SAT problem and Limit of Solomonoff's inductive reasoning theory (2504.00318v1)

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.