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Consistency of circuit lower bounds with bounded theories
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Oliveira, Igor C., Bydzovsky, Jan and Krajicek, Jan (2020) Consistency of circuit lower bounds with bounded theories. Logical Methods in Computer Science, 16 (2). 12:1-12:16. doi:10.23638/LMCS-16(2:12)2020 ISSN 1860-5974.
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Official URL: https://doi.org/10.23638/LMCS-16(2:12)2020
Abstract
Proving that there are problems in PNP 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⊈i.o.SIZE(nk), where (T,C) is one of the pairs: T=T12 and C=PNP, 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. These consistency statements are unconditional and improve on earlier theorems of [KO17] and [BM18] on the consistency of lower bounds with PV.
Item Type: | Journal Article | ||||||||||||
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Subjects: | Q Science > QA Mathematics | ||||||||||||
Divisions: | Faculty of Science, Engineering and Medicine > Science > Computer Science | ||||||||||||
Library of Congress Subject Headings (LCSH): | Computational complexity, Constructive mathematics | ||||||||||||
Journal or Publication Title: | Logical Methods in Computer Science | ||||||||||||
Publisher: | International Federation for Computational Logic | ||||||||||||
ISSN: | 1860-5974 | ||||||||||||
Official Date: | 18 June 2020 | ||||||||||||
Dates: |
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Volume: | 16 | ||||||||||||
Number: | 2 | ||||||||||||
Page Range: | 12:1-12:16 | ||||||||||||
DOI: | 10.23638/LMCS-16(2:12)2020 | ||||||||||||
Status: | Peer Reviewed | ||||||||||||
Publication Status: | Published | ||||||||||||
Access rights to Published version: | Open Access (Creative Commons) | ||||||||||||
Date of first compliant deposit: | 6 March 2020 | ||||||||||||
Date of first compliant Open Access: | 6 March 2020 | ||||||||||||
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