# 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 are computable functions that are not optimal with respect to that measure. 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 and a total computable function with two parameters, then there exists a total computable predicate (a boolean valued computable function) so that for every program for , there exists a program for so that for almost all

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).

## See also

## References

- Blum, Manuel (1967). "A Machine-Independent Theory of the Complexity of Recursive Functions" (PDF).
*Journal of the ACM*.**14**(2): 322–336. doi:10.1145/321386.321395. - Van Emde Boas, Peter (1975). Bečvář, Jíří, ed. "Ten years of speedup".
*Proceedings of Mathematical Foundations of Computer Science, 4th Symposium, Mariánské Lázně, September 1-5, 1975*. Lecture Notes in Computer Science. Springer-Verlag.**32**: 13–29. doi:10.1007/3-540-07389-2_179..