Annals of Pure and Applied Logic最新文献

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Equivalence of multiset-based consequence relations 基于多集的结果关系的等价性

IF 0.6 2区数学 Q2 LOGIC

Annals of Pure and Applied Logic

Pub Date : 2025-08-05 DOI: 10.1016/j.apal.2025.103646

Ali Madanshekaf , Adam Přenosil , Zeinab Khanjanzadeh Seresti , Constantine Tsinakis

The pioneering work of Blok and Jónsson, and its further development by Galatos and Tsinakis, initiated an abstract study of consequence relations through the lens of module theory, treating consequence relations over all types of syntactic objects on an equal footing. Despite this generality, their framework retains the assumption that premises in a consequence relation form a mere set, rather than a more structured collection. An attempt to extend this framework to account for inferentially substructural generalizations of consequence relations, where the premises have the structure of a finite multiset, was recently made by Cintula, Gil-Férez, Moraschini, and Paoli. In this paper, we propose a different substructural generalization of the Galatos–Tsinakis approach, where the premises are instead taken to form a set of finite multisets. This yields a smoother and more flexible framework that, unlike the approach of Cintula et al., subsumes the original theory of Galatos and Tsinakis as a special case.

布洛克和Jónsson的开创性工作，以及加拉托斯和齐纳基斯的进一步发展，通过模块理论的视角开始了对结果关系的抽象研究，平等地对待所有类型的句法对象的结果关系。尽管有这种通用性，但它们的框架保留了一个假设，即结果关系中的前提形成了一个纯粹的集合，而不是一个更结构化的集合。Cintula, gil - fsamurez， Moraschini和Paoli最近尝试将这个框架扩展到推论的子结构推广，其中前提具有有限多集的结构。在本文中，我们提出了Galatos-Tsinakis方法的一种不同的子结构推广，其中的前提被用来形成有限多集的集合。这产生了一个更平滑和更灵活的框架，与Cintula等人的方法不同，它将加拉托斯和齐纳基斯的原始理论作为一个特例。

引用次数: 0

Countable ordered groups and Weihrauch reducibility 可数有序群与Weihrauch可约性

IF 0.6 2区数学 Q2 LOGIC

Annals of Pure and Applied Logic

Pub Date : 2025-07-25 DOI: 10.1016/j.apal.2025.103644

Ang Li

This paper continues to study the connection between reverse mathematics and Weihrauch reducibility. In particular, we study the problems formed from Maltsev's theorem [11] on the order types of countable ordered groups. Solomon [14] showed that the theorem is equivalent to

Π_{1}^{1}

C A_{0}

, the strongest of the big five subsystems of second order arithmetic. We show that the strength of the theorem comes from having a dense linear order without endpoints in its order type. Then, we show that for the related Weihrauch problem to be strong enough to be equivalent to

\hat{WF}

(the analog problem of

Π_{1}^{1}

C A_{0}

), an order-preserving function is necessary in the output. Without the order-preserving function, the problems are very much to the side compared to analog problems of the big five.

本文继续研究逆向数学与魏氏约化性之间的联系。特别地，我们研究了由马尔采夫定理[11]形成的关于可数有序群的序型问题。Solomon[14]证明了该定理等价于二阶算法五大子系统中最强的Π11-CA0。我们证明了定理的强度来自于在其阶型中有一个没有端点的密集线性阶。然后，我们证明了为了使相关的Weihrauch问题足够强而等价于WF - （Π11-CA0的模拟问题），输出中必须有一个保序函数。如果没有保序函数，这些问题与五大模拟问题相比就显得微不足道了。

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引用次数: 0

Thick sets and the Central Set Theorem 厚集与中心集定理

IF 0.6 2区数学 Q2 LOGIC

Annals of Pure and Applied Logic

Pub Date : 2025-07-25 DOI: 10.1016/j.apal.2025.103645

Teng Zhang

In 1981, Furstenberg introduced the notion of central sets, and he established the Central Set Theorem. Since then, several generalizations of this result have been found, where a significant version is obtained by De, Hindman and Strauss. In this article, we find that the Central Set Theorem can be improved further. And we observe that there are some connections between thick sets and J-sets. Based on that, we establish a CST-type result for thick sets. Moreover, we introduce a new notion called super thick sets, and find that this notion has rich combinatorial properties. In particular, it contains additive and multiplicative structures, and it has a CST-type result for two operations. In addition, it can be partitioned into κ super thick subsets in very weakly cancellative weak rings with size κ.

1981年，Furstenberg引入了中心集的概念，并建立了中心集定理。从那时起，对这一结果进行了几次推广，其中De， Hindman和Strauss得到了一个重要的版本。在本文中，我们发现中心集定理是可以进一步改进的。我们观察到在厚集和j集之间存在一些联系。在此基础上，我们建立了厚集的cst型结果。此外，我们引入了一个新的概念——超粗集，并发现这个概念具有丰富的组合性质。特别是，它包含加法和乘法结构，并且对于两个操作具有cst类型的结果。此外，它可以在大小为κ的非常弱消弱环中划分为κ超厚子集。

引用次数: 0

Description of structures computable in polynomial time 在多项式时间内可计算的结构描述

IF 0.6 2区数学 Q2 LOGIC

Annals of Pure and Applied Logic

Pub Date : 2025-07-16 DOI: 10.1016/j.apal.2025.103636

Pavel E. Alaev

We describe structures computable in polynomial time (P-computable) by a simple and short criterion. We prove that every substructure of a P-computable structure generated by a c.e. set also has a P-computable presentation. For example, we easily prove that every computable torsion-free Abelian group has a P-computable presentation.

我们用一个简单而简短的准则描述了多项式时间内可计算的结构（p -可计算）。证明了由c.e.集生成的p -可计算结构的每个子结构也具有p -可计算表示。例如，我们很容易证明每个可计算的无扭阿贝尔群都具有p -可计算的表示。

引用次数: 0

Unified inverse correspondence for LE-logics le逻辑的统一逆对应

IF 0.6 2区数学 Q2 LOGIC

Annals of Pure and Applied Logic

Pub Date : 2025-07-09 DOI: 10.1016/j.apal.2025.103635

Alessandra Palmigiano , Mattia Panettiere

We generalize Kracht's theory of internal describability from classical modal logic to the family of all logics canonically associated with varieties of normal lattice expansions (LE algebras). We work in the purely algebraic setting of perfect LEs; the formulas playing the role of Kracht's formulas in this generalized setting pertain to a first order language whose atoms are special inequalities between terms of perfect algebras. Via duality, formulas in this language can be equivalently translated into first order conditions in the frame correspondence languages of several types of relational semantics for LE-logics.

我们将Kracht的内可描述性理论从经典模态逻辑推广到与各种正规格展开（LE代数）正规相关的所有逻辑族。我们研究的是纯代数的完备最小二乘；在这个广义集合中扮演Kracht公式角色的公式属于一阶语言，它的原子是完全代数项之间的特殊不等式。通过对偶性，该语言中的公式可以等效地转换为le逻辑中几种关系语义的框架对应语言中的一阶条件。

引用次数: 0

The DeMorganization of a locale 场所的非组织化

IF 0.6 2区数学 Q2 LOGIC

Annals of Pure and Applied Logic

Pub Date : 2025-07-08 DOI: 10.1016/j.apal.2025.103634

Igor Arrieta

In 2009, Caramello proved that each topos has a largest dense subtopos whose internal logic satisfies De Morgan law (also known as the law of the weak excluded middle). This finding implies that every locale has a largest dense extremally disconnected sublocale, referred to as its DeMorganization. In this paper, we take the first steps in exploring the DeMorganization in the localic context, shedding light on its geometric nature by showing that it is always a fitted sublocale and by providing a concrete description. Explicit examples of DeMorganizations for toposes that do not satisfy De Morgan law are rather difficult to find. We present a contribution in that direction, with the main result of the paper showing that for any metrizable locale (without isolated points), its DeMorganization coincides with its Booleanization. This, in particular, implies that any extremally disconnected metric locale (without isolated points) must be Boolean, generalizing a well-known result for topological spaces to the localic setting.

2009年，Caramello证明了每个拓扑都有一个最大的稠密子拓扑，其内部逻辑满足De Morgan定律（又称弱排中律）。这一发现意味着每个区域都有一个最大的密集的极度不相连的子区域，称为其非组织。在本文中，我们采取了第一步，探索DeMorganization在当地的背景下，通过显示它总是一个合适的子区域，并提供具体的描述，揭示其几何性质。对于不满足德摩根定律的主题，很难找到明确的反组织例子。我们在这个方向上做出了贡献，论文的主要结果表明，对于任何可度量的区域（没有孤立点），其非组织性与布尔化一致。特别是，这意味着任何极度不连接的度量区域设置（没有孤立点）都必须是布尔值，从而将拓扑空间的一个众所周知的结果推广到局部设置。

引用次数: 0

The unstable formula theorem revisited via algorithms 通过算法重新考察了不稳定公式定理

IF 0.6 2区数学 Q2 LOGIC

Annals of Pure and Applied Logic

Pub Date : 2025-07-07 DOI: 10.1016/j.apal.2025.103633

M. Malliaris , S. Moran

This paper is about the surprising interaction of a foundational result from model theory, about stability of theories, with algorithmic stability in learning. First, in response to gaps in existing learning models, we introduce a new statistical learning model, called “Probably Eventually Correct” or PEC. We characterize Littlestone (stable) classes in terms of this model. As a corollary, Littlestone classes have frequent short definitions in a natural statistical sense. In order to obtain a characterization of Littlestone classes in terms of frequent definitions, we build an equivalence theorem highlighting what is common to many existing approximation algorithms, and to the new PEC. This is guided by an analogy to definability of types in model theory, but has its own character. Drawing on these theorems and on other recent work, we present a complete algorithmic analogue of Shelah's celebrated Unstable Formula Theorem, with algorithmic properties taking the place of the infinite.

这篇论文是关于模型理论的一个基础结果，关于理论的稳定性，与学习中的算法稳定性之间惊人的相互作用。首先，为了应对现有学习模型中的差距，我们引入了一种新的统计学习模型，称为“可能最终正确”或PEC。我们根据这个模型来描述Littlestone（稳定）类。作为推论，Littlestone类在自然统计意义上经常有简短的定义。为了在频繁定义方面获得Littlestone类的特征，我们建立了一个等价定理，突出了许多现有近似算法和新PEC的共同之处。这与模型理论中类型的可定义性类似，但有其自身的特点。利用这些定理和其他最近的工作，我们提出了一个完整的算法模拟Shelah的著名的不稳定公式定理，用算法性质代替无穷。

引用次数: 0

On definable Skolem functions and trichotomy 关于可定义Skolem函数和三分法

IF 0.6 2区数学 Q2 LOGIC

Annals of Pure and Applied Logic

Pub Date : 2025-07-04 DOI: 10.1016/j.apal.2025.103632

Bruno Dinis , Mário J. Edmundo

In this paper we give an explicit characterization of o-minimal structures with definable Skolem functions/definable choice. Such structures are, after naming finitely many elements from the prime model, a union of finitely many trivial points each defined over ∅ and finitely many open intervals each a union of a ∅-definable family of group-intervals with fixed positive elements.

本文给出了具有可定义Skolem函数/可定义选择的0 -极小结构的显式刻划。在从素数模型命名有限多元素之后，这样的结构是有限多平凡点的联合，每个平凡点定义在∅上，有限多开放区间每个开放区间是∅可定义的群区间族的联合，这些群区间具有固定的正元素。

引用次数: 0

On Recurrence Axioms 关于递归公理

IF 0.6 2区数学 Q2 LOGIC

Annals of Pure and Applied Logic

Pub Date : 2025-06-30 DOI: 10.1016/j.apal.2025.103631

Sakaé Fuchino , Toshimichi Usuba

The Recurrence Axiom for a class

P

of posets and a set A of parameters is an axiom scheme in the language of ZFC asserting that if a statement with parameters from A is forced by a poset in

P

, then there is a ground containing the parameters and satisfying the statement.

The tightly super-

C^{(\infty)}

P

-Laver generic hyperhuge continuum implies the Recurrence Axiom for

P

and

H (2^{ℵ_{0}})

. The consistency strength of this assumption can be decided thanks to our main theorems asserting that the minimal ground (bedrock) exists under a tightly

P

-generic hyperhuge cardinal κ, and that κ in the bedrock is genuinely hyperhuge, or even super

C^{(\infty)}

hyperhuge if κ is a tightly super-

C^{(\infty)}

P

-Laver generic hyperhuge definable cardinal.

The Laver Generic Maximum (LGM), one of the strongest combinations of axioms in our context, integrates practically all known set-theoretic principles and axioms in itself, either as its consequences or as theorems holding in (many) grounds of the universe. For instance, double plus version of Martin's Maximum is a consequence of LGM while Cichoń's Maximum is a phenomenon in many grounds of the universe under LGM.

一类偏序集P和参数集a的递归公理是ZFC语言中的一个公理格式，它断言如果一个参数来自a的命题被P中的偏序集强制，则存在一个包含参数并满足该命题的根据。紧超- c(∞)-P-Laver型超巨型连续体蕴涵了P和H（2 ~ 0）的递推公理。这一假设的一致性可以通过我们的主要定理来确定，这些定理断言最小基底（基岩）存在于一个紧密的p -泛型超大基数κ下，并且基岩中的κ是真正的超大，甚至是超级C（∞）超大，如果κ是一个紧密的超级C(∞)-P-Laver泛型超大可定义基数。Laver泛极大值（LGM）是我们的背景中最强大的公理组合之一，它实际上整合了所有已知的集合论原理和公理，无论是作为它的结果还是作为在宇宙（许多）基础上成立的定理。例如，Martin极大值的双正版本是LGM的结果，而cichoski极大值是在LGM下宇宙的许多地方都存在的现象。

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引用次数: 0

Topological games in Ramsey spaces Ramsey空间中的拓扑对策

IF 0.6 2区数学 Q2 LOGIC

Annals of Pure and Applied Logic

Pub Date : 2025-06-26 DOI: 10.1016/j.apal.2025.103630

Julián C. Cano , Carlos A. Di Prisco

Topological Ramsey theory studies a class of combinatorial topological spaces, known as topological Ramsey spaces, unifying the essential features of those combinatorial frames where the Ramsey property is equivalent to the Baire property. In this article, we present a general overview of the combinatorial structure of topological Ramsey spaces and their main properties, and we propose an alternative proof of the abstract Ellentuck theorem for a large family of axiomatized topological Ramsey spaces. Additionally, we introduce the notion of selective axiomatized topological Ramsey space, and generalize Kastanas games in order to characterize the Ramsey property for this broad family of topological Ramsey spaces through topological games.

拓扑Ramsey理论研究了一类组合拓扑空间，即拓扑Ramsey空间，它统一了那些Ramsey性质等价于Baire性质的组合框架的基本特征。本文概述了拓扑Ramsey空间的组合结构及其主要性质，并给出了一大族公理化拓扑Ramsey空间的抽象Ellentuck定理的另一种证明。此外，我们引入了选择性公理化的拓扑Ramsey空间的概念，并推广了Kastanas对策，以便通过拓扑对策来表征这一广义的拓扑Ramsey空间族的Ramsey性质。

引用次数: 0

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