Sensitivity of Boolean formulas

The sensitivity set of a Boolean function at a particular input is the set of input positions where changing that one bit changes the output. Analogously we define the sensitivity set of a Boolean formula in a conjunctive normal form at a particular truth assignment, it is the set of positions where changing that one bit of the truth assignment changes the evaluation of at least one of the conjunct in the formula. We consider Boolean formulas in a generalized conjunctive normal form. Given a set ℱ of Boolean functions, an ℱ-constraint is an application of a function from ℱ to a tuple of literals built upon distinct variables, an ℱ-formula is then a conjunction of ℱ-constraints. In this framework, given a truth assignment I and a set of positions S, we are able to enumerate all ℱ-formulas that are satisfied by I and that have S as the sensitivity set at I. We prove that this number depends on the cardinality of S only, and can be expressed according to the sensitivity of the Boolean functions in ℱ.