A New Structural Property of SAT

TL;DR

This paper introduces the concept of complete internal independence in SAT, demonstrating it as a maximal form of independence and suggesting it could imply SAT's exponential complexity.

Contribution

It defines and analyzes a new structural property of SAT called complete internal independence, which is stronger than previous notions and may be key to proving SAT's exponential complexity.

Findings

SAT exhibits complete internal independence

This property is strictly stronger than previous independence notions

It provides a new perspective on SAT's potential exponential complexity

Abstract

We review a minimum set of notions from our previous paper on structural properties of SAT at arXiv:0802.1790 that will allow us to define and discuss the "complete internal independence" of a decision problem. This property is strictly stronger than the independence property that was called "strong internal independence" in cited paper. We show that SAT exhibits this property. We argue that this form of independence of a decision problem is the strongest possible for a problem. By relying upon this maximally strong form of internal independence, we reformulate in more strict terms the informal remarks on possible exponentiality of SAT that concluded our previous paper. The net result of that reformulation is a hint for a proof for SAT being exponential. We conjecture that a complete proof of that proposition can be obtained by strictly following the line of given hint of proof.