# Blum's speedup theorem

In computational complexity theory, Blum's speedup theorem, first stated by Manuel Blum in 1967, is a fundamental theorem about the complexity of computable functions.

Each computable function has an infinite number of different program representations in a given programming language. In the theory of algorithms one often strives to find a program with the smallest complexity for a given computable function and a given complexity measure (such a program could be called optimal). Blum's speedup theorem shows that for any complexity measure, there exists a computable function, such that there is no optimal program computing it, because every program has a program of lower complexity. This also rules out the idea there is a way to assign to arbitrary functions their computational complexity, meaning the assignment to any f of the complexity of an optimal program for f. This does of course not exclude the possibility of finding the complexity of an optimal program for certain specific functions.

## Speedup theorem

Given a Blum complexity measure $(\varphi ,\Phi )$  and a total computable function $f$  with two parameters, then there exists a total computable predicate $g$  (a boolean valued computable function) so that for every program $i$  for $g$ , there exists a program $j$  for $g$  so that for almost all $x$

$f(x,\Phi _{j}(x))\leq \Phi _{i}(x)\,$

$f$  is called the speedup function. The fact that it may be as fast-growing as desired (as long as it is computable) means that the phenomenon of always having a program of smaller complexity remains even if by "smaller" we mean "significantly smaller" (for instance, quadratically smaller, exponentially smaller).