Sub-exponential Upper Bound for #XSAT of some CNF Classes

We derive a worst upper bound on the number of models for exact satisfiability (XSAT) of arbitrary CNF formulas F. The bound can be calculated solely from the distribution of positive and negated literals in the formula. For certain subsets of CNF instances the new bound can be computed in subexponential time, namely in at most ( ) n O n , where n is the number of variables of F. A wider class of SAT problems beyond XSAT is defined to which the method can be extended. Introduction. Hard instances of SAT and SAT variants are characterized by exponential decision times. Therefore methods to determine whether a specific instance can be expected to be hard or not are of interest. We will introduce such a method in the following and apply it to some SAT variants. A frequently investigated SAT variant is XSAT, short for exact satisfiability. XSAT is the problem of deciding whether assignments exist which satisfy exactly one literal in each clause.The study of this class of problems is interesting, because it is known to be NP-complete on conjunctive normal form formulas (CNF). The natural quantity to test the conditions imposed by XSAT is the number of true (i.e. satisfied) literals F  , because in XSAT the number of true literals necessarily equals the number of clauses, F m   . Not all assignments fulfilling this condition need to be XSAT models, i.e. XSAT