Tod91] ([MV97] also gave a new, self-contained proof). See also the corresponding decision class DET
The class of functions f(x) such that for some NP machine, f(x) is the number of accepting paths minus the number of rejecting paths.
Equivalently, the closure of the #P functions under subtraction.
Defined in [FFK94] and independently [Gup95].
The class of decision problems such that for some #P function f, the answer on input x is 'yes' if and only if the middle bit of f(x) is 1.
Defined in [GKR+95].
Contains AmpMP, and ModPH, which is low for both AmpMP and MP. Contained in P#P[1].
MP with ModP oracle equals MP with #P oracle [KT96].
I decided this class is so important that it deserves an entry of its own, apart from #P.
Contains PH [Tod89], and is contained in CH and in PSPACE.
Equals PPP (exercise for the visitor).
Contains PH [Tod89]. Indeed, contains MP, which in turn contains ModPH [GKR+95].
A level of the counting hierarchy CH.
It is not known whether there exists an oracle relative to which PPP does not equal PSPACE.
Equals P#P (exercise for the visitor).
Since the permanent of a matrix is #P-complete [Val79], Toda's theorem implies that any problem in the polynomial hierarchy can be solved by computing a sequence of permanents.
The class of decision problems solvable by a Turing machine in polynomial space.
Equals NPSPACE [Sav70], AP [CKS81], and CZK assuming the existence of one-way functions [BGG+90].
Equals IP [Sha90], but PSPACE strictly contains IP with probability 1 [CCG+94].
Contains P#P (P with a #P oracle).
A canonical PSPACE-complete problem is QBF.
Relative to a random oracle, PSPACE strictly contains PH with probability 1 [Cai86].
PSPACE has a complete problem that is both downward self-reducible and random self-reducible [TV02]. It is the largest class with such a complete problem.
Contained in EXP. There exists an oracle relative to which this containment is proper [Dek76].
In descriptive complexity, PSPACE can be defined with 4 differents kind of formulae, FO() which is also FO(PFP) and SO() which is also SO(TC).
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].
The class of functions computable as |S|, where S is the set of output values returned by the accepting paths of an NL machine.
Defined in [AJ93], where it is also shown that span-L is a hard class in the sense that span-L is contained in FP if and only if FP = #P.
Span-L is contained in #P, and if span-L = #P, then NL = NP [AJ93].
Span-L is contained in FPRAS [ACJ+21].
The class of functions computable as |S|, where S is the set of output values returned by the accepting paths of an NP machine.
Defined in [KST+89], where it is also shown that span-P contains #P and OptP; and that span-P = #P if and only if UP = NP.
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.
The class of decision problems for which there exists a #P function f, a polynomial p, and an ε > 0, such that for all inputs x,
Defined in [BGM02], where it is also shown that WAPP is contained in AWPP and SBP.
