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.[1]

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

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A Blum complexity measure is a pair   with   a numbering of the partial computable functions   and a computable function

 

which satisfies the following Blum axioms. We write   for the i-th partial computable function under the Gödel numbering  , and   for the partial computable function  .

  • the domains of   and   are identical.
  • the set   is recursive.

Examples

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  •   is a complexity measure, if   is either the time or the memory (or some suitable combination thereof) required for the computation coded by i.
  •   is not a complexity measure, since it fails the second axiom.

Complexity classes

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For a total computable function   complexity classes of computable functions can be defined as

 
 

  is the set of all computable functions with a complexity less than  .   is the set of all boolean-valued functions with a complexity less than  . If we consider those functions as indicator functions on sets,   can be thought of as a complexity class of sets.

References

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  1. ^ Blum, Manuel (1967). "A Machine-Independent Theory of the Complexity of Recursive Functions" (PDF). Journal of the ACM. 14 (2): 322–336. doi:10.1145/321386.321395. S2CID 15710280.