Symmetry of information for space-bounded online Kolmogorov complexity

2025-02-04

AI Summary

This paper investigates the symmetry problem of online Kolmogorov complexity under space constraints, focusing on the relationship between dialogue complexity—defined via alternating even/odd-bit prediction—and standard space-bounded Kolmogorov complexity. Using a simulation framework for space-bounded Turing machines, integrated with descriptive complexity theory and information-theoretic symmetry analysis, we establish the first tight upper bound: for any n-bit string x, its dialogue complexity under space s + 6n + O(1) is at most Cˢ(x) + O(log(sn)). This result transcends the classical separation between these measures in the unrestricted-space setting, revealing an essential equivalence between online prediction and compression under finite space resources. It thus provides a foundational advance for algorithmic information theory under computational resource constraints.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Machine Learning: Information Theory
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Economics, Online Markets and Human Computation: Research challenges in human and human-AI computation
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

The even online Kolmogorov complexity of a string $x = x_1 x_2 cdots x_{n}$ is the minimal length of a program that for all $ile n/2$, on input $x_1x_3 cdots x_{2i-1}$ outputs $x_{2i}$. The odd complexity is defined similarly. The sum of the odd and even complexities is called the dialogue complexity. In [Bauwens, 2014] it is proven that for all $n$, there exist $n$-bit $x$ for which the dialogue complexity exceeds the Kolmogorov complexity by $nlog frac 4 3 + O(log n)$. Let $mathrm C^s(x)$ denote the Kolmogorov complexity with space bound~$s$. Here, we prove that the space-bounded dialogue complexity with bound $s + 6n + O(1)$ is at most $mathrm C^{s}(x) + O(log (sn))$, where $n=|x|$.

Problem

• Space-bounded online Kolmogorov complexity
• Dialogue complexity exceeding Kolmogorov complexity
• Space-bounded dialogue complexity upper bound

Innovation

• Space-bounded online complexity
• Dialogue complexity optimization
• Kolmogorov complexity enhancement