Phase Transition in Unrestricted Random SAT

For random CNF formulae with m clauses, n variables and an unrestricted number of literals per clause the transition from high to low satisfiability can be determined exactly for large n. The critical density m/n turns out to be strongly n-dependent, ccr ln(2)/(1-p) , where pn is the mean number of positive literals per clause.This is in contrast to restricted random SAT problems (random K-SAT), where the critical ratio m/n is a constant. In a biased model, where variables aj und their complements āj occur with different probabilities p > q, the critical line which separates the (m,n)-plane into regions of high and low satisfiability lies wthin a narrow strip between lower and upper bounds given by mlbnln(2)(1-q) -n and mubnln(2)(1-p) . All transition lines are calculated by the second moment method applied to the number of solutions N of a formula. Again in contrast to random K-SAT, the method does not fail for the unrestricted model, and it is not necessary to construct custom made order parameters. It is argued that the difference to K-SAT stems from long range interactions between solutions which are not cut off by disorder. We also point out that models with a fixed number of literals per clause, i.e. random K-SAT, may give restricted information on correlations in solution space, because they suffer from a limited sample space: the set of all K-SAT CNF-formulae with n variables contains only a tiny fraction of all possible logically inequivalent formulae with n variables.