About Inverse 3-SAT

The Inverse 3–SAT problem is known to be coNP Complete: Given φ a set of models on n variables, is there a 3–CNF formula such that φ is its exact set of models ? An immediate candidate formula F 3 φ arises, which is the conjunction of all 3–clauses satisfied by all models in φ. The (co)Inverse 3–SAT problem can then be resumed: Given φ a set of models on n variables, is there a model of F 3 φ / ∈ φ ? This article uses two important intermediate results: 1-The candidate formula can be easily (i.e. in polynomial time) transformed into an equivalent formula F φ which is 3–closed under resolution. A crucial property of F φ is that the induced formula F φ|I by applying any partial assignment I of the n variables to F φ is unsatisfiable iff its 3–closure contains the empty clause. 2-A set of partial assignments (of polynomial size) which subsume all assignments / ∈ φ can be easily computed. The (co)Inverse 3–SAT question is then equivalent to decide whether it exists a partial assignment I / ∈ φ such that the 3–closure of F φ|I does not contain the empty clause.