SAT problem and statistical distance

TL;DR

This paper explores the complexity of SAT problems through Kolmogorov complexity and statistical distance, revealing fundamental limits on their distinguishability and implications for quantum computing and P versus NP.

Contribution

It introduces a novel connection between SAT problem complexity, Kolmogorov complexity, and quantum statistical distance, proposing a proof that NP does not equal P.

Findings

SAT strings can be highly compressed based on Kolmogorov complexity.

Distinguishing distributions in SAT problems is fundamentally hard, akin to quantum state discrimination.

NP != P is supported by the invariance of statistical distance in program and probability spaces.

Abstract

In this paper with two equivalent representations of the information contained by a SAT formula, the reason why string generated by succinct SAT formula can be greatly compressed is firstly presented based on Kolmogorov complexity theory. Then what strings can be greatly compressed were classified and discussed. In this way we discovered the SAT problem was composed of a basic distinguish problem: distinguish two different distributions induced under the computer with certain SAT formula ensemble. We then tried to map this problem into quantum mechanics, or the quantum version basic distinguish problem: this time two different distributions are induced under quantum mechanics. Based on the equivalence of statistical distance between probability space and Hilbert space, in the same time this distance is invariant under all unitary transformations. The quantum version basic problem cannot be efficiently solved by any quantum computer. In the worst case, any quantum computer must perform exponential times measurement in order to solve it. In the end we proposed the main theorem : The statistical distance in program space and probability space are identical. We tried to prove it using the relationship of Kolmogorov complexity and entropy. It showed there is no difference to solve the basic problem in SAT formula space or probability space. In the worst case, exponential trials must be performed to solve it. NP!=P.