# 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.

See computation for a chart showing which classes are subsets of other classes.

Many of these classes have a 'Co' partner which consists of the complements of all languages in the original class. For example if L is in NP then the complement of L is in Co-NP. (This doesn't 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.)

If you don't see a class listed (such as Co-UP) you should look under its partner (such as UP).

 #P Count solutions to an NP problem #P-complete The hardest problems in #P APX Optimization problems that have approximation algorithms with constant approximation ratio AM Solvable in polynomial time by an Arthur-Merlin protocol 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 Co-NP-complete The hardest problems in Co-NP DSPACE(f(n)) Solvable by a deterministic machine in 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 in exponential space with linear exponent EXP Same as EXPTIME EXPSPACE Solvable in exponential space EXPTIME Solvable with 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 IP Solvable in polynomial time by an interactive proof system L Solvable in logarithmic (small) space 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 in exponential space with linear exponent NEXP Same as NEXPTIME NEXPSPACE Solvable by a non-deterministic machine in exponential space NEXPTIME Solvable by a non-deterministic machine in exponential time NL "YES" answers checkable in logarithmic space 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 Either NP-complete or harder NSPACE(f(n)) Solvable by a non-deterministic machine in 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 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 ½) PSPACE Solvable with polynomial memory and unlimited time PSPACE-complete The hardest problems in PSPACE RL Solvable in 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) RLP Solvable in logarithmic space and 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. UP Unambiguous Non-Deterministic Polytime functions. ZPP Solvable by randomized algorithms (answer is always right, average running time is polynomial)

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