The Fastest and Shortest Algorithm for All Well-Defined Problems

TL;DR

The paper introduces an algorithm that nearly matches the fastest possible solution for any well-defined problem, optimizing resource allocation and avoiding known speed-up limitations.

Contribution

It presents a universal algorithm that solves well-defined problems nearly as fast as the fastest known algorithms, improving upon previous methods like Levin's search.

Findings

Algorithm achieves near-optimal speed for well-defined problems.

Broader applicability than Levin's universal search.

Shows shortest provably correct programs are among the most efficient.

Abstract

An algorithm $M$ is described that solves any well-defined problem $p$ as quickly as the fastest algorithm computing a solution to $p$, save for a factor of 5 and low-order additive terms. $M$ optimally distributes resources between the execution of provably correct $p$-solving programs and an enumeration of all proofs, including relevant proofs of program correctness and of time bounds on program runtimes. $M$ avoids Blum's speed-up theorem by ignoring programs without correctness proof. $M$ has broader applicability and can be faster than Levin's universal search, the fastest method for inverting functions save for a large multiplicative constant. An extension of Kolmogorov complexity and two novel natural measures of function complexity are used to show that the most efficient program computing some function $f$ is also among the shortest programs provably computing $f$.