TCS+ Talk

Abstract: Lifting theorems in complexity theory are a method of transferring lower bounds in a weak computational model into lower bounds for a more powerful computational model, via function composition. There has been an explosion of lifting theorems in the last ten years, essentially reducing communication lower bounds to query complexity lower bounds. These theorems only hold for composition with very specific "gadgets" such as indexing and inner product.

In this talk, we will present a generalization of the theorem lifting Nullstellensatz degree to monotone span program size by Pitassi and Robere (2018) so that it works for any gadget with high enough rank, in particular, for useful gadgets such as equality and greater-than. We will then explain how to apply our generalized theorem to solve three open problems:

- We present the first result that demonstrates a separation in proof power for cutting planes with unbounded versus polynomially bounded coefficients. Specifically, we exhibit CNF formulas that can be refuted in quadratic length and constant line space in cutting planes with unbounded coefficients, but for which there are no refutations in subexponential length and subpolynomial line space if coefficients are restricted to be of polynomial magnitude.
- We give the strongest separation to-date between monotone Boolean formulas and monotone Boolean circuits. Namely, we show that the classical GEN problem, which has polynomial-size monotone Boolean circuits, requires monotone Boolean formulas of size 2^{\Omega(n / polylog n)}.
- We give the first explicit separation between monotone Boolean formulas and monotone real formulas. Namely, we give an explicit family of functions that can be computed with monotone real formulas of nearly linear size but require monotone Boolean formulas of exponential size. Previously only a non-explicit separation was known.

This talk is based on joint work with Or Meir, Jakob Nordström, Toniann Pitassi, Robert Robere, and Marc Vinyals, available at https://arxiv.org/abs/2001.02144

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