Acyclic Formulas and Minimal Unsatisfiability⋆

We call a boolean formula (in CNF) stable acyclic if the associated digraph is free of a certain type of cycles. We show that satisfiability and recognition are polynomial–time solvable problems for stable acyclic formulas. Further, we show that a minimal unsatisfiable formula is stable acyclic if and only if the number of clauses exceeds the number of variables exactly by one (this subclass of minimal unsatisfiable formulas, denoted by MU(1), is well known). Applying this result, we are able to state an algorithm for recognizing MU(1). Our algorithm is asymptotically faster than a known algorithm.