Blums axioms
Blum's axioms Two axioms in complexity theory, formulated by M. Blum. Let M_{1},M_{2},…,M_{n},…
be an effective enumeration of the Turing machines and let f_{i} be the partial recursive function of a single variable that is computed by M_{i}. (For technical reasons it is simpler to think in terms of partial recursive functions than set (or language) recognizers.) If F_{1},F_{2},…,F_{n},…
is a sequence of partial recursive functions satisfying
axiom 1:
f_{i}(n) is defined if and only if F_{i}(n) is defined,
and axiom 2:
F_{i}(x) ← y is a recursive predicate of i, x, and y,
then F_{i}(n) is a computational complexity measure and can be thought of as the amount of some “resource” consumed by M_{i} in computing f_{i}(n). This notion represents a useful abstraction of the basic resources – time and space. Several remarkable theorems about computational complexity have been proved for any measure of resources satisfying the two axioms (see gap theorem, speedup theorem).
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