Lifting with simple gadgets and applications to circuit and proof complexity
The result's identifiers
Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F67985840%3A_____%2F20%3A00539558" target="_blank" >RIV/67985840:_____/20:00539558 - isvavai.cz</a>
Result on the web
<a href="http://dx.doi.org/10.1109/FOCS46700.2020.00011" target="_blank" >http://dx.doi.org/10.1109/FOCS46700.2020.00011</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.1109/FOCS46700.2020.00011" target="_blank" >10.1109/FOCS46700.2020.00011</a>
Alternative languages
Result language
angličtina
Original language name
Lifting with simple gadgets and applications to circuit and proof complexity
Original language description
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.
Czech name
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Czech description
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Classification
Type
D - Article in proceedings
CEP classification
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OECD FORD branch
10201 - Computer sciences, information science, bioinformathics (hardware development to be 2.2, social aspect to be 5.8)
Result continuities
Project
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Continuities
I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace
Others
Publication year
2020
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů
Data specific for result type
Article name in the collection
2020 IEEE 61st Annual Symposium on Foundations of Computer Science
ISBN
978-1-7281-9622-0
ISSN
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e-ISSN
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Number of pages
7
Pages from-to
24-30
Publisher name
IEEE
Place of publication
Los Alamitos
Event location
Durham
Event date
Nov 16, 2020
Type of event by nationality
WRD - Celosvětová akce
UT code for WoS article
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