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The probabilistic satisfiability of a logical expression is a fundamentalconcept known as the partition function in statistical physics and fieldtheory, an evaluation of a related graph's Tutte polynomial in mathematics, andthe Moore-Shannon network reliability of that graph in engineering. It is thecrucial element for decision-making under uncertainty. Not surprisingly, it isprovably hard to compute exactly or even to approximate. Many of theseapplications are concerned only with a subset of problems for which thesolutions are monotonic functions. Here we extend the weak- and strong-couplingmethods of statistical physics to heterogeneous satisfiability problems andintroduce a novel approach to constructing lower and upper bounds on theapproximation error for monotonic problems. These bounds combine informationfrom both perturbative analyses to produce bounds that are tight in the sensethat they are saturated by some problem instance that is compatible with allthe information contained in either approximation.

Perturbative methods for mostly monotonic probabilistic satisfiability problems