Important Complexity Classes
Many important complexity classes can be defined by bounding the time or space used by the algorithm. Some important complexity classes of decision problems defined in this manner are the following:
| Complexity class | Model of computation | Resource constraint |
|---|---|---|
| DTIME(f(n)) | Deterministic Turing machine | Time f(n) |
| P | Deterministic Turing machine | Time poly(n) |
| EXPTIME | Deterministic Turing machine | Time 2poly(n) |
| NTIME(f(n)) | Non-deterministic Turing machine | Time f(n) |
| NP | Non-deterministic Turing machine | Time poly(n) |
| NEXPTIME | Non-deterministic Turing machine | Time 2poly(n) |
| DSPACE(f(n)) | Deterministic Turing machine | Space f(n) |
| L | Deterministic Turing machine | Space O(log n) |
| PSPACE | Deterministic Turing machine | Space poly(n) |
| EXPSPACE | Deterministic Turing machine | Space 2poly(n) |
| NSPACE(f(n)) | Non-deterministic Turing machine | Space f(n) |
| NL | Non-deterministic Turing machine | Space O(log n) |
| NPSPACE | Non-deterministic Turing machine | Space poly(n) |
| NEXPSPACE | Non-deterministic Turing machine | Space 2poly(n) |
It turns out that PSPACE = NPSPACE and EXPSPACE = NEXPSPACE by Savitch's theorem.
Other important complexity classes include BPP, ZPP and RP, which are defined using probabilistic Turing machines; AC and NC, which are defined using boolean circuits and BQP and QMA, which are defined using quantum Turing machines. #P is an important complexity class of counting problems (not decision problems). Classes like IP and AM are defined using Interactive proof systems. ALL is the class of all decision problems.
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