Blum's speedup theorem

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In computational complexity theory Blum's speedup theorem, first stated by Manuel Blum in 1967, is an important 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 needs to find the program with the smallest complexity for a given computable function and a given complexity measure. Blum's speedup theorem states that for any complexity measure there are computable functions which have no smallest program.

1 Speedup theorem
2 See also
3 References
4 External links

[edit] 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 $x$