Exact enumeration of satisfiable 2-SAT formulae

TL;DR

This paper derives exact formulas for counting satisfiable 2-SAT formulas using novel generating functions, providing insights into the structure and phase transition behavior of the problem.

Contribution

It introduces a new Implication generating function and exact enumeration formulas for satisfiable 2-SAT instances, advancing combinatorial understanding.

Findings

Exact enumeration formulas for satisfiable 2-SAT formulas

Accurate predictions of the phase transition curve

Introduction of a new Implication generating function

Abstract

We obtain exact expressions counting the satisfiable 2-SAT formulae and describe the structure of associated implication digraphs. Our approach is based on generating function manipulations. To reflect the combinatorial specificities of the implication digraphs, we introduce a new kind of generating function, the Implication generating function, inspired by the Graphic generating function used in digraph enumeration. Using the underlying recurrences, we make accurate numerical predictions of the phase transition curve of the 2-SAT problem inside the critical window. We expect these exact formulae to be amenable to rigorous asymptotic analysis using complex analytic tools, leading to a more detailed picture of the 2-SAT phase transition in the future.