Investigations on autark assignments

The structure of the monoid of autarkies for clause-sets is investigated (autarkies are generalizations of satisfying (truth) assignments). The notion of a \plain clause-set" is introduced, where every clause is usable by some resolution refutation, and it is shown that a clause-set is plain ii it has no non-trivial autarky. Every clause-set has a unique decomposition into a plain sub-clause-set and a satissable sub-clause-set. As a special case of autarkies we introduce the notion of \linear autarky," deenable by a system of linear inequalities over the rational numbers (and thus linear autarkies can be searched for in polynomial time by means of linear programming). Clause-sets without non-trivial linear autarkies we call \ weakly plain," and clause-sets satissable by a linear autarky we call \linearly satissable." As before, every clause-set has a unique decomposition into a weakly plain sub-clause-set and a linearly satissable sub-clause-set, but this time the decomposition is computable in polynomial time. The class of linearly satissable clause-sets seems to be a new and interesting polynomially decidable class of satissable clause-sets, generalizing the notion of a \matched clause-set" introduced in a recent paper by J. Franco and A. Van Gelder. The class of weakly plain clause-sets has some interesting combinatorial properties with respect to the \deeciency" = c ?n, the diierence of the number of clauses and the number of variables, introduced in the same paper: Among others (non-empty) weakly plain clause-sets fullll 1, where this property has been known before only for minimally unsatissable clause-sets.