A Threshold for Unsatisfiability

W h e n analyzing the probabil ist ic per formance of an a lgor i thm one usually considers this a lgor i thm w.r.t , an infinite family of probabi l i ty spaces of inputs. In case of satisfiability a lgor i thms for proposi t ional formulas main ly two types of families of input spaces are considered: One is the constant density model dealt with for example in [6] the other one is the constant clause size model t reated among others in [2], [3], [4], and [5]. The constant clause size model is defined as the family Form,(q, k) where Form,~(q, k) is the probabi l i ty space consisting of all formulas which are sequences of exact ly q clauses (with repeti t ion) each of which has exactly k literals over n variables such tha t each formula is equally likely. Our result constr ibutes to an analysis of this model (modified such tha t each formula is set of q clauses for simplicity). I f the clause size is k = 3 and the number of clauses is (asymptot ica l ly) C . n for a constant C < 1 it is known tha t a lmost all formulas are satisfiable [5] and if In ~ t 5.19) a lmost all formulas are unsatisfiable. Not much is known for C C > -1-g:7~/8~ between these two values. Exper iments show tha t the formulas become unsatisfiable for C a round 4. If the number of literals per clause is k = 2, it is easy to see (cf. In 2 r 2.49) the average number of satisfying assigmnents also [4]) tha t for C > -ln--57-q/4vv of formulas with C 9 n clauses approaches 0 as n --* oo. Hence a lmost no formula is In 2 satisfiable. If C < I n 3/-----~ this average goes to infinity. However, this does not mean tha t almost all formulas are satisfiable, and we show tha t they are not as long as C > I . In proving our result we build on the well known observat ion tha t a proposi t ional