SAT Has No Wizards

TL;DR

This paper explores the structural complexity of SAT and demonstrates that all SAT-solving programs share the same kernel, implying limitations on collective certificates and the nature of SAT solutions.

Contribution

It introduces the concept of logograms and kernels in the context of SAT, proving that SAT cannot have collective certificate strings and all solvers have identical kernels.

Findings

SAT lacks collective certificate strings in its logogram.

All SAT-solving programs share the same kernel.

Internal independence property of SAT is crucial to the results.

Abstract

An (encoded) decision problem is a pair (E, F) where E=words that encode instances of the problem, F=words to be accepted. We use "strings" in a technical sense. With an NP problem (E, F) we associate the "logogram" of F relative to E, which conveys structural information on E, F, and how F is embedded in E. The kernel Ker(P) of a program P that solves (E, F) consists of those strings in the logogram that are used by P. There are relations between Ker(P) and the complexity of P. We develop an application to SAT that relies upon a property of internal independence of SAT. We show that SAT cannot have in its logogram strings serving as collective certificates. As consequence, all programs that solve SAT have same kernel.