Datasets:
wesley7137
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quantum

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yes
∈
accepts(x,y)forsomestringy Ξ£p( x ),andforeverystringx A ,itholdsthat
| | no
∈ ∈
Mrejects(x,y) forallstringsy Ξ£p( x ).
| |
∈
BPP A promise problem A = (A ,A ) is in BPP if and only if there exists a
yes no
polynomial-timeprobabilisticTuringmachine Mthatacceptseverystringx A
yes
∈
with probability at least 2/3, and accepts everystring x A with probability at
no
∈
most1/3.
PP ApromiseproblemA = (A ,A )isinPPifandonlyifthereexistsapolynomial-
yes no
timeprobabilisticTuringmachine M thatacceptseverystring x A withprob-
yes
∈
ability strictlygreaterthan 1/2, and accepts everystring x A with probability
no
∈
atmost1/2.
MA A promise problem A = (A ,A ) is in MA if and only if there exists a
yes no
polynomial-bounded function p and a probabilistic polynomial-time Turing ma-
chine M with the following properties. For every string x A , it holds that
yes
∈
Pr[M accepts(x,y)] 2 for some string y Ξ£p( x ); and for every string x A ,
β‰₯ 3 ∈ | | ∈ no
itholdsthatPr[M accepts(x,y)] 1 forallstringsy Ξ£p( x ).
≀ 3 ∈ | |
AM Apromiseproblem A = (A ,A )isinAMifandonlyifthereexistpolynomial-
yes no
bounded functions p and q and a polynomial-time deterministic Turing machine
M with the following properties. For everystring x A , and at least 2/3 ofall
yes
∈
stringsy Ξ£p( x ),thereexistsastringz Ξ£q( x ) suchthat M accepts(x,y,z); and
| | | |
∈ ∈
for every string x A , and at least 2/3 of all strings y Ξ£p( x ), there are no
no | |
∈ ∈
stringsz Ξ£q( x ) suchthat M accepts(x,y,z).
| |
∈
SZK Apromiseproblem A = (A ,A )isinSZKifandonlyifithasastatisticalzero-
yes no
knowledgeinteractiveproofsystem.
PSPACE A promise problem A = (A ,A ) is in PSPACE if and only if there exists a
yes no
deterministic Turing machine M running in polynomial space that accepts every
string x A andrejectseverystring x A .
yes no
∈ ∈
EXP Apromiseproblem A = (A ,A )isinEXPifandonlyifthereexistsadetermin-
yes no
isticTuringmachine Mrunninginexponentialtime(meaningtimeboundedby2p,
for somepolynomial-bounded function p), that accepts every string x A and
yes
∈
rejectseverystring x A .
no
∈
NEXP A promise problem A = (A ,A ) is in NEXP if and only if there exists an
yes no
exponential-timenon-deterministicTuringmachine N for A.
4
NEXP
EXP
PSPACE
AM PP
SZK MA
BPP NP
P
NC
PL
Figure1: Adiagramillustratingknowninclusionsamongmostoftheclassicalcomplexityclasses
discussedinthispaper. Linesindicatecontainmentsgoingupward;forexample,AMiscontained
inPSPACE.
PL ApromiseproblemA = (A ,A )isinPLifandonlyifthereexistsaprobabilistic
yes no
Turingmachine M runninginpolynomialtimeandlogarithmicspacethataccepts
every string x A with probability strictly greater than 1/2 and accepts every
yes
∈
string x A withprobabilityatmost1/2.
no
∈
NC A promise problem A = (A ,A ) is in NC if and only if there exists a
yes no
logarithmic-space generatedfamily C = C : n N ofpoly-logarithmic depth
n
{ ∈ }
BooleancircuitssuchthatC(x) = 1forall x A andC(x) = 0forall x A .