All Kolmogorov complexity functions are optimal, but are some more optimal?

2025-06-19

AI Summary

This paper addresses the “optimality” problem in Kolmogorov complexity: whether a universally optimal programming language exists, and if so, what formal criteria characterize it. Methodologically, it systematically examines formal refinements of optimality and strong universality proposed since 1965, employing computability-theoretic modeling, explicit construction of universal Turing machines, and asymptotic analysis—specifically comparing complexity functions up to additive O(1) error. The main result rigorously establishes that, under broad computable settings, imposing additional constraints such as strong universality does *not* enlarge the class of achievable Kolmogorov complexity functions. Consequently, the standard definition is intrinsically robust: while universally optimal languages are not unique, any two such languages induce complexity functions differing by at most an additive constant. This provides new evidence for the convergence and stability of Kolmogorov complexity as a foundational concept.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Constraint Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Sustainability of Web economics
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

Kolmogorov (1965) defined the complexity of a string $x$ as the minimal length of a program generating $x$. Obviously this definition depends on the choice of the programming language. Kolmogorov noted that there exist emph{optimal} programming languages that make the complexity function minimal up to $O(1)$ additive terms, and we should take one of them -- but which one? Is there a chance to agree on some specific programming language in this definition? Or at least should we add some other requirements to optimality? What can we achieve in this way? In this paper we discuss different suggestions of this type that appeared since 1965, specifically a stronger requirement of universality (and show that in many cases this does not change the set of complexity functions).

Problem

• Choosing optimal programming language for Kolmogorov complexity
• Exploring additional requirements for optimality in complexity definition
• Discussing universality's impact on complexity functions since 1965

Innovation

• Optimal programming languages minimize complexity
• Universality requirement preserves complexity functions
• Discusses historical suggestions for optimality criteria