Complexity class

In computational complexity theory, a complexity class is a set of problems of related complexity. A typical complexity class has a definition of the form:

the set of problems that can be solved by abstract machine M using O(f(n)) of resource R (n is the size of the input)

For example, the class NP is the set of decision problems that can be solved by a non-deterministic Turing machine in polynomial time, while the class PSPACE is the set of decision problems that can be solved by a deterministic Turing machine in polynomial space. Some complexity classes are sets of function problems, such as FP.

Many complexity classes can be characterized in terms of the mathematical logic needed to express them; see descriptive complexity.

The Blum axioms can be used to define complexity classes without referring to a concrete computational model.

Relationships between complexity classes

The following table shows some of the classes of problems (or languages, or grammars) that are considered in complexity theory. If class X is a strict subset of Y, then X is shown below Y, with a dark line connecting them. If X is a subset, but it is unknown whether they are equal sets, then the line is lighter and is dotted. Technically, the breakdown into solvable and unsolvable belongs more in computability theory but it helps put complexity classes in perspective.

Decision Problem

Type 0 (Recursively enumerable)

Type 1 (Context Sensitive)