Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 1502.07545

SAT problem and statistical distance

Feng Pan
View original source ↗Read source document ↗Export metadata ↓

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.

Explore related subjects

cs.CC

Keep this discovery

BibTeXRIS

Feng Pan. 2015-02-26. SAT problem and statistical distance. https://arxiv.org/abs/1502.07545

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

The Computational Complexity of Holant Problems on 4-regular Graphs from the Stable Subgroup Sequence of $SL(2,\mathbb{C})$

The Holant framework provides a general setting for studying counting problems and includes graph homomorphisms (\#GH) and counting constraint satisfaction problems (\#CSP) as special cases. Over the past twenty years, a series of computational complexity dichotomies have been established for Holant problems, but the classification for complex-valued signatures is still open. The main obstacle is the case in which all signatures have even arity. In this paper, we establish a dichotomy for Holant problems with a complex-valued 4-ary signature, which is a key base case for the full classification of Holant problems. We present a new strategy by introducing Schur's theorem, the classification of finite subgroups of $\mathrm{SL}(2,\mathbb{C})$ and stable subgroup sequences into the proof. These new techniques are of independent interest.

cs.CC↗
arXiv2026

Topology inside NC$^1$

We show that ACC$^0$ is precisely what can be computed with constant-width circuits of polynomial size and polylogarithmic genus. This extends a characterization given by Hansen, showing that planar constant-width circuits also characterize ACC$^0$. Thus polylogarithmic genus provides no additional computational power in this model. We consider other generalizations of planarity, including crossing number and thickness. We show that constant-width circuits of polynomial size and thickness two already suffice to capture all of NC$^1$.

cs.CC↗