Exponential lower bound for static semi-algebraic proofs - Grigoriev, Dima. (2014)
Semi-algebraic proof systems were introduced in [1] as extensions of Lovász-Schrijver proof systems [2,3]. These systems are very strong; in particular, they have short proofs of Tseitin’s tautologies, the pigeonhole principle, the symmetric knapsack problem and the clique-coloring tautologies [1]. In this paper we study static versions of these systems. We prove an exponential lower bound on the length of proofs in one such system. The same bound for two tree-like (dynamic) systems follows. The proof is based on a lower bound on the “Boolean degree” of Positivstellensatz Calculus refutations of the symmetric knapsack problem.