On ϵ-sensitive monotone computations

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F67985840%3A_____%2F20%3A00531552" target="_blank" >RIV/67985840:_____/20:00531552 - isvavai.cz</a>

Výsledek na webu

<a href="https://doi.org/10.1007/s00037-020-00196-6" target="_blank" >https://doi.org/10.1007/s00037-020-00196-6</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1007/s00037-020-00196-6" target="_blank" >10.1007/s00037-020-00196-6</a>

Jazyk výsledku

angličtina

Název v původním jazyce

On ϵ-sensitive monotone computations

Popis výsledku v původním jazyce

We show that strong-enough lower bounds on monotone arithmetic circuits or the nonnegative rank of a matrix imply unconditional lower bounds in arithmetic or Boolean circuit complexity. First, we show that if a polynomial f∈ R[x1, ⋯ , xn] of degree d has an arithmetic circuit of size s then (x1+⋯+xn+1)d+ϵf has a monotone arithmetic circuit of size O(sd2+ nlog n) , for some ϵ> 0. Second, if f: { 0 , 1 } n→ { 0 , 1 } is a Boolean function, we associate with f an explicit exponential-size matrix M(f) such that the Boolean circuit size of f is at least Ω(min ϵ>(rk +(M(f) - ϵJ)) - 2 n) , where J is the all-ones matrix and rk + denotes the nonnegative rank of a matrix. In fact, the quantity min ϵ>(rk +(M(f) - ϵJ)) characterizes how hard is it to distinguish rejecting and accepting inputs of f by means of a linear program. Finally, we introduce a proof system resembling the monotone calculus of Atserias et al. (J Comput Syst Sci 65:626–638, 2002) and show that similar ϵ-sensitive lower bounds on monotone arithmetic circuits imply lower bounds on proof-size in the system.

Název v anglickém jazyce

On ϵ-sensitive monotone computations

Popis výsledku anglicky

We show that strong-enough lower bounds on monotone arithmetic circuits or the nonnegative rank of a matrix imply unconditional lower bounds in arithmetic or Boolean circuit complexity. First, we show that if a polynomial f∈ R[x1, ⋯ , xn] of degree d has an arithmetic circuit of size s then (x1+⋯+xn+1)d+ϵf has a monotone arithmetic circuit of size O(sd2+ nlog n) , for some ϵ> 0. Second, if f: { 0 , 1 } n→ { 0 , 1 } is a Boolean function, we associate with f an explicit exponential-size matrix M(f) such that the Boolean circuit size of f is at least Ω(min ϵ>(rk +(M(f) - ϵJ)) - 2 n) , where J is the all-ones matrix and rk + denotes the nonnegative rank of a matrix. In fact, the quantity min ϵ>(rk +(M(f) - ϵJ)) characterizes how hard is it to distinguish rejecting and accepting inputs of f by means of a linear program. Finally, we introduce a proof system resembling the monotone calculus of Atserias et al. (J Comput Syst Sci 65:626–638, 2002) and show that similar ϵ-sensitive lower bounds on monotone arithmetic circuits imply lower bounds on proof-size in the system.

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

OECD FORD obor

10201 - Computer sciences, information science, bioinformathics (hardware development to be 2.2, social aspect to be 5.8)

Projekt

<a href="/cs/project/GX19-27871X" target="_blank" >GX19-27871X: Efektivní aproximační algoritmy a obvodová složitost</a><br>

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

Computational Complexity

ISSN

1016-3328

e-ISSN

—

Svazek periodika

29

Číslo periodika v rámci svazku

2

Stát vydavatele periodika

CH - Švýcarská konfederace

Počet stran výsledku

38

Strana od-do

6

Kód UT WoS článku

000552200600001

EID výsledku v databázi Scopus

2-s2.0-85088507263