From proof complexity to circuit complexity via interactive protocols

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F67985840%3A_____%2F24%3A00588537" target="_blank" >RIV/67985840:_____/24:00588537 - isvavai.cz</a>

Výsledek na webu

<a href="https://doi.org/10.4230/LIPIcs.ICALP.2024.12" target="_blank" >https://doi.org/10.4230/LIPIcs.ICALP.2024.12</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.4230/LIPIcs.ICALP.2024.12" target="_blank" >10.4230/LIPIcs.ICALP.2024.12</a>

Jazyk výsledku

angličtina

Název v původním jazyce

From proof complexity to circuit complexity via interactive protocols

Popis výsledku v původním jazyce

Folklore in complexity theory suspects that circuit lower bounds against NC1 or P/poly, currently out of reach, are a necessary step towards proving strong proof complexity lower bounds for systems like Frege or Extended Frege. Establishing such a connection formally, however, is already daunting, as it would imply the breakthrough separation NEXP ⊈ P/poly, as recently observed by Pich and Santhanam [58]. We show such a connection conditionally for the Implicit Extended Frege proof system (iEF) introduced by Krajíček [45], capable of formalizing most of contemporary complexity theory. In particular, we show that if iEF proves efficiently the standard derandomization assumption that a concrete Boolean function is hard on average for subexponential-size circuits, then any superpolynomial lower bound on the length of iEF proofs implies #P ⊈ FP/poly (which would in turn imply, for example, PSPACE ⊈ P/poly). Our proof exploits the formalization inside iEF of the soundness of the sum-check protocol of Lund, Fortnow, Karloff, and Nisan [54]. This has consequences for the self-provability of circuit upper bounds in iEF. Interestingly, further improving our result seems to require progress in constructing interactive proof systems with more efficient provers.

Název v anglickém jazyce

From proof complexity to circuit complexity via interactive protocols

Popis výsledku anglicky

Folklore in complexity theory suspects that circuit lower bounds against NC1 or P/poly, currently out of reach, are a necessary step towards proving strong proof complexity lower bounds for systems like Frege or Extended Frege. Establishing such a connection formally, however, is already daunting, as it would imply the breakthrough separation NEXP ⊈ P/poly, as recently observed by Pich and Santhanam [58]. We show such a connection conditionally for the Implicit Extended Frege proof system (iEF) introduced by Krajíček [45], capable of formalizing most of contemporary complexity theory. In particular, we show that if iEF proves efficiently the standard derandomization assumption that a concrete Boolean function is hard on average for subexponential-size circuits, then any superpolynomial lower bound on the length of iEF proofs implies #P ⊈ FP/poly (which would in turn imply, for example, PSPACE ⊈ P/poly). Our proof exploits the formalization inside iEF of the soundness of the sum-check protocol of Lund, Fortnow, Karloff, and Nisan [54]. This has consequences for the self-provability of circuit upper bounds in iEF. Interestingly, further improving our result seems to require progress in constructing interactive proof systems with more efficient provers.

Druh

D - Stať ve sborníku

CEP obor

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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

2024

Kód důvěrnosti údajů

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

Název statě ve sborníku

51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)

ISBN

978-3-95977-322-5

ISSN

1868-8969

e-ISSN

1868-8969

Počet stran výsledku

20

Strana od-do

12

Název nakladatele

Schloss Dagstuhl, Leibniz-Zentrum für Informatik

Místo vydání

Dagstuhl

Místo konání akce

Tallin

Datum konání akce

8. 7. 2024

Typ akce podle státní příslušnosti

WRD - Celosvětová akce

Kód UT WoS článku

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