yawiki.org entry for Blum axioms

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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).

    Blum axioms
        Blum axioms
        Examples
        Complexity classes

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Blum axioms

A Blum complexity measure is a tuple

(varphi, Phi)

with

varphi

a Gödel numbering of the partial computable functions

mathbf^

and a computable function

$Phi: mathbb o mathbf^$

so that the following Blum axioms are satisfied

for all

i in mathbb

mathrm(varphi_i) = mathrm(Phi_i)

the set

is recursive

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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.

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Complexity classes

For a total computable function

f

complexity classes of computable functions can be defined as

$C(f)$
=

$C^0(f)$
=

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.

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