Random Formulas, Monotone Circuits, and Interpolation

We prove new lower bounds on the sizes of proofs in the Cutting Plane proof system, using a concept that we call unsatisfiability certificate. This approach is, essentially, equivalent to the well-known feasible interpolation method, but is applicable to CNF formulas that do not seem suitable for interpolation. Specifically, we prove exponential lower bounds for random k-CNFs, where k is the logarithm of the number of variables, and for the Weak Bit Pigeon Hole Principle. Furthermore, we prove a monotone variant of a hypothesis of Feige [12]. We give a superpolynomial lower bound on monotone real circuits that approximately decide the satisfiability of k-CNFs, where k = ω(1). For k ≈ logn, the lower bound is exponential.