COMPLEXITY OF SEMI-ALGEBRAIC PROOFS

Dima Grigoriev (1, 2) , Edward A Hirsch , Dmitrii V Pasechnik

1 CNRS - Centre National de la Recherche Scientifique
2 IRMAR - Institut de Recherche Mathématique de Rennes

It is a known approach to translate propositional formulas into systems of polynomial inequalities and to consider proof systems for the latter ones. The well-studied proof systems of this kind are the Cutting Planes proof system (CP) utilizing linear inequalities and the Lovasz-Schrijver calculi (LS) utilizing quadratic inequalities. We introduce generalizations LS d of LS that operate with polynomial inequalities of degree at most d. It turns out that the obtained proof systems are very strong. We construct polynomial-size bounded degree LS d proofs of the clique-coloring tautologies (which have no polynomial-size CP proofs), the symmetric knapsack problem (which has no bounded degree Positivstellensatz Calculus proofs), and Tseitin's tautologies (which are hard for many known proof systems). Extending our systems with a division rule yields a polynomial simulation of CP with polynomially bounded coe cients, while other extra rules further reduce the proof degrees for the aforementioned examples. Finally, we prove lower bounds on Lov asz-Schrijverranks and on the \Boolean degree" of Positivstellensatz Calculus refutations. We use the latter bound to obtain an exponential lower bound on the size of static LS d and tree-like LS d refutations. Contents

Domaines: Mathématiques [math]

Dima Grigoriev, Edward A Hirsch, Dmitrii V Pasechnik. COMPLEXITY OF SEMI-ALGEBRAIC PROOFS. Moscow Mathematical Journal, 2002. ⟨hal-03047115⟩