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
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| #P-complete | The hardest problems in #P
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| AM | Solvable in polynomial time by an Arthur-Merlin protocol
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| BPP | Solvable in polynomial time by randomized algorithms (answer is probably right)
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| BQP | Solvable in polynomial time on a quantum computer (answer is probably right)
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| Co-NP | NO answers checkable in polynomial time
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| Co-NP-complete | The hardest problems in Co-NP
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| DSPACE(f(n)) | Solvable by a deterministic machine in space O(f(n)).
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| DTIME(f(n)) | Solvable by a deterministic machine in time O(f(n)).
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| E | Solvable in exponential time with linear exponent
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| ELEMENTARY | The union of the classes in the exponential hierarchy
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| ESPACE | Solvable in exponential space with linear exponent
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| EXP | Same as EXPTIME
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| EXPSPACE | Solvable in exponential space
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| EXPTIME | Solvable with exponential time
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| FNP | The analogue of NP for function problems
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| FP | The analogue of P for function problems
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| FPNP | The analogue of PNP for function problems; the home of the traveling salesman problem
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| IP | Solvable in polynomial time by an interactive proof system
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| MA | Solvable in polynomial time by an Merlin-Arthur protocol
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| NC | Solvable efficiently (in polylogarithmic time) on parallel computers
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| NE | Solvable by a non-deterministic machine in exponential time with linear exponent
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| NESPACE | Solvable by a non-deterministic machine in exponential space with linear exponent
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| NEXP | Same as NEXPTIME
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| NEXPSPACE | Solvable by a non-deterministic machine in exponential space
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| NEXPTIME | Solvable by a non-deterministic machine in exponential time
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| NP | YES answers checkable in polynomial time (see complexity classes P and NP)
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| NP-complete | The hardest problems in NP
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| NP-easy | Analogue to PNP for function problems; another name for FPNP
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| NP-equivalent | The hardest problems in FPNP
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| NP-hard | Either NP-complete or harder
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| NSPACE(f(n)) | Solvable by a non-deterministic machine in space O(f(n)).
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| NTIME(f(n)) | Solvable by a non-deterministic machine in time O(f(n)).
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| P | Solvable in polynomial time
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| P-complete | The hardest problems in P to solve on parallel computers
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| PCP | Probabilistically Checkable Proof
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| PH | The union of the classes in the polynomial hierarchy
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| PNP | Solvable in polynomial time with an oracle for a problem in NP; also known as Δ2P
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| PP | Probabilistically Polynomial (answer is right with probability slightly more than ½)
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| PSPACE | Solvable with polynomial memory and unlimited time
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| PSPACE-complete | The hardest problems in PSPACE
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| RP | Solvable in polynomial time by randomized algorithms (NO answer is probably right, YES is certainly right)
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| UP | Unambiguous Non-Deterministic Polytime functions.
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| ZPP | Solvable by randomized algorithms (answer is always right, average running time is polynomial)
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