Annals of Pure and Applied Logic最新文献

英文中文

The logic of cardinality comparison without the axiom of choice 没有选择公理的基数比较逻辑

IF 0.6 2区数学 Q2 LOGIC

Annals of Pure and Applied Logic

Pub Date : 2024-12-10 DOI: 10.1016/j.apal.2024.103549

Matthew Harrison-Trainor , Dhruv Kulshreshtha

We work in the setting of Zermelo-Fraenkel set theory without assuming the Axiom of Choice. We consider sets with the Boolean operations together with the additional structure of comparing cardinality (in the Cantorian sense of injections). What principles does one need to add to the laws of Boolean algebra to reason not only about intersection, union, and complementation of sets, but also about the relative size of sets? We give a complete axiomatization.

A particularly interesting case is when one restricts to the Dedekind-finite sets. In this case, one needs exactly the same principles as for reasoning about imprecise probability comparisons, the central principle being Generalized Finite Cancellation (which includes, as a special case, division-by-m). In the general case, the central principle is a restricted version of Generalized Finite Cancellation within Archimedean classes which we call Covered Generalized Finite Cancellation.

我们在Zermelo-Fraenkel集合理论的背景下工作，而不假设选择公理。我们考虑具有布尔运算的集合以及比较基数的附加结构（在Cantorian意义上的注入）。我们需要在布尔代数的法则中加入什么原则来推理集合的相交、并和互补，以及集合的相对大小？我们给出了一个完全的公理化。一个特别有趣的例子是当我们限制dedekind有限集的时候。在这种情况下，人们需要与不精确概率比较的推理完全相同的原理，中心原理是广义有限消去（作为特殊情况，它包括除以m的除法）。在一般情况下，中心原理是阿基米德类中广义有限消去的一个限制版本，我们称之为覆盖广义有限消去。

引用次数: 0

Ordered transexponential fields 有序转幂域

IF 0.6 2区数学 Q2 LOGIC

Annals of Pure and Applied Logic

Pub Date : 2024-12-02 DOI: 10.1016/j.apal.2024.103541

Lothar Sebastian Krapp , Salma Kuhlmann

We develop a first-order theory of ordered transexponential fields in the language

{+, \cdot, 0, 1, <, e, T}

, where e and T stand for unary function symbols. While the archimedean models of this theory are readily described, the study of the non-archimedean models leads to a systematic examination of the induced structure on the residue field and the value group under the natural valuation. We establish necessary and sufficient conditions on the value group of an ordered exponential field

(K, e)

to admit a transexponential function T compatible with e. Moreover, we give a full characterisation of all countable ordered transexponential fields in terms of their valuation theoretic invariants.

我们在{+，⋅,0,1,<,e，T}语言中建立了有序转幂域的一阶理论，其中e和T代表一元函数符号。虽然该理论的阿基米德模型很容易描述，但对非阿基米德模型的研究导致了对剩余场和自然估值下的值群的诱导结构的系统检查。在有序指数域（K,e）的值群上建立了允许转幂函数T与e相容的充要条件，并给出了所有可数有序转幂域的赋值理论不变量的完整刻画。

引用次数: 0

Tame topology in Hensel minimal structures Hensel最小结构中的驯服拓扑

IF 0.6 2区数学 Q2 LOGIC

Annals of Pure and Applied Logic

Pub Date : 2024-11-29 DOI: 10.1016/j.apal.2024.103540

Krzysztof Jan Nowak

We are concerned with topology of Hensel minimal structures on non-trivially valued fields K, whose axiomatic theory was introduced in a recent paper by Cluckers–Halupczok–Rideau. We additionally require that every definable subset in the imaginary sort RV, binding together the residue field Kv and value group vK, be already definable in the plain valued field language. This condition is satisfied by several classical tame structures on Henselian fields, including Henselian fields with analytic structure, V-minimal fields, and polynomially bounded o-minimal structures with a convex subring. In this article, we establish many results concerning definable functions and sets. These are, among others, existence of the limit for definable functions of one variable, a closedness theorem, several non-Archimedean versions of the Łojasiewicz inequalities, an embedding theorem for regular definable spaces, and the definable ultranormality and ultraparacompactness of definable Hausdorff LC-spaces.

本文研究了非平凡值域K上的Hensel极小结构的拓扑结构，该结构的公理化理论已在cluckers - halupczk - rideau最近的一篇论文中提出。我们还要求虚排序RV中的每个可定义子集，将剩余域Kv和值群vK结合在一起，在纯值域语言中已经是可定义的。这一条件被Henselian域上的几个经典驯服结构所满足，包括解析结构Henselian域、v -极小域和带凸子的多项式有界o-极小结构。在本文中，我们建立了许多关于可定义函数和可定义集的结果。其中包括：单变量可定义函数极限的存在性、闭性定理、Łojasiewicz不等式的几个非阿基米德版本、正则可定义空间的嵌入定理、可定义Hausdorff lc空间的可定义超不规则性和超紧性。

引用次数: 0

Strength and limitations of Sherali-Adams and Nullstellensatz proof systems Sherali-Adams和Nullstellensatz证明系统的优势和局限性

IF 0.6 2区数学 Q2 LOGIC

Annals of Pure and Applied Logic

Pub Date : 2024-11-28 DOI: 10.1016/j.apal.2024.103538

Ilario Bonacina, Maria Luisa Bonet

We compare the strength of the algebraic proof systems Sherali-Adams (

SA

) and Nullstellensatz (

NS

) with Frege-style proof systems. Unlike bounded-depth Frege,

SA

has polynomial-size proofs of the pigeonhole principle (PHP). A natural question is whether adding PHP to bounded-depth Frege is enough to simulate

SA

. We show that

SA

, with unary integer coefficients, lies strictly between tree-like depth-1

Frege + PHP

and tree-like

Resolution

. We introduce a levelled version of PHP (

L PHP

) and we show that

SA

with integer coefficients lies strictly between tree-like depth-1

Frege + L PHP

and

Resolution

. Analogous results are shown for

NS

using the bijective (i.e. onto and functional) pigeonhole principle and a leveled version of it.

我们比较了Sherali-Adams （SA）和Nullstellensatz （NS）代数证明系统与Frege-style证明系统的强度。与有界深度的Frege不同，SA对鸽子洞原理（PHP）有多项式大小的证明。一个自然的问题是，向有界深度弗雷格添加PHP是否足以模拟SA。我们证明了具有一元整数系数的SA严格位于树状深度-1 Frege+PHP和树状分辨率之间。我们引入了一个级别版本的PHP (LPHP)，并证明了具有整数系数的SA严格位于树状深度-1 Frege+LPHP和分辨率之间。使用双射（即映上和泛函）鸽子洞原理及其水平版本的NS显示了类似的结果。

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引用次数: 0

More about the cofinality and the covering of the ideal of strong measure zero sets 进一步讨论了强测度零集理想的共性和覆盖性

IF 0.6 2区数学 Q2 LOGIC

Annals of Pure and Applied Logic

Pub Date : 2024-11-28 DOI: 10.1016/j.apal.2024.103537

Miguel A. Cardona , Diego A. Mejía

We improve the previous work of Yorioka and the first author about the combinatorics of the ideal

SN

of strong measure zero sets of reals. We refine the notions of dominating systems of the first author and introduce the new combinatorial principle

DS (δ)

that helps to find simple conditions to deduce

d_{κ} \leq cof (SN)

(where

d_{κ}

is the dominating number on

κ^{κ}

). In addition, we find a new upper bound of

cof (SN)

by using products of relational systems and cardinal characteristics associated with Yorioka ideals.

In addition, we dissect and generalize results from Pawlikowski to force upper bounds of the covering of

SN

, particularly for finite support iterations of precaliber posets.

Finally, as applications of our main theorems, we prove consistency results about the cardinal characteristics associated with

SN

and the principle

DS (δ)

. For example, we show that

cov (SN) < non (SN) = c < cof (SN)

holds in Cohen model, and we refine a result (and the proof) of the first author about the consistency of

cov (SN) < non (SN) < cof (SN)

, with

c

in any desired position with respect to

cof (SN)

, and the improvement that

non (SN)

can be singular here.

我们改进了Yorioka和第一作者之前关于实数的强测度零集的理想SN的组合的工作。我们改进了第一作者的支配系统概念，并引入了新的组合原理DS(δ)，该原理有助于找到简单的条件来推导dκ≤cof(SN)（其中dκ是κκ的支配数）。此外，我们利用关系系统的乘积和与Yorioka理想相关的基数特征，找到了cof（SN）的一个新的上界。此外，我们剖析和推广了Pawlikowski的结果，以强制SN覆盖的上界，特别是对于预校准集的有限支持迭代。最后，作为主要定理的应用，我们证明了SN与DS（δ）原理相关的基本特征的一致性结果。例如，我们证明了cov(SN)<non(SN)=c<cof（SN）在Cohen模型中成立，并改进了第一作者关于cov(SN)<non(SN)<cof（SN）在c相对于cof（SN）处于任意位置时的一致性的一个结果（和证明），并改进了non（SN）可以是奇异的。

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引用次数: 0

Positive definability patterns 正可定义性模式

IF 0.6 2区数学 Q2 LOGIC

Annals of Pure and Applied Logic

Pub Date : 2024-11-28 DOI: 10.1016/j.apal.2024.103539

Ori Segel

We reformulate Hrushovski's definability patterns from the setting of first order logic to the setting of positive logic. Given an h-universal theory T we put two structures on the type spaces of models of T in two languages,

L

and

L_{π}

. It turns out that for sufficiently saturated models, the corresponding h-universal theories

T

and

T_{π}

are independent of the model. We show that there is a canonical model

J

T

, and in many interesting cases there is an analogous canonical model

J_{π}

T_{π}

, both of which embed into every type space. We discuss the properties of these canonical models, called cores, and give some concrete examples.

我们将赫鲁晓夫斯基的可定义模式从一阶逻辑的设定重新表述到正逻辑的设定。给定一个h-全称理论T，我们在L和Lπ两种语言的T模型的类型空间上放置了两个结构。结果表明，对于充分饱和的模型，相应的h-泛理论T和Tπ是独立于模型的。我们证明了存在一个正则模型J (T)，并且在许多有趣的情况下存在一个类似的正则模型Jπ (Tπ)，这两个模型都嵌入到每个类型空间中。我们讨论了这些称为核的规范模型的性质，并给出了一些具体的例子。

引用次数: 0

Strong standard completeness theorems for S5-modal Łukasiewicz logics 5-模态Łukasiewicz逻辑的强标准完备性定理

IF 0.6 2区数学 Q2 LOGIC

Annals of Pure and Applied Logic

Pub Date : 2024-11-22 DOI: 10.1016/j.apal.2024.103529

Diego Castaño , José Patricio Díaz Varela , Gabriel Savoy

We study the S5-modal expansion of the Łukasiewicz logic. We exhibit a finitary propositional calculus and show that it is finitely strongly complete with respect to this logic. This propositional calculus is then expanded with an infinitary rule to achieve strong completeness. These results are derived from properties of monadic MV-algebras: functional representations of simple and finitely subdirectly irreducible algebras, as well as the finite embeddability property. We also show similar completeness theorems for the extension of the logic based on models with bounded universe.

我们研究了Łukasiewicz逻辑的s5模态展开。我们展示了一个有限命题演算，并证明了它对于这个逻辑是有限强完备的。然后用一个无限规则展开这个命题演算，以达到强完备性。这些结果来源于一元mv -代数的性质：简单和有限次直接不可约代数的泛函表示，以及有限可嵌入性。对于基于有界宇宙模型的逻辑扩展，我们也给出了类似的完备性定理。

引用次数: 0

Semiconic idempotent logic II: Beth definability and deductive interpolation 半符号幂等逻辑II：贝丝可定义性与演绎插值

IF 0.6 2区数学 Q2 LOGIC

Annals of Pure and Applied Logic

Pub Date : 2024-11-15 DOI: 10.1016/j.apal.2024.103528

Wesley Fussner , Nikolaos Galatos

Semiconic idempotent logic sCI is a common generalization of intuitionistic logic, semilinear idempotent logic sLI, and in particular relevance logic with mingle. We establish the projective Beth definability property and the deductive interpolation property for many extensions of sCI, and identify extensions where these properties fail. We achieve these results by studying the (strong) amalgamation property and the epimorphism-surjectivity property for the corresponding algebraic semantics, viz. semiconic idempotent residuated lattices. Our study is made possible by the structural decomposition of conic idempotent models achieved in the prequel, as well as a detailed analysis of the structure of idempotent residuated chains serving as index sets in this decomposition. Here we study the latter on two levels: as certain enriched Galois connections and as enhanced monoidal preorders. Using this, we show that although conic idempotent residuated lattices do not have the amalgamation property, the natural class of stratified and conjunctive conic idempotent residuated lattices has the strong amalgamation property, and thus has surjective epimorphisms. This extends to the variety generated by stratified and conjunctive conic idempotent residuated lattices, and we establish the (strong) amalgamation and epimorphism-surjectivity properties for several important subvarieties. Using the algebraizability of sCI, this yields the deductive interpolation property and the projective Beth definability property for the corresponding substructural logics extending sCI.

半符号幂等逻辑sCI是对直觉逻辑、半线性幂等逻辑sLI，特别是混合关联逻辑的一般推广。我们建立了sCI的许多扩展的射影Beth可定义性和演绎插值性，并确定了这些性质失效的扩展。我们通过研究相应代数语义，即半符号幂等残格的（强）合并性质和附子满射性质得到了这些结果。我们的研究是通过在前文中实现的二次幂等模型的结构分解，以及在此分解中作为指标集的幂等剩余链的结构的详细分析而得以实现的。在这里，我们从两个层面研究后者：作为某些丰富的伽罗瓦连接和作为增强的一元序。利用这一点，我们证明了虽然二次幂等剩余格不具有合并性质，但自然类的层合二次幂等剩余格具有强合并性质，因而具有满射外胚。这扩展到由层合和合二次幂等剩余格生成的簇，并建立了几个重要子簇的（强）合并和上泛满性。利用sCI的可代数性，给出了相应的扩展sCI的子结构逻辑的演绎插值性质和投影Beth可定义性。

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引用次数: 0

Failure of the Blok–Esakia Theorem in the monadic setting 一元情况下block - esakia定理的失效

IF 0.6 2区数学 Q2 LOGIC

Annals of Pure and Applied Logic

Pub Date : 2024-11-14 DOI: 10.1016/j.apal.2024.103527

G. Bezhanishvili , L. Carai

The Blok–Esakia Theorem establishes that the lattice of superintuitionistic logics is isomorphic to the lattice of extensions of Grzegorczyk's logic. We prove that the Blok–Esakia isomorphism σ does not extend to the fragments of the corresponding predicate logics of already one fixed variable. In other words, we prove that σ is no longer an isomorphism from the lattice of extensions of the monadic intuitionistic logic to the lattice of extensions of the monadic Grzegorczyk logic.

block - esakia定理证明了超直觉逻辑的格与Grzegorczyk逻辑的扩展格是同构的。证明了block - esakia同构σ不能推广到已有一个固定变量的对应谓词逻辑的片段上。也就是说，我们证明了σ不再是一元直觉逻辑的扩展格到一元Grzegorczyk逻辑的扩展格的同构。

引用次数: 0

Universal proof theory: Feasible admissibility in intuitionistic modal logics 通用证明理论：直觉模态逻辑中的可行可接受性

IF 0.6 2区数学 Q2 LOGIC

Annals of Pure and Applied Logic

Pub Date : 2024-10-23 DOI: 10.1016/j.apal.2024.103526

Amirhossein Akbar Tabatabai , Raheleh Jalali

We introduce a general and syntactically defined family of sequent-style calculi over the propositional language with the modalities

{□, ◇}

and its fragments as a formalization for constructively acceptable systems. Calling these calculi constructive, we show that any strong enough constructive sequent calculus, satisfying a mild technical condition, feasibly admits all Visser's rules. This means that there exists a polynomial-time algorithm that, given a proof of the premise of a Visser's rule, provides a proof for its conclusion. As a positive application, we establish the feasible admissibility of Visser's rules in sequent calculi for several intuitionistic modal logics, including

CK

IK

, their extensions by the modal axioms T, B, 4, 5, and the axioms for bounded width and depth and their fragments

{CK}_{□}

, propositional lax logic and

IPC

. On the negative side, we show that if a strong enough intuitionistic modal logic (satisfying a mild technical condition) does not admit at least one of Visser's rules, it cannot have a constructive sequent calculus. Consequently, no intermediate logic other than

IPC

has a constructive sequent calculus.

我们在模态为{□,◇}的命题语言及其片段上引入了一系列通用的、语法上定义的序列式计算，作为构造性可接受系统的形式化。我们称这些计算为构造式计算，并证明任何足够强的构造式顺序微积分，只要满足一个温和的技术条件，就可以接受所有的维塞尔规则。这意味着存在一种多项式时间算法，只要给定一个维塞尔规则的前提证明，就能为其结论提供证明。作为正面应用，我们为几种直觉模态逻辑建立了维塞尔规则在时序计算中的可行可接受性，这些模态逻辑包括 CK、IK 及其模态公理 T、B、4、5 的扩展，以及有界宽度和深度公理及其片段 CK□、命题宽松逻辑和 IPC。从反面来看，我们证明了如果一个足够强的直观模态逻辑（满足一个温和的技术条件）不接受至少一条维塞尔规则，它就不可能有构造时序微积分。因此，除了 IPC 之外，没有任何中间逻辑具有构造时序微积分。

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