An effective Procedure for Speeding up Algorithms

Abstract

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.