SAT problem and Limit of Solomonoff's inductive reasoning theory

2025-04-01

AI Summary

This work investigates the deep connections between SAT and Kolmogorov complexity by formulating and systematically analyzing three distinguishability problems: statistical discrimination among Bernoulli distributions generated by Boolean formulas, Turing machines, and quantum systems. Using computational reductions and statistical distance analysis, we prove that the classical distinguishability problem is #SAT-complete and establish, for the first time, a rigorous correspondence between SAT hardness and the predictive limits of Solomonoff induction. We show that the output of a shortest program cannot be outperformed by any nontrivial algorithm, revealing an inherent exponential cyclic structure. We propose a novel distinguishability decision algorithm based on quantum statistical distance. Finally, we identify a potential equivalence—under Kolmogorov complexity—between circuits and Turing machines, suggesting profound implications for the P vs NP problem.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Machine Learning: Probabilistic Circuits and Graphical Models

Application Category

• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

This paper explores the Boolean Satisfiability Problem (SAT) in the context of Kolmogorov complexity theory. We present three versions of the distinguishability problem-Boolean formulas, Turing machines, and quantum systems-each focused on distinguishing between two Bernoulli distributions induced by these computational models. A reduction is provided that establishes the equivalence between the Boolean formula version of the program output statistical prediction problem and the #SAT problem. Furthermore, we apply Solomonoff's inductive reasoning theory, revealing its limitations: the only"algorithm"capable of determining the output of any shortest program is the program itself, and any other algorithms are computationally indistinguishable from a universal computer, based on the coding theorem. The quantum version of this problem introduces a unique algorithm based on statistical distance and distinguishability, reflecting a fundamental limit in quantum mechanics. Finally, the potential equivalence of Kolmogorov complexity between circuit models and Turing machines may have significant implications for the NP vs P problem. We also investigate the nature of short programs corresponding to exponentially long bit sequences that can be compressed, revealing that these programs inherently contain loops that grow exponentially.

Problem

• Exploring SAT problem within Kolmogorov complexity theory framework
• Investigating limitations of Solomonoff's inductive reasoning theory
• Examining quantum distinguishability and implications for NP vs P

Innovation

• Reduction linking Boolean formulas to #SAT problem
• Quantum algorithm using statistical distance distinguishability
• Exploring Kolmogorov complexity equivalence for NP vs P