Complexity class

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

Complexity classes are concerned with the rate of growth of the requirement in resources as the input n increases. It is an abstract measurement, and does not give time or space in requirements in terms of seconds or bytes, which would require knowledge of implementation specifics. The function inside the O(...) expression could be a constant, for algorithms which are unaffected by the size of n, or an expression involving a logarithm, an expression involving a power of n, i.e. a polynomial expression, and many others. The O is read as "order of..". For the purposes of computational complexity theory, some of the details of the function can be ignored, for instance many possible polynomials can be grouped together as a class.

The resource in question can either be time, essentially the number of primitive operations on an abstract machine, or (storage) space. For example, the class NP is the set of decision problems whose solutions can be determined 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.

The simplest complexity classes are defined by the type of computational problem, the model of computation, and the the resource (or resources) that are being bounded and the bounds. The resource and bounds are usually stated together, such as "polynomial time", "logarithmic space", "constant depth", etc.

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

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