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 >= 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 >= 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