On the strength of Sherali-Adams and Nullstellensatz as propositional proof systems

We characterize the strength of the algebraic proof systems Sherali-Adams () and Nullstellensatz () in terms of Frege-style proof systems. Unlike bounded-depth Frege, has polynomial-size proofs of the pigeonhole principle (). A natural question is whether adding to bounded-depth Frege is enough to simulate . We show that , with unary integer coefficients, lies strictly between tree-like and tree-like Resolution. We introduce a weighted version of () and we show that with integer coefficients lies strictly between tree-like and Resolution. Analogous results are shown for using the bijective (i.e. onto and functional) pigeonhole principle and a weighted version of it.