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CONSISTENCY OF CIRCUIT LOWER BOUNDS WITH BOUNDED THEORIES

Identifikátory výsledku

  • Kód výsledku v IS VaVaI

    <a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216208%3A11320%2F20%3A10420835" target="_blank" >RIV/00216208:11320/20:10420835 - isvavai.cz</a>

  • Výsledek na webu

    <a href="https://verso.is.cuni.cz/pub/verso.fpl?fname=obd_publikace_handle&handle=RcpDf4ka5Z" target="_blank" >https://verso.is.cuni.cz/pub/verso.fpl?fname=obd_publikace_handle&handle=RcpDf4ka5Z</a>

  • DOI - Digital Object Identifier

    <a href="http://dx.doi.org/10.23638/LMCS-16(2:12)2020" target="_blank" >10.23638/LMCS-16(2:12)2020</a>

Alternativní jazyky

  • Jazyk výsledku

    angličtina

  • Název v původním jazyce

    CONSISTENCY OF CIRCUIT LOWER BOUNDS WITH BOUNDED THEORIES

  • Popis výsledku v původním jazyce

    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 &gt;= 1 it is consistent with theory T that computational class C not subset of i.o.SIZE(n(k)), where (T, C) is one of the pairs: T=T-2(1) and C = P-NP , T=S-2(1) 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 [Pic15b]. These consistency statements are unconditional and improve on earlier theorems of [KO17] and [BM18] on the consistency of lower bounds with PV.

  • Název v anglickém jazyce

    CONSISTENCY OF CIRCUIT LOWER BOUNDS WITH BOUNDED THEORIES

  • Popis výsledku anglicky

    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 &gt;= 1 it is consistent with theory T that computational class C not subset of i.o.SIZE(n(k)), where (T, C) is one of the pairs: T=T-2(1) and C = P-NP , T=S-2(1) 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 [Pic15b]. These consistency statements are unconditional and improve on earlier theorems of [KO17] and [BM18] on the consistency of lower bounds with PV.

Klasifikace

  • Druh

    J<sub>imp</sub> - Článek v periodiku v databázi Web of Science

  • CEP obor

  • OECD FORD obor

    10101 - Pure mathematics

Návaznosti výsledku

  • Projekt

  • Návaznosti

    I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Ostatní

  • Rok uplatnění

    2020

  • Kód důvěrnosti údajů

    S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Údaje specifické pro druh výsledku

  • Název periodika

    Logical Methods in Computer Science

  • ISSN

    1860-5974

  • e-ISSN

  • Svazek periodika

    16

  • Číslo periodika v rámci svazku

    2

  • Stát vydavatele periodika

    DE - Spolková republika Německo

  • Počet stran výsledku

    16

  • Strana od-do

    12

  • Kód UT WoS článku

    000549432200010

  • EID výsledku v databázi Scopus

    2-s2.0-85087104274