What is the satisfiability threshold of random balanced Boolean expressions?. Issue 3 (18th December 2021) - Lindenstrauss, Naomi; Talagrand, Michel
Abstract: We consider the model of random Boolean expressions based on balanced binary trees with 2 N $$ {2}^N $$ leaves, to which are randomly attributed one of k N $$ {k}_N $$ Boolean variables or their negations. We prove that if for every c > 0 $$ c>0 $$ it holds that k N exp ( − c N ) → 0 $$ {k}_N\exp \left(-c\sqrt{N}\right)→ 0 $$ then asymptotically with high probability the Boolean expression is either a tautology or an antitautology. Our methods are based on the study of a certain binary operation on the set of probability measures on { 0, 1 } I $$ {\left\{0, 1\right\}}^I $$ for a finite set I .