# List of 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. Furthermore, the reduction is also a problem of the given class, or its subset.

#P | Count solutions to an NP problem |

#P-complete | The hardest problems in #P |

2-EXPTIME | Solvable in doubly exponential time |

AC^{0} | A circuit complexity class of bounded depth |

ACC^{0} | 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 |

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 |

FP^{NP} | The analogue of P^{NP} 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 |

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 P^{NP} for function problems; another name for FP^{NP} |

NP-equivalent | The hardest problems in FP^{NP} |

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 |

P^{NP} | Solvable in polynomial time with an oracle for a problem in NP; also known as Δ_{2}P |

PP | Probabilistically Polynomial (answer is right with probability slightly more than ½) |

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). |

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. |

S_{2}P | 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

- Sanjeev Arora, Boaz Barak (2009),
*Computational Complexity: A Modern Approach*, Cambridge University Press; 1 edition, ISBN 978-0-521-42426-4 - "S
_{2}P: Second Level of the Symmetric Hierarchy". Stanford University Complexity Zoo.

## External links

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

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