Annals of Pure and Applied Logic最新文献_第2页

First-order reasoning and efficient semi-algebraic proofs 一阶推理和高效半代数证明

IF 0.6 2区数学

Annals of Pure and Applied Logic Pub Date : 2024-07-14 DOI: 10.1016/j.apal.2024.103496

{"title":"First-order reasoning and efficient semi-algebraic proofs","authors":"","doi":"10.1016/j.apal.2024.103496","DOIUrl":"10.1016/j.apal.2024.103496","url":null,"abstract":"<div>Semi-algebraic proof systems such as sum-of-squares (<math><mi>SoS</mi></math>) 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 <math><mi>NP</mi></math>-hard optimization problems. Motivated by the need to allow a more streamlined and uniform framework for working with <math><mi>SoS</mi></math> 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 <math><mi>SoS</mi></math> 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 <math><mi>SoS</mi></math>. Moreover, we prove new results for propositional proofs, and specifically extend Berkholz's dynamic-by-static simulation of polynomial calculus (PC) by <math><mi>SoS</mi></math> to PC with the radical rule.</div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-07-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0168007224001003/pdfft?md5=0c4bf895df8d8576c65657d12bc0c25e&pid=1-s2.0-S0168007224001003-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141781080","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Owings-like theorems for infinitely many colours or finite monochromatic sets 无限多色或有限单色集的类欧文定理

IF 0.6 2区数学

Annals of Pure and Applied Logic Pub Date : 2024-07-10 DOI: 10.1016/j.apal.2024.103495

{"title":"Owings-like theorems for infinitely many colours or finite monochromatic sets","authors":"","doi":"10.1016/j.apal.2024.103495","DOIUrl":"10.1016/j.apal.2024.103495","url":null,"abstract":"<div>Inspired by Owings's problem, we investigate whether, for a given an Abelian group G and cardinal numbers <math><mi>κ</mi><mo>,</mo><mi>θ</mi></math>, every colouring <math><mi>c</mi><mo>:</mo><mi>G</mi><mo>⟶</mo><mi>θ</mi></math> yields a subset <math><mi>X</mi><mo>⊆</mo><mi>G</mi></math> with <math><mo>|</mo><mi>X</mi><mo>|</mo><mo>=</mo><mi>κ</mi></math> such that <math><mi>X</mi><mo>+</mo><mi>X</mi></math> is monochromatic. (Owings's problem asks this for <math><mi>G</mi><mo>=</mo><mi>Z</mi></math>, <math><mi>θ</mi><mo>=</mo><mn>2</mn></math> and <math><mi>κ</mi><mo>=</mo><msub><mrow><mi>ℵ</mi></mrow><mrow><mn>0</mn></mrow></msub></math>; this is known to be false for the same G and κ but <math><mi>θ</mi><mo>=</mo><mn>3</mn></math>.) We completely settle the question for κ and θ both finite (by obtaining sufficient and necessary conditions for a positive answer) and for κ and θ both infinite (with a negative answer). Also, in the case where θ is infinite but κ is finite, we obtain some sufficient conditions for a negative answer as well as an example with a positive answer.</div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141717895","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Computably and punctually universal spaces 可计算和准时通用空间

IF 0.6 2区数学

Annals of Pure and Applied Logic Pub Date : 2024-07-10 DOI: 10.1016/j.apal.2024.103491

引用次数: 0

Concerning Keisler measures over ultraproducts 关于超积物上的凯斯勒量纲

IF 0.6 2区数学

Annals of Pure and Applied Logic Pub Date : 2024-07-09 DOI: 10.1016/j.apal.2024.103492

引用次数: 0

Elimination of quantifiers for a theory of real closed rings 实闭环理论的量词消除

IF 0.6 2区数学

Annals of Pure and Applied Logic Pub Date : 2024-07-06 DOI: 10.1016/j.apal.2024.103494

引用次数: 0

Model-completeness and decidability of the additive structure of integers expanded with a function for a Beatty sequence 用一个函数扩展的整数加法结构的模型完备性和可判定性的比蒂序列

IF 0.6 2区数学

Annals of Pure and Applied Logic Pub Date : 2024-07-06 DOI: 10.1016/j.apal.2024.103493

引用次数: 0

Club stationary reflection and other combinatorial principles at ℵω+2 ℵω+2处的俱乐部静止反射和其他组合原理

IF 0.6 2区数学

Annals of Pure and Applied Logic Pub Date : 2024-07-01 DOI: 10.1016/j.apal.2024.103489

引用次数: 0

Bi-intermediate logics of trees and co-trees 树和共树的双中间逻辑

IF 0.6 2区数学

Annals of Pure and Applied Logic Pub Date : 2024-07-01 DOI: 10.1016/j.apal.2024.103490

{"title":"Bi-intermediate logics of trees and co-trees","authors":"","doi":"10.1016/j.apal.2024.103490","DOIUrl":"10.1016/j.apal.2024.103490","url":null,"abstract":"<div>A bi-Heyting algebra validates the Gödel-Dummett axiom <math><mo>(</mo><mi>p</mi><mo>→</mo><mi>q</mi><mo>)</mo><mo>∨</mo><mo>(</mo><mi>q</mi><mo>→</mo><mi>p</mi><mo>)</mo></math> iff the poset of its prime filters is a disjoint union of co-trees (i.e., order duals of trees). Bi-Heyting algebras of this kind are called bi-Gödel algebras and form a variety that algebraizes the extension <math><mi>bi-GD</mi></math> of bi-intuitionistic logic axiomatized by the Gödel-Dummett axiom. In this paper we initiate the study of the lattice <math><mi>Λ</mi><mo>(</mo><mi>bi-GD</mi><mo>)</mo></math> of extensions of <math><mi>bi-GD</mi></math>.We develop the methods of Jankov-style formulas for bi-Gödel algebras and use them to prove that there are exactly continuum many extensions of <math><mi>bi-GD</mi></math>. We also show that all these extensions can be uniformly axiomatized by canonical formulas. Our main result is a characterization of the locally tabular extensions of <math><mi>bi-GD</mi></math>. We introduce a sequence of co-trees, called the finite combs, and show that a logic in <math><mi>Λ</mi><mo>(</mo><mi>bi-GD</mi><mo>)</mo></math> is locally tabular iff it contains at least one of the Jankov formulas associated with the finite combs. It follows that there exists the greatest nonlocally tabular extension of <math><mi>bi-GD</mi></math> and consequently, a unique pre-locally tabular extension of <math><mi>bi-GD</mi></math>. These results contrast with the case of the intermediate logic axiomatized by the Gödel-Dummett axiom, which is known to have only countably many extensions, all of which are locally tabular.</div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0168007224000940/pdfft?md5=c7604d9cf135b7d72a099447fc38fed7&pid=1-s2.0-S0168007224000940-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141573355","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Searching problems above arithmetical transfinite recursion 搜索算术无穷递归以上的问题

IF 0.6 2区数学

Annals of Pure and Applied Logic Pub Date : 2024-06-19 DOI: 10.1016/j.apal.2024.103488

Yudai Suzuki , Keita Yokoyama

引用次数: 0

Decidability bounds for Presburger arithmetic extended by sine 由正弦扩展的普雷斯伯格算术的可判定性边界

IF 0.6 2区数学

Annals of Pure and Applied Logic Pub Date : 2024-06-12 DOI: 10.1016/j.apal.2024.103487

引用次数: 0