List of complexity classes

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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

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#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)
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
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

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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.

## External links

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* Complexity Zoo \- list of over 500 complexity classes and their properties