Lifting with Simple Gadgets and Applications to Circuit and Proof Complexity

We significantly strengthen and generalize 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 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 first explicit separation between monotone Boolean formulas and monotone real formulas. Specifically, 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. •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 $2^{\Omega(n/\text{polylog}(n))}$. An important technical ingredient, which may be of independent interest, is that we show that the Nullstellensatz degree of refuting the pebbling formula over a DAG $G$ over any field coincides exactly with the reversible pebbling price of $G$. In particular, this implies that the standard decision tree complexity and the parity decision tree complexity of the corresponding falsified clause search problem are equal. This is an extended abstract. The full version of the paper is available at https://arxiv.org/abs/2001.02144.

[1]  Robert Robere,et al.  Unified Lower Bounds for Monotone Computation , 2018 .

[2]  Russell Impagliazzo,et al.  Communication complexity towards lower bounds on circuit depth , 2001, computational complexity.

[3]  Ran Raz,et al.  Separation of the Monotone NC Hierarchy , 1999, Comb..

[4]  Daniel Dadush,et al.  On the complexity of branching proofs , 2020, CCC.

[5]  Alexander A. Sherstov The Pattern Matrix Method , 2009, SIAM J. Comput..

[6]  Marc Vinyals,et al.  How Limited Interaction Hinders Real Communication (and What It Means for Proof and Circuit Complexity) , 2016, 2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS).

[7]  Yuval Filmus,et al.  Semantic Versus Syntactic Cutting Planes , 2016, STACS.

[8]  Dmitry Sokolov Dag-Like Communication and Its Applications , 2016, CSR.

[9]  Toniann Pitassi,et al.  Exponential Lower Bounds for Monotone Span Programs , 2016, 2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS).

[10]  Alexander A. Razborov,et al.  Applications of matrix methods to the theory of lower bounds in computational complexity , 1990, Comb..

[11]  Russell Impagliazzo,et al.  Stabbing Planes , 2017, ITCS.

[12]  Alexander A. Razborov,et al.  Majority gates vs. general weighted threshold gates , 1992, [1992] Proceedings of the Seventh Annual Structure in Complexity Theory Conference.

[13]  Or Meir,et al.  Toward the KRW Composition Conjecture: Cubic Formula Lower Bounds via Communication Complexity , 2016, Electron. Colloquium Comput. Complex..

[14]  Arkadev Chattopadhyay,et al.  Simulation beats richness: new data-structure lower bounds , 2018, Electron. Colloquium Comput. Complex..

[15]  Arnold Rosenbloom,et al.  Monotone Real Circuits are More Powerful than Monotone Boolean Circuits , 1997, Inf. Process. Lett..

[16]  Russell Impagliazzo,et al.  Homogenization and the polynomial calculus , 2000, computational complexity.

[17]  Or Meir,et al.  Toward Better Formula Lower Bounds: The Composition of a Function and a Universal Relation , 2017, SIAM J. Comput..

[18]  Pavel Pudlák,et al.  Lower bounds for resolution and cutting plane proofs and monotone computations , 1997, Journal of Symbolic Logic.

[19]  Toniann Pitassi,et al.  Randomized Communication vs. Partition Number , 2015, Electron. Colloquium Comput. Complex..

[20]  Peter Clote,et al.  Cutting planes, connectivity, and threshold logic , 1996, Arch. Math. Log..

[21]  Anna Gál A characterization of span program size and improved lower bounds for monotone span programs , 1998, STOC '98.

[22]  Or Meir,et al.  Nullstellensatz size-degree trade-offs from reversible pebbling , 2019, Computational Complexity Conference.

[23]  Rahul Jain,et al.  Extension Complexity of Independent Set Polytopes , 2016, 2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS).

[24]  Eli Ben-Sasson,et al.  Short proofs are narrow—resolution made simple , 2001, JACM.

[25]  Toniann Pitassi,et al.  Lifting Nullstellensatz to monotone span programs over any field , 2018, Electron. Colloquium Comput. Complex..

[26]  Christoph Berkholz,et al.  Near-Optimal Lower Bounds on Quantifier Depth and Weisfeiler–Leman Refinement Steps , 2016, 2016 31st Annual ACM/IEEE Symposium on Logic in Computer Science (LICS).

[27]  Noga Alon,et al.  The monotone circuit complexity of boolean functions , 1987, Comb..

[28]  Ankit Garg,et al.  Monotone circuit lower bounds from resolution , 2018, Electron. Colloquium Comput. Complex..

[29]  Maria Luisa Bonet,et al.  On the Relative Complexity of Resolution Refinements and Cutting Planes Proof Systems , 2000, SIAM J. Comput..

[30]  Toniann Pitassi,et al.  Strongly exponential lower bounds for monotone computation , 2017, Electron. Colloquium Comput. Complex..