Length of Prime Implicants and Number of Solutions of Random CNF Formulae

Consider a uniform distribution of r-CNF formulae (in Conjunctive Normal Form) with cn clauses, each with r distinct literals, over a set of n variables. A prime implicant J of a formula Φ is a consistent conjunction of literals which implies Φ but ceases to imply when deprived of any one literal. The normalized length of J is the ratio of the number of its literals to the number of variables occurring in Φ. We show that for any e > 0 and for some range of values of c depending on r, almost every r-CNF formula: 1. either is satisfiable and any one of its prime implicants has a normalized length at least equal to (αmr(c)(1 - e-rc)) - e and at most equal to (αMr(c)(1 - e-rc)) + e, αmr(c) and αMr(c) being well-defined as functions of c, 2. or is unsatisfiable. A first practical consequence is when testing the satisfiability of r-CNF formulae by procedures such as the well-known Davis, Putnam and Loveland Procedure, for almost every r-CNF formula, when it is satisfiable, the proportion of variables which must be assigned a value by such procedures, in order to find a solution, is at least equal to (αmr(c)(1 - e-rc)) - e. A second consequence is that almost every r-CNF formula, when it is satisfiable, has an exponential actual number of solutions (i.e. the number of solutions defined on the variables occurring in the formula) at least equal to 2(1 - e-rc - αMr(c) - e)n. Moreover for r = 2, 3 we show th for any c it is at least equal to 20.03n,20.012n, respectively.