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).
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 .
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.