(Semi)Algebraic Proofs Over {+/- 1} Variables

One of the major open problems in proof complexity is to prove lower bounds on AC(0) [p]-Frege proof systems. As a step toward this goal Impagliazzo, Mouli and Pitassi in a recent paper suggested to prove lower bounds on the size for Polynomial Calculus over the {+/- 1} basis. In this paper we show a technique for proving such lower bounds and moreover we also give lower bounds on the size for Sum-of-Squares over the {+/- 1} basis.We show lower bounds on random Delta-CNF formulas and formulas composed with a gadget. As a byproduct, we establish a separation between Polynomial Calculus and Sum-of-Squares over the {+/- 1} basis by proving a lower bound on the Pigeonhole Principle.