  We propose a new encoding of the first-order connection method as a Boolean
satisfiability problem. The encoding eschews tree-like presentations of the
connection method in favour of matrices, as we show that tree-like calculi have
a number of drawbacks in the context of satisfiability solving. The matrix
setting permits numerous global refinements of the basic connection calculus.
We also show that a suitably-refined calculus is a decision procedure for the
Bernays-Sch\"onfinkel class.
