In computational complexity theory the Blum axioms or Blum complexity axioms are axioms that specify desirable properties of complexity measures on the set of computable functions. The axioms were first defined by Manuel Blum in 1967. Importantly, Blum's speedup theorem and the Gap theorem hold for any complexity measure satisfying these axioms. The most well-known measures satisfying these axioms are those of time (i.e., running time) and space (i.e., memory usage).

Definitions

A Blum complexity measure is a pair with \varphi a numbering of the partial computable functions and a computable function which satisfies the following Blum axioms. We write \varphi_i for the i-th partial computable function under the Gödel numbering \varphi, and \Phi_i for the partial computable function \Phi(i).

Examples

Complexity classes

For a total computable function f complexity classes of computable functions can be defined as C(f) is the set of all computable functions with a complexity less than f. C^0(f) is the set of all boolean-valued functions with a complexity less than f. If we consider those functions as indicator functions on sets, C^0(f) can be thought of as a complexity class of sets.