We study the survey propagation algorithm~\cite{MZ02, BMZ03,BMWZ03}, which is an iterative technique that appears to be very effective in solving random $k$-SAT problems even with densities close to threshold. We first describe how any SAT formula can be associated with a novel family of Markov random fields (MRFs), parameterized by a real number $\rho \in [0,1]$. We then show that applying belief propagation---a well-known ``message-passing'' technique---to this family of MRFs recovers various algorithms, ranging from pure survey propagation at one extreme ($\rho = 1$) to standard belief propagation on the uniform distribution over SAT assignments at the other extreme ($\rho = 0$). Configurations in these MRFs have a natural interpretation as generalized satisfiability assignments, on which a partial order can be defined. We isolate \emph{cores} as minimal elements in this partial ordering, and prove that any core is a fixed point of survey propagation. We investigate the associated lattice structure, and prove a weight-preserving identity that shows how any MRF with $\rho > 0$ can be viewed as a ``smoothed'' version of the naive factor graph representation of the $k$-SAT problem ($\rho = 0$). Our experimental results suggest that random formulas typically do not possess non-trivial cores. This result and additional experiments indicate that message-passing on our family of MRFs is most effective for values of $\rho \neq 1$ (i.e., distinct from survey propagation). Finally, we isolate properties of Gibbs sampling and message-passing algorithms that are typical for an ensemble of $k$-SAT problems.