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There is a deep 1-generic set 有一个深度 1 般集
Pub Date : 2024-09-01 DOI: arxiv-2409.00631
Ang Li
An infinite binary sequence is Bennett deep if, for any computable timebound, the difference between the time-bounded prefix-free Kolmogorovcomplexity and the prefix-free Kolmogorov complexity of its initial segments iseventually unbounded. It is known that weakly 2-generic sets are shallow, i.e.not deep. In this paper, we show that there is a deep 1-generic set.
如果对任何可计算的时间边界而言,无限二元序列的时间边界无前缀科尔莫哥罗夫复杂度与其初始段的无前缀科尔莫哥罗夫复杂度之间的差值最终是无边界的,那么这个序列就是贝内特深度序列。众所周知,弱 2 代集是浅集,即不深集。在本文中,我们将证明存在一个深度 1 代集。
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引用次数: 0
Limit Groups and Automorphisms of $κ$-Existentially Closed Groups κ$-存在封闭群的极限群和自动形
Pub Date : 2024-08-31 DOI: arxiv-2409.00545
Burak Kaya, Mahmut Kuzucuoğlu, Patrizia Longobardi, Mercede Maj
The structure of automorphism groups of $kappa$-existentially closed groupsare studied by Kaya-Kuzucuou{g}lu in 2022. It was proved that Aut(G) is the union of subgroups of level preservingautomorphisms and $|Aut(G)|=2^kappa$ whenever $kappa$ is an inaccessible cardinal and $G$ is the unique $kappa$-existentially closed group of cardinality $kappa$. The cardinality of the automorphism group of a$kappa$-existentially closed group of cardinality $lambda>kappa$ is asked in Kourovka NotebookQuestion 20.40. Here we answer positively the promised case $kappa=lambda$ namely: If $G$ is a $kappa$-existentially closed group of cardinality $kappa$, then $|Aut(G)|=2^{kappa}$. We also answer Kegel's question on universal groups, namely: For any uncountable cardinal $kappa$, there exist universal groups of cardinality $kappa$.
Kaya-Kuzucuou{g}lu 在 2022 年研究了$kappa$-存在封闭群的自变群结构。研究证明,Aut(G)是保留自变量的级子群的联合,并且当$kappa$是不可访问的心数且$G$是心数为$kappa$的唯一的$kappa$-存在封闭群时,$|Aut(G)|=2^kappa$。在库洛夫卡笔记本问题 20.40 中提出了心性为 $lambda>kappa$ 的 $kappa$-existentially closed group 的自变群的心性问题。在此,我们对承诺的$kappa=lambda$情况作正面回答:如果$G$是一个心性为$kappa$的存在封闭群,那么$|Aut(G)|=2^{kappa}$。我们还回答了凯格尔关于普遍群的问题,即对于任何不可数的心数 $kappa$, 都存在心数为 $kappa$ 的普遍群。
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引用次数: 0
Strong reducibilities and set theory 强还原性与集合论
Pub Date : 2024-08-30 DOI: arxiv-2408.17393
Noah Schweber
We study Medvedev reducibility in the context of set theory -- specifically,forcing and large cardinal hypotheses. Answering a question of Hamkins and Licite{HaLi}, we show that the Medvedev degrees of countable ordinals are farfrom linearly ordered in multiple ways, our main result here being that thereis a club of ordinals which is an antichain with respect to Medvedevreducibility. We then generalize these results to arbitrary``reasonably-definable" reducibilities, under appropriate set-theoretichypotheses. We then turn from ordinals to general structures. We show that some of theresults above yield characterizations of counterexamples to Vaught'sconjecture; another applies to all situations, assigning an ordinal to anyreasonable class of structures and ``measure" on that class. We end bydiscussing some directions for future research.
我们在集合论的背景下研究梅德韦杰夫可还原性--特别是强迫假设和大贲门假设。我们回答了哈姆金斯和李的一个问题,证明了可数序元的梅德韦杰夫度在多个方面远不是线性有序的,我们在此的主要结果是,有一个序元俱乐部在梅德韦杰夫可还原性方面是一个反链。然后,在适当的集合论假设下,我们将这些结果推广到任意的 "合理定义的 "还原性。然后,我们从序数转向一般结构。我们证明,上述一些结果产生了沃特猜想反例的特征;另一个结果则适用于所有情况,即给任何合理的结构类分配一个序数,并对该类进行 "度量"。最后,我们讨论了未来研究的一些方向。
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引用次数: 0
Algebraic structure theory and interpolation failures in semilinear logics 半线性逻辑中的代数结构理论和插值失败
Pub Date : 2024-08-30 DOI: arxiv-2408.17400
Valeria Giustarini, Sara Ugolini
In this work we study integral residuated chains, and we solve some openproblems related to the study of the amalgamation property in varieties ofresiduated lattices, or equivalently, about the deductive interpolationproperty in substructural logics. More precisely, we find a V-formationconsisting of 2-potent finite commutative integral chains that does not have anamalgam, nor a one-amalgam, in residuated chains; as most relevantconsequences, this entails that the following varieties do not have theamalgamation property: semilinear commutative (integral) residuated lattices,MTL-algebras, involutive and pseudocomplemented MTL-algebras, and all of theirn-potent subvarieties for n strictly larger than 1. These results entail thefailure of the deductive interpolation property for the correspondingsubstructural logics.
在这项工作中,我们研究了积分残差链,并解决了一些与研究残差格的品种中的汞齐性质有关的开放问题,或者等价于研究子结构逻辑中的演绎插值性质的开放问题。更确切地说,我们发现了一个由 2 能有限交换积分链构成的 V 形,它在残差链中不存在混汞(amalgam),也不存在一汞(one-amalgam);作为最相关的结果,这意味着下列品种不存在混汞属性:半线性交换(积分)残差格、MTL-代数、非累加和伪补全的 MTL-代数,以及它们的 n 严格大于 1 的所有 n 能子品种。这些结果导致相应的结构逻辑的演绎内插性失效。
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引用次数: 0
Relational Companions of Logics 逻辑学的关系伴侣
Pub Date : 2024-08-30 DOI: arxiv-2408.17019
Sankha S. Basu, Sayantan Roy
The variable inclusion companions of logics have lately been thoroughlystudied by multiple authors. There are broadly two types of these companions:the left and the right variable inclusion companions. Another type ofcompanions of logics induced by Hilbert-style presentations (Hilbert-stylelogics) were introduced in a recent paper. A sufficient condition for therestricted rules companion of a Hilbert-style logic to coincide with its leftvariable inclusion companion was proved there, while a necessary conditionremained elusive. The present article has two parts. In the first part, we givea necessary and sufficient condition for the left variable inclusion and therestricted rules companions of a Hilbert-style logic to coincide. In the restof the paper, we recognize that the variable inclusion restrictions used todefine variable inclusion companions of a logic$langlemathcal{L},vdashrangle$ are relations from$mathcal{P}(mathcal{L})$ to $mathcal{L}$. This leads to a more general ideaof a relational companion of a logical structure, a framework that we borrowfrom the field of universal logic. We end by showing that even Hilbert-stylelogics and the restricted rules companions of these can be brought under theumbrella of the general notions of logical structures and their relationalcompanions that are discussed here.
最近,多位学者对逻辑的变包含同伴进行了深入研究。这些伴随体大致分为两类:左变量包含伴随体和右变量包含伴随体。最近的一篇论文介绍了由希尔伯特式陈述(希尔伯特式逻辑)诱导的另一种逻辑的同伴。该文证明了希尔伯特式逻辑的受限规则同伴与其左变量包含同伴重合的充分条件,而一个必要条件仍然难以捉摸。本文分为两部分。在第一部分中,我们给出了希尔伯特式逻辑的左变量包含和受限规则同伴重合的必要条件和充分条件。在本文的其余部分,我们认识到用于定义逻辑$langlemathcal{L},vdashrangle$的变量包含限制是从$mathcal{P}(mathcal{L})$到$mathcal{L}$的关系。这就引出了逻辑结构的关系伴生的更一般的概念,我们从普遍逻辑领域借用了这一框架。最后,我们将证明,即使是希尔伯特式逻辑及其受限规则同伴,也可以纳入本文所讨论的逻辑结构及其关系同伴的一般概念之中。
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引用次数: 0
The complexity of classifying continuous t-norms up to isomorphism 直到同构为止的连续 t-norms 分类的复杂性
Pub Date : 2024-08-29 DOI: arxiv-2408.16456
Jialiang He, Lili Shen, Yi Zhou
It is shown that the isomorphism relation between continuous t-norms is Borelbireducible with the relation of order isomorphism between linear orders on theset of natural numbers, and therefore, it is Borel bireducible with every Borelcomplete equivalence relation.
结果表明,连续 t-norms 之间的同构关系与自然数集上线性阶之间的阶同构关系是伯尔双向可简化关系,因此,它与每个伯尔完全等价关系都是伯尔双向可简化关系。
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引用次数: 0
Collapsing Constructive and Intuitionistic Modal Logics 折叠构造模态逻辑和直觉模态逻辑
Pub Date : 2024-08-29 DOI: arxiv-2408.16428
Leonardo Pacheco
In this note, we prove that the constructive and intuitionistic variants ofthe modal logic $mathsf{KB}$ coincide. This result contrasts with a recentresult by Das and Marin, who showed that the constructive and intuitionisticvariants of $mathsf{K}$ do not prove the same diamond-free formulas.
在这篇论文中,我们证明了模态逻辑 $mathsf{KB}$ 的构造变体和直觉变体是重合的。这一结果与达斯和马林最近的一个结果形成了对比,后者证明了$mathsf{K}$的构造变体和直觉变体并不能证明相同的无菱形公式。
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引用次数: 0
Every Polish group has a non-trivial topological group automorphism 每个波兰群都有一个非三维拓扑群自形变
Pub Date : 2024-08-28 DOI: arxiv-2408.16162
Carlos Pérez Estrada, Ulises Ariet Ramos-García
We prove that every Polish group admits a non-trivial topological groupautomorphism. This answers a question posed by Forte Shinko. As a consequence,we prove that there are no uniquely homogeneous Polish groups.
我们证明了每个波兰群都有一个非难拓扑群同构。这回答了福特-新科(Forte Shinko)提出的一个问题。因此,我们证明不存在唯一同质的波兰群。
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引用次数: 0
The equational theory of the Weihrauch lattice with (iterated) composition 具有(迭代)构成的魏赫劳赫网格的等式理论
Pub Date : 2024-08-27 DOI: arxiv-2408.14999
Cécilia Pradic
We study the equational theory of the Weihrauch lattice with composition anditerations, meaning the collection of equations between terms built fromvariables, the lattice operations $sqcup$, $sqcap$, the composition operator$star$ and its iteration $(-)^diamond$ , which are true however we substitute(slightly extended) Weihrauch degrees for the variables. We characterize themusing B"uchi games on finite graphs and give a complete axiomatization thatderives them. The term signature and the axiomatization are reminiscent ofKleene algebras, except that we additionally have meets and the latticeoperations do not fully distributes over composition. The game characterizationalso implies that it is decidable whether an equation is universally valid. Wegive some complexity bounds; in particular, the problem is Pspace-hard ingeneral and we conjecture that it is solvable in Pspace.
我们研究的是具有组成和迭代的魏赫劳赫格的等式理论,即由变量、格运算 $sqcup$, $sqcap$, 组成算子$star$及其迭代$(-)^diamond$构成的项之间的等式集合。我们用有限图上的渊博博弈来描述它们,并给出了一个完整的公理化来解释它们。术语签名和公理化都让人想起克莱因代数,只是我们另外有了相遇,而且网格操作并不完全分布于组成。博弈表征也意味着方程是否普遍有效是可以判定的。我们给出了一些复杂性边界;特别是,这个问题一般来说是 Pspace-hard(Pspace-hard)的,我们猜想它在 Pspace 中是可解的。
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引用次数: 0
Simple Models of Randomization and Preservation Theorems 随机化和保存定理的简单模型
Pub Date : 2024-08-27 DOI: arxiv-2408.15014
Karim Khanaki, Massoud Pourmahdian
The main purpose of this paper is to present new and more uniformmodel-theoretic/combinatorial proofs of the theorems (in [5] and [4]): Therandomization $T^{R}$ of a complete first-order theory $T$ with $NIP$/stabilityis a (complete) first-order continuous theory with $NIP$/stability. The proofmethod for both theorems is based on the significant use of a particular typeof models of $T^{R}$, namely simple models, and certain indiscernible arrays.Using simple models of $T^R$ gives the advantage of re-proving these theoremsin a simpler and quantitative manner. We finally turn our attention to $NSOP$in randomization. We show that based on the definition of $NSOP$ given [11],$T^R$ is stable if and only if it is $NIP$ and $NSOP$.
本文的主要目的是对这些定理(见 [5] 和 [4])提出新的、更统一的模型理论/组合证明:具有 $NIP$/stability 的完整一阶理论 $T$ 的随机化 $T^{R}$ 是具有 $NIP$/stability 的(完整)一阶连续理论。这两个定理的证明方法都基于对 $T^{R}$ 的一种特殊模型,即简单模型和某些不可辨别阵列的大量使用。最后,我们将注意力转向随机化中的 $NSOP$。我们证明,根据 [11] 给出的 $NSOP$ 定义,当且仅当 $NIP$ 和 $NSOP$ 时,$T^R$ 是稳定的。
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arXiv - MATH - Logic
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