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Consistency of Circuit Lower Bounds with Bounded Theories Carboni Oliveira, Igor
Description
Proving that there are problems in $P^{NP}$ that require boolean circuits of super-linear size is a major frontier in complexity theory. While such lower bounds are known for larger complexity classes, existing results only show that the corresponding problems are hard on infinitely many input lengths. For instance, proving almost-everywhere circuit lower bounds is open even for problems in MAEXP. Giving the notorious difficulty of proving lower bounds that hold for all large input lengths, we ask the following question: Can we show that a large set of techniques cannot prove that NP is easy infinitely often Motivated by this and related questions about the interaction between mathematical proofs and computations, we investigate circuit complexity from the perspective of logic. Among other results, we prove that for any parameter $k > 1$ it is consistent with theory T that computational class $C \not\subseteq i.o.SIZE(n^k)$, where $(T,C)$ is one of the pairs: $T=T12$ and $C=P^{NP}$, $T=S12$ and $C=NP$, $T=PV$ and $C=P$. In other words, these theories cannot establish infinitely often circuit upper bounds for the corresponding problems. This is of interest because the weaker theory PV already formalizes sophisticated arguments, such as a proof of the PCP Theorem (Pich, 2015). These consistency statements are unconditional and improve on earlier theorems of Krajicek and Oliveira (2017) and Bydzovsky and Muller (2018) on the consistency of lower bounds with PV. https://arxiv.org/abs/1905.12935
Item Metadata
Title |
Consistency of Circuit Lower Bounds with Bounded Theories
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Creator | |
Publisher |
Banff International Research Station for Mathematical Innovation and Discovery
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Date Issued |
2020-01-21T11:22
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Description |
Proving that there are problems in $P^{NP}$ that require boolean circuits of super-linear size is a major frontier in complexity theory. While such lower bounds are known for larger complexity classes, existing results only show that the corresponding problems are hard on infinitely many input lengths. For instance, proving almost-everywhere circuit lower bounds is open even for problems in MAEXP. Giving the notorious difficulty of proving lower bounds that hold for all large input lengths, we ask the following question:
Can we show that a large set of techniques cannot prove that NP is easy infinitely often
Motivated by this and related questions about the interaction between mathematical proofs and computations, we investigate circuit complexity from the perspective of logic. Among other results, we prove that for any parameter $k > 1$ it is consistent with theory T that computational class $C \not\subseteq i.o.SIZE(n^k)$, where $(T,C)$ is one of the pairs:
$T=T12$ and $C=P^{NP}$, $T=S12$ and $C=NP$, $T=PV$ and $C=P$.
In other words, these theories cannot establish infinitely often circuit upper bounds for the corresponding problems. This is of interest because the weaker theory PV already formalizes sophisticated arguments, such as a proof of the PCP Theorem (Pich, 2015). These consistency statements are unconditional and improve on earlier theorems of Krajicek and Oliveira (2017) and Bydzovsky and Muller (2018) on the consistency of lower bounds with PV.
https://arxiv.org/abs/1905.12935
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Extent |
28.0 minutes
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Subject | |
Type | |
File Format |
video/mp4
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Language |
eng
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Notes |
Author affiliation: University of Oxford
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Series | |
Date Available |
2020-07-20
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Provider |
Vancouver : University of British Columbia Library
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Rights |
Attribution-NonCommercial-NoDerivatives 4.0 International
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DOI |
10.14288/1.0392457
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URI | |
Affiliation | |
Peer Review Status |
Unreviewed
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Scholarly Level |
Postdoctoral
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Rights URI | |
Aggregated Source Repository |
DSpace
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Rights
Attribution-NonCommercial-NoDerivatives 4.0 International