Annals of Pure and Applied Logic最新文献

英文中文

Constructing models of small ordered theories with maximal countable spectrum 构造具有极大可数谱的小有序理论模型

IF 0.6 2区数学 Q2 LOGIC

Annals of Pure and Applied Logic

Pub Date : 2025-10-01 DOI: 10.1016/j.apal.2025.103659

B. Baizhanov , T. Zambarnaya

We propose a method for the construction of countable models of small theories. We then apply it to prove theorems concerning the maximal number of countable non-isomorphic models of linearly ordered theories.

我们提出了一种构造小理论可数模型的方法。然后将其应用于线性有序理论的可数非同构模型的最大数目定理的证明。

引用次数: 0

Fundamental sequences based on localization 基于定位的基本序列

IF 0.6 2区数学 Q2 LOGIC

Annals of Pure and Applied Logic

Pub Date : 2025-09-24 DOI: 10.1016/j.apal.2025.103658

Gunnar Wilken

Building on Buchholz' assignment for ordinals below Bachmann-Howard ordinal, see [2], we introduce systems of fundamental sequences for two kinds of relativized ϑ-function-based notation systems of strength

Π_{1}^{1} {-CA}_{0}

and prove Bachmann property for these systems, which is essential for monotonicity properties of subrecursive hierarchies defined on the basis of fundamental sequences. The central notion of our construction is the notion of localization, which was introduced in [12].

The first kind of stepwise defined ϑ-functions over ordinal addition as basic function fits the framework of the ordinal arithmetical toolkit developed in [12], whereas the second kind of ϑ-functions is defined simultaneously and will allow for further generalization to larger proof-theoretic ordinals, see [10].

The systems of fundamental sequences given here enable the investigation of fundamental sequences and independence phenomena also in the context of patterns of resemblance, an approach to ordinal notations that is both semantic and combinatorial and was first introduced by Carlson in [4] and further analyzed in [11], [13], [14], [5].

Our exposition is put into the context of the abstract approach to fundamental sequences developed by Buchholz, Cichon, and Weiermann in [3]. The results of this paper will be applied to the theory of Goodstein sequences, extending results of [7].

在Buchholz对低于Bachmann- howard序数的赋值（见[2]）的基础上，我们引入了两种强度为Π11-CA0的相对ϑ-function-based符号系统的基本序列系统，并证明了这些系统的Bachmann性质，这对于基于基本序列定义的子递归层次的单调性是必不可少的。我们构建的中心概念是本地化的概念，这是在b[12]中引入的。第一类在序数加法上逐步定义的ϑ-functions作为基本函数符合[12]中开发的序数算术工具包的框架，而第二类ϑ-functions是同时定义的，并且将允许进一步推广到更大的证明论序数，参见[10]。这里给出的基本序列系统使得在相似模式的背景下研究基本序列和独立现象成为可能，相似模式是一种语义和组合的有序符号方法，由Carlson在[4]中首次引入，并在[11]，[13],[14]，[5]中进一步分析。我们的阐述被置于由Buchholz， Cichon和Weiermann在1986年开发的基本序列的抽象方法的背景下。本文的结果将应用于Goodstein序列理论，推广了[7]的结果。

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引用次数: 0

Proof-theoretic investigation of λβ-reduction in the simply typed λ-calculus 单型λ微积分中λβ约简的证明理论研究

IF 0.6 2区数学 Q2 LOGIC

Annals of Pure and Applied Logic

Pub Date : 2025-09-23 DOI: 10.1016/j.apal.2025.103657

William Stirton

The paper defines a function f from simply typed λ-terms to natural numbers and proves that, if

M^{⁎}

is a simply typed λ-term formed by contracting an arbitrary λβ-redex within another term M, then

f (M^{⁎}) < f (M)

. Unlike previous proofs of similar theorems, the redex contracted may be completely arbitrary, i.e. without any restriction on rule (ξ). The function f itself is related to, and no more computationally difficult than, a similar-looking function defined in Schütte's Proof Theory (1977).

本文定义了一个由单型λ项到自然数的函数f，并证明了，如果M是由任意λβ-索引缩并到另一项M内形成的单型λ项，则f(M) <f(M)。与之前类似定理的证明不同，收缩的索引可以是完全任意的，即不受规则（ξ）的任何限制。函数f本身与sch特的证明理论（1977）中定义的一个类似的函数相关，并且在计算上并不困难。

引用次数: 0

Uncountable homogeneous structures 不可数同构结构

IF 0.6 2区数学 Q2 LOGIC

Annals of Pure and Applied Logic

Pub Date : 2025-08-27 DOI: 10.1016/j.apal.2025.103649

Adam Bartoš, Wiesław Kubiś

We study the existence of uncountable first-order structures that are homogeneous with respect to their finitely generated substructures. In many classical cases this is either well-known or follows from general facts, for example, if the language is finite and relational then ultrapowers provide arbitrarily large such structures. On the other hand, there are no general results saying that uncountable homogeneous structures with a given age exist. We examine the monoid of self-embeddings of a fixed countable homogeneous structure and, using abstract Fraïssé theory, we present a method of constructing an uncountable homogeneous structure, based on the amalgamation property of this monoid.

研究了不可数一阶结构的存在性，这些结构对其有限生成的子结构是齐次的。在许多经典情况下，这要么是众所周知的，要么是遵循一般事实的，例如，如果语言是有限的和关系的，那么超能力提供任意大的这样的结构。另一方面，没有一般的结果表明，具有给定年龄的不可数同质结构存在。我们研究了固定可数齐次结构的自嵌入单群，并利用抽象的Fraïssé理论，提出了一种构造不可数齐次结构的方法，该方法基于该单群的合并性质。

引用次数: 0

Transposition of variables is hard to axiomatize 变量的变换是很难公理化的

IF 0.6 2区数学 Q2 LOGIC

Annals of Pure and Applied Logic

Pub Date : 2025-08-26 DOI: 10.1016/j.apal.2025.103650

Hajnal Andréka, István Németi, Zsolt Tuza

The function

p_{x y}

that interchanges two logical variables

x, y

in formulas is hard to describe in the following sense. Let F denote the Lindenbaum–Tarski formula-algebra of a finite-variable first-order logic, endowed with

p_{x y}

as a unary function. We prove that each equational axiom system for the equational theory of F has to contain, for each finite n, an equation that contains together with

p_{x y}

at least n algebraic variables, and each of the operations

\exists, =, \lor

. This gives an answer to a problem raised by Johnson (1969) [30]: the class

R P E A_{α}

of representable polyadic equality algebras of a finite dimension

α \geq 3

cannot be axiomatized by adding finitely many equations to the equational theory of representable cylindric algebras of dimension α. Consequences for proof systems of finite-variable logic and for defining equations of polyadic equality algebras are given.

The proof uses a family of nonrepresentable polyadic equality algebras

A_{n}

that are more and more nearly representable as n increases: their n-generated subalgebras as well as their proper reducts are representable. The lattice of subvarieties of

R P E A_{α}

is investigated and new open problems are asked about the interplay between the transposition operations and about generalizability of the results to infinite dimensions.

在公式中交换两个逻辑变量x，y的函数pxy很难用下面的意义来描述。设F表示有限变量一阶逻辑的Lindenbaum-Tarski公式代数，赋予pxy为一元函数。我们证明F的方程理论的每个方程公理系统必须包含，对于每一个有限n，一个方程与pxy至少包含n个代数变量，并且每个操作∃，=,∨。这就回答了Johnson(1969)[30]提出的一个问题：有限维α≥3的可表示多进代数类RPEAα不能通过在α维可表示圆柱代数的方程理论中添加有限多个方程来公理化。给出了有限变量逻辑的证明系统和定义多进等式代数方程的结果。该证明使用了一组随着n的增加而越来越接近可表示的不可表示的多进相等代数：它们的n生成子代数和它们的适当约化都是可表示的。研究了RPEAα的子变格，并提出了一些新的开放问题，包括转置运算之间的相互作用和结果在无限维上的可推广性。

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引用次数: 0

Classifying different criteria for learning algebraic structures 分类学习代数结构的不同标准

IF 0.6 2区数学 Q2 LOGIC

Annals of Pure and Applied Logic

Pub Date : 2025-08-22 DOI: 10.1016/j.apal.2025.103648

Nikolay Bazhenov , Vittorio Cipriani , Sanjay Jain , Luca San Mauro , Frank Stephan

In the last years there has been a growing interest in the study of learning problems associated with algebraic structures. The framework we use models the scenario in which a learner is given larger and larger fragments of a structure from a given target family and is required to output an hypothesis about the structure's isomorphism type. So far researchers focused on Ex-learning, in which the learner is asked to eventually stabilize to the correct hypothesis, and on restrictions where the learner is allowed to change the hypothesis a fixed number of times. Yet, other learning paradigms coming from classical algorithmic learning theory remained unexplored. We study the ‘‘learning power’’ of such criteria, comparing them via descriptive-set-theoretic tools thanks to the novel notion of E-learnability. The main outcome of this paper is that such criteria admit natural syntactic characterizations in terms of infinitary formulas analogous to the one given for Ex-learning in [8]. Such characterizations give a powerful method to understand whether a family of structures is learnable with respect to the desired criterion.

在过去的几年里，人们对与代数结构相关的学习问题的研究越来越感兴趣。我们使用的框架模拟了这样的场景：学习者从给定的目标族中获得越来越大的结构片段，并被要求输出关于结构同构类型的假设。到目前为止，研究人员主要集中在前学习（Ex-learning）和限制上，前者要求学习者最终稳定在正确的假设上，后者允许学习者在固定的次数内改变假设。然而，来自经典算法学习理论的其他学习范式仍未被探索。我们研究了这些标准的“学习能力”，通过描述集理论工具对它们进行比较，这要归功于e-可学习性的新概念。本文的主要结果是，这些准则可以用类似于[8]中给出的前学习的无穷公式来自然地描述句法特征。这样的描述提供了一种强大的方法来理解一组结构是否可以根据期望的标准学习。

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引用次数: 0

On the spectrum of limit models 在极限模型的谱上

IF 0.6 2区数学 Q2 LOGIC

Annals of Pure and Applied Logic

Pub Date : 2025-08-21 DOI: 10.1016/j.apal.2025.103647

Jeremy Beard , Marcos Mazari-Armida

We study the spectrum of limit models assuming the existence of a nicely behaved independence notion. Under reasonable assumptions, we show that all ‘long’ limit models are isomorphic, and all ‘short’ limit models are non-isomorphic.

Theorem

Let K be a

ℵ_{0}

-tame abstract elementary class stable in

λ \geq LS (K)

with amalgamation, joint embedding and no maximal models. Let

κ < λ^{+}

be regular. Suppose

is an independence relation on the models of size λ that satisfies uniqueness, extension, non-forking amalgamation, universal continuity, and

(\geq κ)

-local character.

Suppose

δ_{1}, δ_{2} < λ^{+}

with

cf (δ_{1}) < cf (δ_{2})

. Then for any

N_{1}, N_{2}, M \in K_{λ}

where

N_{l}

is a

(λ, δ_{l})

-limit model over M for

l = 1, 2

Both implications in the conclusion have improvements. High cofinality limits are isomorphic without the

ℵ_{0}

-tameness assumption and assuming

is defined only on high cofinality limit models. Low cofinality limits are non-isomorphic without assuming non-forking amalgamation.

We show how our results can be used to study limit models in both abstract settings and in natural examples of abstract elementary classes.

我们研究了假设存在一个很好的独立性概念的极限模型的谱。在合理的假设下，我们证明了所有的“长”极限模型都是同构的，所有的“短”极限模型都是非同构的。定理设K是一个在λ≥LS(K)中稳定的、具有合并、联合嵌入和无极大模型的λ 0-驯服的抽象初等类。设κ<；λ+是正则的。假设大小为λ的模型上存在一个独立关系，该关系满足唯一性、可拓性、非分叉合并、普遍连续性和(≥κ)-局部特征。假设δ1 δ2<；λ+ cf(δ1)<cf（δ2）然后，对于任意N1，N2,M∈Kλ，其中当l=1,2时，Nl是M上的（λ,δl）极限模型。高共度极限是同构的，不需要假设，假设只在高共度极限模型上定义。低共通性限制是非同构的，不假设非分叉合并。我们展示了如何将我们的结果用于研究抽象设置和抽象初等类的自然示例中的极限模型。

引用次数: 0

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

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