# Blum axioms

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Blum axioms

In computational complexity theory the Blum axioms or Blum complexity axioms are axioms which specify desirable properties of complexity measures on the set of computable functions. The axioms were first defined by Manuel Blum in 1967.

Importantly, the Speedup and Gap theorems 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 tuple (φ,Φ) with φ a Gödel numbering of the partial computable functions $\mathbf{P}^{(1)}$ and a computable function $\Phi: \mathbb{N} \to \mathbf{P}^{(1)}$

which satisfies the following Blum axioms. We write φi for the i-th partial computable function under the Gödel numbering φ, and Φi for the partial computable function Φ(i).

• the domain of φi and the domain of Φi is identical.
• the set $\{(i,x,t) \in \mathbb{N}^3 | \Phi_i(x) = t\}$ is recursive.

### Examples

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

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