(1,2)-QSAT: A Good Candidate for Understanding Phase Transitions Mechanisms

Abstract

We explore random Boolean quantified CNF formulas of the form ∀ X ∃ Y ϕ(X,Y), where X has $m=\lfloor \alpha \log n\rfloor$ variables (α> 0), Y has n variables and each clause in ϕ has one literal from X and two from Y. These (1,2)-QCNF-formulas, which can be seen as quantified extended 2-CNF formulas, were introduced in SAT’08. It was proved that the threshold phenomenon associated to the satisfiability of such random formulas, (1,2)-QSAT, is controlled by the ratio c between the number of clauses and the number n of existential variables. In this paper, we prove that the threshold is sharp. For any value of α, we give the exact location of the associated critical ratio, a(α). At this ratio, our study highlights the sudden emergence of unsatisfiable formulas with a very specific shape. From the experimental point of view (1,2)-QSAT is challenging. Indeed, while for small values of m the critical ratio can be observed experimentally, it is not anymore the case for bigger values of m. For small values of m we give precise numerical estimates of the probability of satisfiability for critical (1,2)-QCNF-formulas. These experiments give evidence that the asymptotical regime is difficult to reach and provide some indication on the behavior of random instances. Moreover, experiments show that the computational effort, which is increasing with m, is maximized within the phase transition.