Mathematical Structures in Computer Science最新文献

英文中文

On Hofmann–Streicher universes 关于霍夫曼-斯特赖歇尔宇宙

IF 0.5 4区计算机科学 Q4 COMPUTER SCIENCE, THEORY & METHODS

Mathematical Structures in Computer Science

Pub Date : 2024-09-19 DOI: 10.1017/s0960129524000203

Steve Awodey

We take another look at the construction by Hofmann and Streicher of a universe

$(U,{mathcal{E}l})$

for the interpretation of Martin-Löf type theory in a presheaf category

$[{{{mathbb{C}}}^{textrm{op}}},textsf{Set}]$

. It turns out that

$(U,{mathcal{E}l})$

can be described as the nerve of the classifier

$dot{{textsf{Set}}}^{textsf{op}} rightarrow{{textsf{Set}}}^{textsf{op}}$

for discrete fibrations in

$textsf{Cat}$

, where the nerve functor is right adjoint to the so-called “Grothendieck construction” taking a presheaf

$P :{{{mathbb{C}}}^{textrm{op}}}rightarrow{textsf{Set}}$

to its category of elements

$int _{mathbb{C}} P$

. We also consider change of base for such universes, as well as universes of structured families, such as fibrations.

我们再来看一下霍夫曼和施特莱歇尔构建的宇宙 $(U,{mathcal{E}l})$ 对于马丁-洛夫类型理论在预设范畴 $[{{{mathbb{C}}^{textrm{op}}},textsf{Set}]$ 中的解释。事实证明，$(U,{{mathcal{E}l}})$ 可以被描述为分类器 $dot{{textsf{Set}}}^{textsf{op}} 的神经。其中神经函子与所谓的 "格罗thendieck 构造 "是右邻接的，所谓的 "格罗thendieck 构造 "是将预叶 $P :{{{textrm{op}}}^{textsf{Set}}$ 取为其元素类别 $int _{{mathbb{C}} 。P$ .我们还考虑了这类宇宙的基底变化，以及结构族的宇宙，如纤维。

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引用次数: 0

T0-spaces and the lower topology T0 空间和下拓扑

IF 0.5 4区计算机科学 Q4 COMPUTER SCIENCE, THEORY & METHODS

Mathematical Structures in Computer Science

Pub Date : 2024-09-19 DOI: 10.1017/s0960129524000240

Jimmie Lawson, Xiaoquan Xu

The authors’ primary goal in this paper is to enhance the study of $T_0$ topological spaces by using the order of specialization of a $T_0$ -space to introduce the lower topology (with a subbasis of closed sets $mathord{uparrow } x$ ) and studying the interaction of the original topology and the lower topology. Using the lower topology, one can define and study new properties of the original space that provide deeper insight into its structure. One focus of study is the property R, which asserts that if the intersection of a family of finitely generated sets $mathord{uparrow } F$ , $F$ finite, is contained in an open set $U$ , then the same is true for finitely many of the family. We first show that property R is equivalent to several other interesting properties, for example, the property that all closed subsets of the original space are compact in the lower topology. We then find conditions under which these spaces are compact, well-filtered, and coherent, a weaker variant of stably compact spaces. We also investigate what have been called strong $d$ -spaces, develop some of their basic properties, and make connections with the earlier considerations involving spaces satisfying property R. Two key results we obtain are that if a dcpo <jats:inline-formula

作者在本文中的主要目标是通过使用 $T_0$ -space 的特化阶引入下拓扑（具有闭集的子基础 $mathord{uparrow } x$ ），并研究原始拓扑与下拓扑的相互作用，从而加强对 $T_0$ 拓扑空间的研究。利用低级拓扑，我们可以定义和研究原始空间的新属性，从而更深入地了解其结构。研究的一个重点是属性 R，它断言如果有限生成集合 $mathord{uparrow } 的交集为F$ ，$F$ 有限，包含在一个开集 $U$ 中，那么这个族中的有限个集也是如此。我们首先证明性质 R 等价于其他几个有趣的性质，例如，原始空间的所有封闭子集在下拓扑中都是紧凑的性质。然后，我们找到了这些空间紧凑、过滤良好且连贯的条件，这是稳定紧凑空间的较弱变体。我们还研究了所谓的强 $d$ -空间，发展了它们的一些基本性质，并把它们与前面涉及满足性质 R 的空间的考虑联系起来。我们得到的两个关键结果是：如果具有斯科特拓扑的 dcpo $P$ 是一个强 $d$ -空间，那么它就是过滤良好的；如果另外的乘积 $Ptimes P$ 的斯科特拓扑是各因子的斯科特拓扑的乘积，那么 $P$ 的斯科特空间就是清醒的。我们还展示了这项工作与德格鲁特对偶性的联系。

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引用次数: 0

GADTs are not (Even partial) functors GADT 不是（甚至不是部分）函数

IF 0.5 4区计算机科学 Q4 COMPUTER SCIENCE, THEORY & METHODS

Mathematical Structures in Computer Science

Pub Date : 2024-08-27 DOI: 10.1017/s0960129524000161

Pierre Cagne, Enrico Ghiorzi, Patricia Johann

Generalized Algebraic Data Types (GADTs) are a syntactic generalization of the usual algebraic data types (ADTs), such as lists, trees, etc. ADTs’ standard initial algebra semantics (IAS) in the category

$mathit{Set}$

of sets justify critical syntactic constructs – such as recursion, pattern matching, and fold – for programming with them. In this paper, we show that semantics for GADTs that specialize to the IAS for ADTs are necessarily unsatisfactory. First, we show that the functorial nature of such semantics for GADTs in

$mathit{Set}$

introduces ghost elements, i.e., elements not writable in syntax. Next, we show how such ghost elements break parametricity. We observe that the situation for GADTs contrasts dramatically with that for ADTs, whose IAS coincides with the parametric model constructed via their Church encodings in System F. Our analysis reveals that the fundamental obstacle to giving a functorial IAS for GADTs is the inherently partial nature of their map functions. We show that this obstacle cannot be overcome by replacing

$mathit{Set}$

with other categories that account for this partiality.

广义代数数据类型（GADT）是通常代数数据类型（ADT）（如列表、树等）的语法广义化。ADTs 在集合类别 $mathit{Set}$ 中的标准初始代数语义（IAS）证明了使用它们编程的关键语法结构--如递归、模式匹配和折叠--的合理性。在本文中，我们证明了专门针对 ADT 的 IAS 的 GADT 语义必然不能令人满意。首先，我们证明了$mathit{Set}$中GADT的这种语义的函数性质会引入幽灵元素，即语法中不可写的元素。接下来，我们将展示这些幽灵元素是如何破坏参数性的。我们观察到，GADTs 的情况与 ADTs 形成了鲜明对比，后者的 IAS 与通过系统 F 中的 Church 编码构建的参数模型相吻合。我们的分析揭示出，为 GADTs 提供函数式 IAS 的根本障碍在于其映射函数的固有部分性。我们的分析表明，将 $mathit{Set}$ 替换为其他考虑到这种片面性的范畴是无法克服这一障碍的。

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引用次数: 0

A linear linear lambda-calculus 线性λ演算法

IF 0.5 4区计算机科学 Q2 Mathematics

Mathematical Structures in Computer Science

Pub Date : 2024-05-31 DOI: 10.1017/s0960129524000197

Alejandro Díaz-Caro, Gilles Dowek

We present a linearity theorem for a proof language of intuitionistic multiplicative additive linear logic, incorporating addition and scalar multiplication. The proofs in this language are linear in the algebraic sense. This work is part of a broader research program aiming to define a logic with a proof language that forms a quantum programming language.

我们提出了包含加法和标量乘法的直观乘法加法线性逻辑证明语言的线性定理。这种语言的证明在代数意义上是线性的。这项工作是更广泛研究计划的一部分，该计划旨在定义一种具有证明语言的逻辑，从而形成一种量子编程语言。

引用次数: 0

Countability constraints in order-theoretic approaches to computability 可计算性秩序论方法中的可数性约束

IF 0.5 4区计算机科学 Q2 Mathematics

Mathematical Structures in Computer Science

Pub Date : 2024-05-30 DOI: 10.1017/s0960129524000173

Pedro Hack, Daniel A. Braun, Sebastian Gottwald

Computability on uncountable sets has no standard formalization, unlike that on countable sets, which is given by Turing machines. Some of the approaches to define computability in these sets rely on order-theoretic structures to translate such notions from Turing machines to uncountable spaces. Since these machines are used as a baseline for computability in these approaches, countability restrictions on the ordered structures are fundamental. Here, we show several relations between the usual countability restrictions in order-theoretic theories of computability and some more common order-theoretic countability constraints, like order density properties and functional characterizations of the order structure in terms of multi-utilities. As a result, we show how computability can be introduced in some order structures via countability order density and multi-utility constraints.

不可数集上的可计算性没有标准的形式化，这与图灵机给出的可数集上的可计算性不同。在这些集合中定义可计算性的一些方法依赖于秩序论结构，将这些概念从图灵机转换到不可数空间。由于在这些方法中，这些机器被用作可计算性的基线，因此对有序结构的可计算性限制是至关重要的。在这里，我们展示了可计算性的有序理论中通常的可计算性限制与一些更常见的有序理论可计算性约束之间的关系，如有序密度特性和有序结构在多效用方面的函数特征。因此，我们展示了如何通过可计算性阶密度和多效用约束在某些阶结构中引入可计算性。

引用次数: 0

Logical characterizations of algebraic circuit classes over integral domains 积分域上代数电路类的逻辑特征

IF 0.5 4区计算机科学 Q2 Mathematics

Mathematical Structures in Computer Science

Pub Date : 2024-05-13 DOI: 10.1017/s0960129524000136

Timon Barlag, Florian Chudigiewitsch, Sabrina A. Gaube

We present an adapted construction of algebraic circuits over the reals introduced by Cucker and Meer to arbitrary infinite integral domains and generalize the

$mathrm{AC}_{mathbb{R}}$

and

$mathrm{NC}_{mathbb{R}}^{}$

classes for this setting. We give a theorem in the style of Immerman’s theorem which shows that for these adapted formalisms, sets decided by circuits of constant depth and polynomial size are the same as sets definable by a suitable adaptation of first-order logic. Additionally, we discuss a generalization of the guarded predicative logic by Durand, Haak and Vollmer, and we show characterizations for the

$mathrm{AC}_{R}$

and

$mathrm{NC}_R^{}$

hierarchy. Those generalizations apply to the Boolean

$mathrm{AC}$

and

$mathrm{NC}$

hierarchies as well. Furthermore, we introduce a formalism to be able to compare some of the aforementioned complexity classes with different underlying integral domains.

我们介绍了卡克和米尔引入的对任意无限积分域的实数代数回路的改编构造，并针对这种情形推广了 $mathrm{AC}_{mathbb{R}}$ 和 $mathrm{NC}_{mathbb{R}}^{}$ 类。我们给出了一个类似于伊默曼定理的定理，它表明对于这些经过调整的形式主义，由恒定深度和多项式大小的电路决定的集合与由一阶逻辑的适当调整定义的集合是相同的。此外，我们还讨论了杜兰、哈克和沃尔默对有保护谓词逻辑的广义化，并展示了 $mathrm{AC}_{R}$ 和 $mathrm{NC}_R^{}$ 层次的特征。这些概括也适用于布尔 $mathrm{AC}$ 和 $mathrm{NC}$ 层次。此外，我们还引入了一种形式主义，以便能够比较上述一些具有不同底层积分域的复杂性等级。

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引用次数: 0

On planarity of graphs in homotopy type theory 论同调类型理论中图形的平面性

IF 0.5 4区计算机科学 Q2 Mathematics

Mathematical Structures in Computer Science

Pub Date : 2024-05-08 DOI: 10.1017/s0960129524000100

Jonathan Prieto-Cubides, Håkon Robbestad Gylterud

In this paper, we present a constructive and proof-relevant development of graph theory, including the notion of maps, their faces and maps of graphs embedded in the sphere, in homotopy type theory (HoTT). This allows us to provide an elementary characterisation of planarity for locally directed finite and connected multigraphs that takes inspiration from topological graph theory, particularly from combinatorial embeddings of graphs into surfaces. A graph is planar if it has a map and an outer face with which any walk in the embedded graph is walk-homotopic to another. A result is that this type of planar maps forms a homotopy set for a graph. As a way to construct examples of planar graphs inductively, extensions of planar maps are introduced. We formalise the essential parts of this work in the proof assistant Agda with support for HoTT.

在本文中，我们介绍了同调类型理论（HoTT）中图论的构造性和证明相关性发展，包括图的概念、图的面和嵌入球面的图的映射。这使我们能够为局部有向有限多图和连通多图的平面性提供一个基本特征，这个特征从拓扑图理论，特别是从组合图嵌入曲面中得到启发。如果一个图有一个映射和一个外表面，嵌入图中的任何行走都与另一个行走同构，那么这个图就是平面图。结果是，这类平面映射构成了图的同构集。作为归纳构建平面图实例的一种方法，我们引入了平面映射的扩展。我们在支持 HoTT 的证明助手 Agda 中形式化了这项工作的重要部分。

引用次数: 0

Special Issue on Theory and Applications of Models of Computation TAMC 2022 计算模型的理论与应用特刊 TAMC 2022

IF 0.5 4区计算机科学 Q2 Mathematics

Mathematical Structures in Computer Science

Pub Date : 2024-05-07 DOI: 10.1017/s096012952400015x

Dingzhu Du, Chenchen Wu, Ruiqi Yang

引用次数: 0

You can only be lucky once: optimal gossip for epistemic goals 你只能幸运一次：认识论目标的最佳流言

IF 0.5 4区计算机科学 Q2 Mathematics

Mathematical Structures in Computer Science

Pub Date : 2024-04-19 DOI: 10.1017/s0960129524000082

Hans van Ditmarsch, Malvin Gattinger

It is known that without synchronization via a global clock one cannot obtain common knowledge by communication. Moreover, it is folklore that without communicating higher-level information one cannot obtain arbitrary higher-order shared knowledge. Here, we make this result precise in the setting of gossip where agents make one-to-one telephone calls to share secrets: we prove that “everyone knows that everyone knows that everyone knows all secrets” is unsatisfiable in a logic of knowledge for gossiping. We also prove that, given n agents,

$2n-3$

calls are optimal to reach “someone knows that everyone knows all secrets” and that

$n - 2 + binom{n}{2}$

calls are optimal to reach “everyone knows that everyone knows all secrets.”

众所周知，没有全局时钟的同步，就无法通过通信获得共同知识。此外，众所周知，如果不交流高层信息，就无法获得任意的高阶共享知识。在这里，我们将这一结果精确地应用于代理人通过一对一通话来分享秘密的闲聊中：我们证明 "每个人都知道每个人都知道每个人都知道所有秘密 "在闲聊的知识逻辑中是不可满足的。我们还证明，给定 n 个代理人，要达到 "有人知道每个人都知道所有秘密"，2n-3$ 的通话是最优的；要达到 "每个人都知道每个人都知道所有秘密"，$n - 2 + binom{n}{2}$ 的通话是最优的。

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引用次数: 0

Abstract cyclic proofs 抽象循环证明

IF 0.5 4区计算机科学 Q2 Mathematics

Mathematical Structures in Computer Science

Pub Date : 2024-04-19 DOI: 10.1017/s0960129524000070

Bahareh Afshari, Dominik Wehr

Cyclic proof systems permit derivations that are finite graphs in contrast to conventional derivation trees. The soundness of such proofs is ensured by imposing a soundness condition on derivations. The most common such condition is the global trace condition (GTC), a condition on the infinite paths through the derivation graph. To give a uniform treatment of such cyclic proof systems, Brotherston proposed an abstract notion of trace. We extend Brotherston’s approach into a category theoretical rendition of cyclic derivations, advancing the framework in two ways: first, we introduce activation algebras which allow for a more natural formalisation of trace conditions in extant cyclic proof systems. Second, accounting for the composition of trace information allows us to derive novel results about cyclic proofs, such as introducing a Ramsey-style trace condition. Furthermore, we connect our notion of trace to automata theory and prove that verifying the GTC for abstract cyclic proofs with certain trace conditions is PSPACE-complete.

与传统的推导树不同，循环证明系统允许有限图的推导。这种证明的健全性是通过对推导施加健全性条件来保证的。最常见的此类条件是全局踪迹条件（GTC），这是一个关于通过推导图的无限路径的条件。为了统一处理这种循环证明系统，Brotherston 提出了一个抽象的轨迹概念。我们将兄弟斯顿的方法扩展到循环推导的范畴论演绎中，并从两个方面推进了这一框架：首先，我们引入了激活代数，使现有循环证明系统中的迹条件形式化得更自然。其次，考虑到踪迹信息的构成，我们可以推导出关于循环证明的新结果，比如引入拉姆齐式的踪迹条件。此外，我们将我们的轨迹概念与自动机理论联系起来，并证明验证具有特定轨迹条件的抽象循环证明的 GTC 是 PSPACE-complete的。

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