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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 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.) "The hardest problems" of a class refer to problems, which belong to the class and 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. 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 2-EXPTIME Solvable with 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 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 GapL Logspace-reducible to computing the integer determinant of a matrix IP Solvable in polynomial time by an interactive proof system L Solvable in 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 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 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 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 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 ½) PR Solvable by recursively building up arithmetic functions. PSPACE Solvable with polynomial memory. PSPACE-complete The hardest problems in PSPACE. 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 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) 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 running space is logarithmic) ZPP Solvable by randomized algorithms (answer is always right, average running time is polynomial) |
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