The strength of multilinear proofs

We introduce an algebraic proof system that manipulates multilinear arithmetic formulas. We show that this proof system is fairly strong, even when restricted to multilinear arithmetic formulas of a very small depth. Specifically, we show the following: 1. Algebraic proofs manipulating depth 2 multilinear arithmetic formulas polynomially simulate Resolution, Polynomial Calculus (PC) and Polynomial Calculus with Resolution (PCR) proofs; 2. Polynomial size proofs manipulating depth 3 multilinear arithmetic formulas for the functional pigeonhole principle; 3. Polynomial size proofs manipulating depth 3 multilinear arithmetic formulas for Tseitin's graph tautologies. By known lower bounds, this demonstrates that algebraic proof systems manipulating depth 3 multilinear formulas are strictly stronger than Resolution, PC and PCR, and have an exponential gap over bounded-depth Frege for both the functional pigeonhole principle and Tseitin's graph tautologies. We also illustrate a connection between lower bounds on multilinear proofs and lower bounds on multilinear circuits. In particular, we show that (an explicit) super-polynomial size separation between proofs manipulating general arithmetic circuits and proofs manipulating multilinear circuits implies a super-polynomial size lower bound on multilinear circuits for an explicit family of polynomials.