Biased random satisfiability problems: from easy to hard instances

In this paper we study biased random K -satisfiability ( K -SAT) problems in which each logical variable is negated with probability p . This generalization provides us a crossover from easy to hard problems and would help us in a better understanding of the typical complexity of random K -SAT problems. The exact solution of 1-SAT case is given. The critical point of K -SAT problems and results of replica method are derived in the replica symmetry framework. It is found that in this approximation alpha(c) proportional p(-(K-1)) for p --> 0. Solving numerically the survey propagation equations for K = 3 we find that for p < p* approximately 0.17 there is no replica symmetry breaking and still the SAT-UNSAT transition is discontinuous.