Disproving the Single Level Conjecture

We consider the size of monotone circuits for quadratic Boolean functions, that is, disjunctions of length-2 monomials. Our motivation is that a good (linear in the number of variables) lower bound on the monotone circuit size for a certain type of quadratic function would imply a good (even exponential) lower bound on the general nonmonotone circuit size. To get more insight into the structure of monotone circuits for quadratic functions, we consider the so-called single level conjecture posed explicitly in the early 1990s. The conjecture claims that monotone single level circuits, that is, circuits which have only one level of AND gates, for quadratic functions are not much larger than arbitrary monotone circuits. In this paper we disprove the conjecture as follows: there exist quadratic functions whose monotone circuits have linear size but whose monotone single level circuits require almost quadratic size.