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Complexity Zoo:B - Complexity Zoo

β_kFOLL is the class of decision problems solvable by a uniform family of polynomial-sized FOLL circuits that accept O(log^k n) nondeterministic input bits, with the same acceptance mechanism as FOLL.

βFOLL is the union of β_kFOLL over all constant k.

We note that β_kFOLL = NFOLL(log^k n).

The notion of a non-deterministic NC circuit was introduced in [Wol94]; see NNC(f(n)).

Chattopadhyay, Torán, and Wagner [CTW13] showed that β₂FOLL contains Quasigroup Isomorphism when the inputs are given by their multiplication tables. Additionally, Chattopadhyay, Torán, and Wagner [CTW13] showed that β_kFO((log log n)^c) cannot compute Parity for any constants c and k. As a consequence, this rules out many-to-one AC⁰-computable reduction from Graph Isomorphism to Quasigroup Isomorphism.

β_kMAC⁰ is the class of decision problems solvable by a uniform family of polynomial-sized MAC⁰ circuits that accept O(log^k n) nondeterministic input bits, with the same acceptance mechanism as MAC⁰.

βMAC⁰ is the union of β_kMAC⁰ over all constant k.

For any constant k, β_kMAC⁰ is contained in β_kTC⁰. In particular, β₁MAC⁰ is contained in β₁TC⁰ = TC⁰.

We note that β_kMAC⁰ = NMAC⁰(log^k n).

The notion of a non-deterministic NC circuit was introduced in [Wol94]; see NNC(f(n)).

β_kP is the class of decision problems solvable by a polynomial-time Turing machine that makes O(log^kn) nondeterministic transitions, with the same acceptance mechanism as NP. Equivalently, the machine receives a purported proof of size O(log^kn) that the answer is 'yes.'

Then βP is the union of β_kP over all constant k.

Defined in [KF84]. See also the survey [GLM96].

There exist oracles relative to which basically any consistent inclusion structure among the β_kP's can be realized [BG98].

β₂P contains LOGNP and LOGSNP.

The class of decision problems solvable by an NP machine such that

1. If the answer is 'yes' then exactly 1/2 of the computation paths accept.
2. If the answer is 'no' then either at most 1/4 or at least 3/4 of the computation paths accept.

(Here all computation paths have the same length.)

Defined in [Wat15], where it was shown that BC₌P admits efficient amplification and is closed under union, intersection, disjunction, and conjunction, and that coRP ⊆ BC₌P ⊆ BPP.

The smallest class that contains NP and is closed under union, intersection, and complement.

The levels are defined as follows:

• BH₁ = NP.
• BH_2i is the class of languages that are the intersection of a BH_2i-1 language with a coNP language.
• BH_2i+1 is the class of languages that are the union of a BH_2i language with an NP language.

Then BH is the union over all i of BH_i.

Defined in [WW85].

For more detail see [CGH+88].

Contained in Δ₂P and indeed in P^NP[log].

If BH collapses at any level, then PH collapses to Σ₃P [Kad88].

See also: DP, QH.

Defined in [Weg88].

The class of decision problems solvable by a family of polynomial size branching programs, with the additional condition that each bit of the input is tested at most d times.

BP_d(P) strictly contains BP_d-1(P), for every d > 1 [Tha98].

Contained in PBP.

See also: P-OBDD, k-PBP.

Has the same relation to E as BPP does to P.

EE = BPE if and only if EXP = BPP [IKW01].

Has the same relation to EE as BPP does to P.

The class of decision problems solvable in O(f(n))-space with error probability at most 1/3, by a Turing machine that halts on every input and every random tape setting.

Contains R_HSPACE(f(n)).

Is contained in DSPACE(f(n)^3/2) [SZ95].

Has the same relation to L as BPP does to P. The Turing machine has to halt for every input and every randomness.

The randomness is read once. That is, each random bit is only given once and to be referenced again it must be stored in the working space. This in contrast to the two way randomness of BP•L.

Contained in SC [Nis92], PL, BP•L, ZP•L [Nis93], DET [Coo85], NC² and P [BCP83].

Languages decided with two sided bounded error by a log space machine with a read only tape containing random bits. In contrast with BPL, the random bits can be read multiple times without storing them in working space.

For example, BP•L contains RNC¹ since NC¹ is contained in L. However, this reduction does not hold for BPL.

It is unknown if BP•L is contained in P [Nis93].

Contained in BPP.

Contains BPL and ZP•L.

Equals AM.

The class of decision problems solvable by an NP machine such that

1. If the answer is 'yes' then at least 2/3 of the computation paths accept.
2. If the answer is 'no' then at most 1/3 of the computation paths accept.

(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 ZPP^NP [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 2^Ω(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(n^c) 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: BPP_path.

The analogue of P^cc for bounded-error probabilistic communication complexity.

Does not equal P^cc, and is not contained in NP^cc, because of the EQUALITY problem.

If the classes are defined in terms of partial functions, then BPP^cc

• is not contained in P^NPcc [PPS14];
• does not contain NP^cc ∩ coNP^cc [Kla03].

Has the same relation to BPP^cc and BPP as P _{k {\displaystyle k}} ^cc does to P^cc and P.

NP _{k {\displaystyle k}} ^cc is not contained in BPP _{k {\displaystyle k}} ^cc for k ≤ ( 1 − δ ) ⋅ log ⁡ n {\displaystyle k\leq (1-\delta )\cdot \log n} players, for any constant δ > 0 {\displaystyle \delta >0} [DP08].

BPP with an oracle that, given a string x, returns the minimum over all programs P that output x_i on input i, of the length of P plus the maximum time taken by P on any input.

A similar class was defined in [ABK+02], where it was also shown that in BPP^KT one can factor, compute discrete logarithms, and more generally invert any one-way function on a non-negligible fraction of inputs.

See also: P^K.

The class of problems solvable by a semantic BPP machine with O(log n) advice bits that depend only on the input length n. If the advice is good, the output must be correct with probability at least 2/3. If it is bad, the machine must provide some answer with probability at least 2/3. See the discussion for BQP/poly.

Contained in BPP/mlog.

The class of problems solvable by a syntactic BPP machine with O(log n) advice bits that depend only on the input length n. If the advice is good, the output must be correct with probability at least 2/3. If it is bad, it need not be.

Contained in BPP/rlog.

The class of problems solvable by a BPP machine that is given O(log n) advice bits, which can depend on both the machine's random coin flips and the input length n, but not on the input itself.

Defined in [TV02], where it was also shown that if EXP is in BPP//log then EXP = BPP, and if PSPACE is in BPP//log then PSPACE = BPP.

The class of problems solvable by a syntactic BPP machine with O(log n) random advice bits whose probability distribution depends only on the input length n. For each n, there exists good advice such that the output is correct with probability at least 2/3.

Contains BPP/mlog. The inclusion is strict, because BPP/rlog contains any finitely sparse language by fingerprinting; see the discussion for ALL.

Contained in BPP//log.

Same as P-OBDD, except that probabilistic transitions are allowed and the OBDD need only accept with probability at least 2/3.

Does not contain the integer multiplication problem [AK96].

Strictly contained in BQP-OBDD [NHK00].

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:

• BPP_path contains MA and P^NP[log], and is contained in PP and BPP^NP.
• BPP_path is closed under complementation, intersection, and union.
• If BPP_path = BPP_path^BPP_path, then PH collapses to BPP_path.
• If BPP_path contains Σ₂P, then PH collapses to BPP^NP.

There exists an oracle relative to which BPP_path is not contained in Σ₂P [BGM02].

An alternate characterization of BPP_path uses the idea of post-selection. That is, BPP_path is the class of languages L {\displaystyle L} for which there exists a pair of polynomial-time Turing machines A {\displaystyle A} and B {\displaystyle B} such that the following conditions hold for all x {\displaystyle x} :

• If x ∈ L {\displaystyle x\in L} , Pr r ∈ { 0 , 1 } p o l y ( | x | ) [ A ( x , r ) ∣ B ( x , r ) ] ≥ 2 3 {\displaystyle \Pr _{r\in \{0,1\}^{\mathrm {poly} (\left\vert x\right\vert )}}\left[A(x,r)\mid B(x,r)\right]\geq {\frac {2}{3}}} .
• If x ∉ L {\displaystyle x\notin L} , Pr r ∈ { 0 , 1 } p o l y ( | x | ) [ A ( x , r ) ∣ B ( x , r ) ] < 1 3 {\displaystyle \Pr _{r\in \{0,1\}^{\mathrm {poly} (\left\vert x\right\vert )}}\left[A(x,r)\mid B(x,r)\right]<{\frac {1}{3}}} .
• Pr r ∈ { 0 , 1 } p o l y ( n ) [ B ( x , r ) ] > 0 {\displaystyle \Pr _{r\in \{0,1\}^{\mathrm {poly} (n)}}\left[B(x,r)\right]>0} .

We say that B {\displaystyle B} 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 ⊆ BPP_path nearly trivial.

See Also: PostBQP (quantum analogue).

Equals BPTIME(2^{O((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:

If either (1) #P does not have a subexponential-time bounded-error algorithm, or (2) EXP does not have subexponential-size circuits, then the BPQP hierarchy is strict -- that is, for all a < b at least 1, BPTIME(2^{(log n)^a}) is strictly contained in BPTIME(2^{(log n)^b}).

The class of decision problems solvable in O(f(n))-space with error probability at most 1/3, by a Turing machine that halts with probability 1 on every input.

Contains RSPACE(f(n)) and BP_HSPACE(f(n)).

Same as BPP, but with f(n)-time (for some constructible function f) rather than polynomial-time machines.

Defined in [Gil77].

BPTIME(n^{log n}) does not equal BPTIME(2^{n^ε}) for any ε>0 [KV88]. Proving a stronger time hierarchy theorem for BPTIME is a longstanding open problem; see [BH97] for details.

[Bar02] has shown the following:

• If we allow a small number of advice bits (say log n), then there is a strict hierarchy: for every d at least 1, BPTIME(n^d)/(log n) does not equal BPTIME(n^d+1)/(log n).
• In the uniform setting, if BPP has complete problems then BPTIME(n^d) does not equal BPTIME(n^d+1).
• BPTIME(n) does not equal NP.

Subsequently, [FS04] managed to reduce the number of advice bits to only 1: BPTIME(n^d)/1 does not equal BPTIME(n^d+1)/1. They also proved a hierarchy theorem for HeurBPTIME.

For another bounded-error hierarchy result, see BPQP.

The class of decision problems solvable by a quantum Turing machine using O(log n) many qubits and polynomial time, with at most 1/3 probability of error.

Whether or not intermediate measurements are allowed does not change the resulting class [FR21].

Contained in DET [FR21].

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.

BQP^BQP = 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:

• BQP does not equal to BPP [BV97] (and by similar arguments, is not in P/poly).
• BQP is not contained in MA [Wat00].
• BQP is not contained in PH [RT18] (see also [Wu18]).
• BQP is not contained in Mod_pP for prime p [GV02].
• NP, and indeed NP ∩ coNP, are not contained in BQP with probability 1 relative to a random oracle and a random permutation oracle, respectively [BBB+97].
• SZK is not contained in BQP [Aar02].
• BQP is not contained in SZK (follows easily using the quantum walk problem in [CCD+03]).
• BQP is not contained in BPP_path [Aar10] (see also [Che16]).
• PPAD is not contained in BQP [Li11].
• P=BQP and PH is infinite [FR98].

If P=BQP relative to a random oracle then BQP=BPP [FR98].

Same as BQP/poly except that the advice is O(log n) bits instead of a polynomial number.

Contained in BQP/mlog.

Is to BQP/mpoly as ∃BPP is to MA. Namely, the BQP machine is required to give some answer with probability at least 2/3 even if the advice is bad. Even though BQP/mpoly is a more natural class, BQP/poly follows the standard definition of advice as a class operator [KL82].

Contained in BQP/mpoly and contains BQP/log.

Same as BQP/mpoly except that the advice is O(log n) bits instead of a polynomial number.

Strictly contained in BQP/qlog [NY03].

The class of languages recognized by a syntactic BQP machine with deterministic polynomial advice that depends only on the input length, such that the output is correct with probability 2/3 when the advice is good.

Can also be defined as the class of problems solvable by a nonuniform family of polynomial-size quantum circuits, just as P/poly is the class solvable by a nonuniform family of polynomial-size classical circuits.

Referred to with a variety of other ad hoc names, including BQP/poly on occassion.

Contains BQP/qlog, and is contained in BQP/qpoly.

Does not contain ESPACE [NY03].

Same as BQP/mlog except that the advice is quantum instead of classical.

Strictly contains BQP/mlog [NY03].

Contained in BQP/mpoly.

The class of problems solvable by a BQP machine that receives a quantum state ψ_n as advice, which depends only on the input length n.

As with BQP/mpoly, the acceptance probability does not need to be bounded away from 1/2 if the machine is given bad advice. (Thus, we are discussing the class that [NY03] call BQP/*Qpoly.) Indeed, such a condition would make quantum advice unusable, by a continuity argument.

Does not contain EESPACE [NY03].

[AD14] showed that BQP/qpoly = YQP/poly.

There exists an oracle relative to which BQP/qpoly does not contain NP [Aar04b].

A classical oracle separation between BQP/qpoly and BQP/mpoly is presently unknown, but there is a quantum oracle separation [AK06]. An unrelativized separation is too much to hope for, since it would imply that PP is not contained in P/poly.

Contains BQP/mpoly.

Does not contain PP unless CH collapses [Aar06],[Yir24].

Same as P-OBDD, except that unitary (quantum) transitions are allowed and the OBDD need only accept with probability at least 2/3.

Strictly contains BPP-OBDD [NHK00].

Equals PSPACE and PPSPACE.

Same as BQP, but with f(n)-time (for some constructible function f) rather than polynomial-time machines.

Defined in [BV97].

Same as BQP with access to two sets of qubits: causality-respecting qubits and CTC qubits.

Defined in [Aar05c], where it was shown that PSPACE is contained in BQP_CTC, which in turn is contained in SQG = PSPACE. Therefore, BQP_CTC = PSPACE; this was also shown in [AW09].

See also P_CTC.

Same as BQP/mpoly, except that the machine only gets to make nonadaptive queries to whatever oracle it might have.

Defined in [NY03b], where it was also shown that P is not contained in BQP_tt/poly relative to an oracle.

Alternate name for k-PBP.

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