Blum axioms explained
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
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
.
and
are identical.
\{(i,x,t)\inN3|\Phii(x)=t\}
is recursive.
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.
Complexity classes
complexity classes of computable functions can be defined as
C(f):=\{\varphii\inP(1)|\forallx. \Phii(x)\leqf(x)\}
C0(f):=\{h\inC(f)|codom(h)\subseteq\{0,1\}\}
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.
Notes and References
- Blum . Manuel . Manuel Blum. A Machine-Independent Theory of the Complexity of Recursive Functions . 10.1145/321386.321395 . Journal of the ACM. 14 . 2 . 322–336. 1967 . 15710280 .