List of complexity classes

A representation of the relation among complexity classes

This is a list of complexity classes in computational complexity theory. For other computational and complexity subjects, see list of computability and complexity topics.

Many of these classes have a 'co' partner which consists of the complements of all languages in the original class. For example, if a language L is in NP then the complement of L is in co-NP. (This does not mean that the complement of NP is co-NP—there are languages which are known to be in both, and other languages which are known to be in neither.)

"The hardest problems" of a class refer to problems which belong to the class such that every other problem of that class can be reduced to it.

#P Count solutions to an NP problem
#P-complete The hardest problems in #P
2-EXPTIME Solvable in doubly exponential time
AC0 A circuit complexity class of bounded depth
ACC0 A circuit complexity class of bounded depth and counting gates
AC A circuit complexity class
AH The arithmetic hierarchy
AP The class of problems alternating Turing machines can solve in polynomial time.[1]
APX Optimization problems that have approximation algorithms with constant approximation ratio[1]
AM Solvable in polynomial time by an Arthur–Merlin protocol[1]
BPP Solvable in polynomial time by randomized algorithms (answer is probably right)
BPL problems solvable in logarithmic space and polynomial time with probabilistic Turing machines with two-sided error
BQP Solvable in polynomial time on a quantum computer (answer is probably right)
co-NP "NO" answers checkable in polynomial time by a non-deterministic machine
co-NP-complete The hardest problems in co-NP
DLIN Solvable by a deterministic multitape Turing machine in time O(n).
DSPACE(f(n)) Solvable by a deterministic machine with space O(f(n)).
DTIME(f(n)) Solvable by a deterministic machine in time O(f(n)).
E Solvable in exponential time with linear exponent
ELEMENTARY The union of the classes in the exponential hierarchy
ESPACE Solvable with exponential space with linear exponent
EXP Same as EXPTIME
EXPSPACE Solvable with exponential space
EXPTIME Solvable in exponential time
FNP The analogue of NP for function problems
FP The analogue of P for function problems
FPNP The analogue of PNP for function problems; the home of the traveling salesman problem
FPT Fixed-parameter tractable
GapL Logspace-reducible to computing the integer determinant of a matrix
IP Solvable in polynomial time by an interactive proof system
L Solvable with logarithmic (small) space
LOGCFL Logspace-reducible to a context-free language
MA Solvable in polynomial time by a Merlin–Arthur protocol
NC Solvable efficiently (in polylogarithmic time) on parallel computers
NE Solvable by a non-deterministic machine in exponential time with linear exponent
NESPACE Solvable by a non-deterministic machine with exponential space with linear exponent
NEXP Same as NEXPTIME
NEXPSPACE Solvable by a non-deterministic machine with exponential space
NEXPTIME Solvable by a non-deterministic machine in exponential time
NL "YES" answers checkable with logarithmic space
NLIN Solvable by a nondeterministic multitape Turing machine in time O(n).
NONELEMENTARY Complement of ELEMENTARY.
NP "YES" answers checkable in polynomial time (see complexity classes P and NP)
NP-complete The hardest or most expressive problems in NP
NP-easy Analogue to PNP for function problems; another name for FPNP
NP-equivalent The hardest problems in FPNP
NP-hard At least as hard as every problem in NP but not known to be in the same complexity class
NSPACE(f(n)) Solvable by a non-deterministic machine with space O(f(n)).
NTIME(f(n)) Solvable by a non-deterministic machine in time O(f(n)).
P Solvable in polynomial time
P-complete The hardest problems in P to solve on parallel computers
P/poly Solvable in polynomial time given an "advice string" depending only on the input size
PCP Probabilistically Checkable Proof
PH The union of the classes in the polynomial hierarchy
PL solvable in polynomial time with a logarithmic space randomized machine with probability > 1⁄2
PNP Solvable in polynomial time with an oracle for a problem in NP; also known as Δ2P
PP Probabilistically Polynomial (answer is right with probability slightly more than 1/2)
PPAD Polynomial Parity Arguments on Directed graphs
PR Solvable by recursively building up arithmetic functions.
PSPACE Solvable with polynomial space.
PSPACE-complete The hardest problems in PSPACE.
PTAS Polynomial-time approximation scheme (a subclass of APX).
QIP Solvable in polynomial time by a quantum interactive proof system.
QMA Quantum analog of NP.
R Solvable in a finite amount of time.
RE Problems to which we can answer "YES" in a finite amount of time, but a "NO" answer might never come.
RL Solvable with logarithmic space by randomized algorithms (NO answer is probably right, YES is certainly right)
RP Solvable in polynomial time by randomized algorithms (NO answer is probably right, YES is certainly right)
SL Problems log-space reducible to determining if a path exist between given vertices in an undirected graph. In October 2004 it was discovered that this class is in fact equal to L.
S2P one round games with simultaneous moves refereed deterministically in polynomial time[2]
TFNP Total function problems solvable in non-deterministic polynomial time. A problem in this class has the property that every input has an output whose validity may be checked efficiently, and the computational challenge is to find a valid output.
UP Unambiguous Non-Deterministic Polytime functions.
ZPL Solvable by randomized algorithms (answer is always right, average space usage is logarithmic)
ZPP Solvable by randomized algorithms (answer is always right, average running time is polynomial)

References

  1. ^ a b c Sanjeev Arora, Boaz Barak (2009), Computational Complexity: A Modern Approach, Cambridge University Press; 1 edition, ISBN 978-0-521-42426-4
  2. ^ "S2P: Second Level of the Symmetric Hierarchy". Stanford University Complexity Zoo. Archived from the original on 2012-10-14. Retrieved 2011-10-27.
  • Complexity Zoo - list of over 500 complexity classes and their properties

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