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An effective Procedure for Speeding up Algorithms
The provably asymptotically fastest algorithm within a factor of 5 for
formally described problems will be constructed. The main idea is to enumerate
all programs provably equivalent to the original problem by enumerating all
proofs. The algorithm could be interpreted as a generalization and improvement
of Levin search, which is, within a multiplicative constant, the fastest
algorithm for inverting functions. Blum's speed-up theorem is avoided by taking
into account only programs for which a correctness proof exists. Furthermore,
it is shown that the fastest program that computes a certain function is also
one of the shortest programs provably computing this function. To quantify this
statement, the definition of Kolmogorov complexity is extended, and two new
natural measures for the complexity of a function are defined.Comment: 10 LaTeX page