First-order reasoning and efficient semi-algebraic proofs

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F67985840%3A_____%2F25%3A00598223" target="_blank" >RIV/67985840:_____/25:00598223 - isvavai.cz</a>

Výsledek na webu

<a href="https://doi.org/10.1016/j.apal.2024.103496" target="_blank" >https://doi.org/10.1016/j.apal.2024.103496</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1016/j.apal.2024.103496" target="_blank" >10.1016/j.apal.2024.103496</a>

Jazyk výsledku

angličtina

Název v původním jazyce

First-order reasoning and efficient semi-algebraic proofs

Popis výsledku v původním jazyce

Semi-algebraic proof systems such as sum-of-squares (SoS) have attracted a lot of attention due to their relation to approximation algorithms: constant degree semi-algebraic proofs lead to conjecturally optimal polynomial-time approximation algorithms for important NP-hard optimization problems. Motivated by the need to allow a more streamlined and uniform framework for working with SoS proofs than the restrictive propositional level, we initiate a systematic first-order logical investigation into the kinds of reasoning possible in algebraic and semi-algebraic proof systems. Specifically, we develop first-order theories that capture in a precise manner constant degree algebraic and semi-algebraic proof systems: every statement of a certain form that is provable in our theories translates into a family of constant degree polynomial calculus or SoS refutations, respectively, and using a reflection principle, the converse also holds. This places algebraic and semi-algebraic proof systems in the established framework of bounded arithmetic, while providing theories corresponding to systems that vary quite substantially from the usual propositional-logic ones. We give examples of how our semi-algebraic theory proves statements such as the pigeonhole principle, we provide a separation between algebraic and semi-algebraic theories, and we describe initial attempts to go beyond these theories by introducing extensions that use the inequality symbol, identifying along the way which extensions lead outside the scope of constant degree SoS. Moreover, we prove new results for propositional proofs, and specifically extend Berkholz's dynamic-by-static simulation of polynomial calculus (PC) by SoS to PC with the radical rule.

Název v anglickém jazyce

First-order reasoning and efficient semi-algebraic proofs

Popis výsledku anglicky

Semi-algebraic proof systems such as sum-of-squares (SoS) have attracted a lot of attention due to their relation to approximation algorithms: constant degree semi-algebraic proofs lead to conjecturally optimal polynomial-time approximation algorithms for important NP-hard optimization problems. Motivated by the need to allow a more streamlined and uniform framework for working with SoS proofs than the restrictive propositional level, we initiate a systematic first-order logical investigation into the kinds of reasoning possible in algebraic and semi-algebraic proof systems. Specifically, we develop first-order theories that capture in a precise manner constant degree algebraic and semi-algebraic proof systems: every statement of a certain form that is provable in our theories translates into a family of constant degree polynomial calculus or SoS refutations, respectively, and using a reflection principle, the converse also holds. This places algebraic and semi-algebraic proof systems in the established framework of bounded arithmetic, while providing theories corresponding to systems that vary quite substantially from the usual propositional-logic ones. We give examples of how our semi-algebraic theory proves statements such as the pigeonhole principle, we provide a separation between algebraic and semi-algebraic theories, and we describe initial attempts to go beyond these theories by introducing extensions that use the inequality symbol, identifying along the way which extensions lead outside the scope of constant degree SoS. Moreover, we prove new results for propositional proofs, and specifically extend Berkholz's dynamic-by-static simulation of polynomial calculus (PC) by SoS to PC with the radical rule.

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

OECD FORD obor

10101 - Pure mathematics

Projekt

<a href="/cs/project/GA23-04825S" target="_blank" >GA23-04825S: Logika a nesplnitelnost</a><br>

Návaznosti

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Rok uplatnění

2025

Kód důvěrnosti údajů

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

Název periodika

Annals of Pure and Applied Logic

ISSN

0168-0072

e-ISSN

1873-2461

Svazek periodika

176

Číslo periodika v rámci svazku

1

Stát vydavatele periodika

NL - Nizozemsko

Počet stran výsledku

36

Strana od-do

103496

Kód UT WoS článku

001282125600001

EID výsledku v databázi Scopus

2-s2.0-85199721877