# Blum axioms

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, 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 with a Gödel 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 .

### Examples

- 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.

### Notes

A Blum complexity measure is defined using computable functions without any reference to a specific model of computation. In order to make the definition more accessible we rephrase the Blum axioms in terms of Turing machines:

A *Blum complexity measure* is a function from pairs (Turing machine , input for ) to the natural numbers union infinity. Furthermore, should satisfy the following axioms:

- is finite if and only if halts
- There is an algorithm which, on input decides if

For example, suppose gives the number of time steps that the machine *M* runs for on input *x* before halting. The first axiom is clear; the second follows because a universal Turing machine can simulate *M* on *x* while counting its steps. If *M* exceeds *n* steps, it can halt and reject, so there is no need to determine if *M* halts on *x*.

## Complexity classes

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.