  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.
