Exact location of the phase transition for random (1,2)-QSAT

Abstract

The QSAT problem is the quantified version of the SAT problem. We show the existence of a threshold effect for the phase transition associated with the satisfiability of random quantified boolean CNF formulas of the form ∀X∃Yϕ(X,Y), where X has m variables, Y has n variables and each clause in ϕ has one literal from X and two from Y. For such formulas, we show that the threshold phenomenon is controlled by the ratio between the number of clauses and the number n of existential variables. Then we give the exact location of the associated critical ratio c∗: it is a decreasing function of α, where α is the limiting value of m/ log (n) when n tends to infinity. Thus we give a precise location of the phase transition associated with a coNP-complete problem.