Lifting with simple gadgets and applications to circuit and proof complexity
(2020) 61st IEEE Annual Symposium on Foundations of Computer Science, FOCS 2020 In Proceedings  Annual IEEE Symposium on Foundations of Computer Science, FOCS 2020November. p.2430 Abstract
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 greaterthan. 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... (More)
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 greaterthan. 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 nonexplicit separation was known. •We give the strongest separation todate between monotone Boolean formulas and monotone Boolean circuits. Namely, we show that the classical GEN problem, which has polynomialsize 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.
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 author
 De Rezende, Susanna ^{LU} ; Meir, Or ; Nordstrom, Jakob ^{LU} ; Pitassi, Toniann ; Robere, Robert and Vinyals, Marc
 organization
 publishing date
 202011
 type
 Chapter in Book/Report/Conference proceeding
 publication status
 published
 subject
 keywords
 circuit complexity, communication complexity, cutting planes, pebble games, proof complexity, tradeoffs
 host publication
 Proceedings  2020 IEEE 61st Annual Symposium on Foundations of Computer Science, FOCS 2020
 series title
 Proceedings  Annual IEEE Symposium on Foundations of Computer Science, FOCS
 volume
 2020November
 article number
 9317963
 pages
 7 pages
 publisher
 IEEE Computer Society
 conference name
 61st IEEE Annual Symposium on Foundations of Computer Science, FOCS 2020
 conference location
 Virtual, Durham, United States
 conference dates
 20201116  20201119
 external identifiers

 scopus:85100337762
 ISSN
 02725428
 ISBN
 9781728196220
 9781728196213
 DOI
 10.1109/FOCS46700.2020.00011
 language
 English
 LU publication?
 yes
 additional info
 Funding Information: Or Meir was supported by the Israel Science Foundation (grant No. 1445/16). Toniann Pitassi was supported by NSERC. Susanna F. de Rezende and Jakob Nordström were supported by the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007–2013) / ERC grant agreement no. 279611, as well as by the Knut and Alice Wallenberg grant KAW 2016.0066. Susanna F. de Rezende also received funding fromg Knut and Alice Wallenberg Foundation grant KAW 2018.0371 and Jakob Nordström from the Swedish Research Council grants 62120125645 and 201600782 and from the Independent Research Fund Denmark grant 904000389B. Part of this work was completed while Robert Robere was a postdoctoral researcher at DIMACS and the Institute for Advanced Study. This material is based upon work directly supported by the Charles Simonyi Endowment and indirectly supported by the National Science Foundation Grant No. CCF1900460. Any opinions, findings and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation. Marc Vinyals was supported by the Prof. R Narasimhan postdoctoral award. Funding Information: Part of this work was carried out while several of the authors were visiting the Simons Institute for the Theory of Computing in association with the DIMACS/Simons Collaboration on Lower Bounds in Computational Complexity, which is conducted with support from the National Science Foundation. Publisher Copyright: © 2020 IEEE.
 id
 1d94d8bb899a455e8eadef77ae35b69e
 date added to LUP
 20211214 15:46:18
 date last changed
 20241103 13:07:22
@inproceedings{1d94d8bb899a455e8eadef77ae35b69e, abstract = {{<p>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 greaterthan. 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 nonexplicit separation was known. •We give the strongest separation todate between monotone Boolean formulas and monotone Boolean circuits. Namely, we show that the classical GEN problem, which has polynomialsize 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.</p>}}, author = {{De Rezende, Susanna and Meir, Or and Nordstrom, Jakob and Pitassi, Toniann and Robere, Robert and Vinyals, Marc}}, booktitle = {{Proceedings  2020 IEEE 61st Annual Symposium on Foundations of Computer Science, FOCS 2020}}, isbn = {{9781728196220}}, issn = {{02725428}}, keywords = {{circuit complexity; communication complexity; cutting planes; pebble games; proof complexity; tradeoffs}}, language = {{eng}}, pages = {{2430}}, publisher = {{IEEE Computer Society}}, series = {{Proceedings  Annual IEEE Symposium on Foundations of Computer Science, FOCS}}, title = {{Lifting with simple gadgets and applications to circuit and proof complexity}}, url = {{http://dx.doi.org/10.1109/FOCS46700.2020.00011}}, doi = {{10.1109/FOCS46700.2020.00011}}, volume = {{2020November}}, year = {{2020}}, }