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.

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 $(\varphi, \Phi)$ with $\varphi$ 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 $\varphi_i$ for the i-th partial computable function under the Gödel numbering $\varphi$, and $\Phi_i$ for the partial computable function $\Phi(i)$.

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

Examples

• $(\varphi, \Phi)$ is a complexity measure, if $\Phi$ is either the time or the memory (or some suitable combination thereof) required for the computation coded by i.
• $(\varphi, \varphi)$ 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 $\Phi$ from pairs (Turing machine $M$, input $x$ for $M$) to the natural numbers union infinity. Furthermore, $\Phi$ should satisfy the following axioms:

• $\Phi(M,x)$ is finite if and only if $M(x)$ halts
• There is an algorithm which, on input $(M,x,n)$ decides if $\Phi(M,x)=n$

For example, suppose $\Phi(M,x)$ 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 $f$ complexity classes of computable functions can be defined as

$C(f) := \{ \varphi_i \in \mathbf{P}^{(1)} | \forall x.\ \Phi_i(x) \leq f(x) \}$

$C^0(f) := \{ h \in C(f) | \mathrm{codom}(h) \subseteq \{0,1\} \}$

$C(f)$ 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.