CONSISTENCY OF CIRCUIT LOWER BOUNDS WITH BOUNDED THEORIES

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>

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.

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

OECD FORD obor

10101 - Pure mathematics

Projekt

—

Návaznosti

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Rok uplatnění

2020

Kód důvěrnosti údajů

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

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