Mathematical Structures in Computer Science最新文献

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Approximation Algorithm and FPT Algorithm for Connected-k-Subgraph Cover on Minor-Free Graphs 无小图上连通-k-子图覆盖的逼近算法和 FPT 算法

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

Mathematical Structures in Computer Science

Pub Date : 2024-01-10 DOI: 10.1017/s0960129523000439

Pengcheng Liu, Zhao Zhang, Yingli Ran, Xiaohui Huang

Given a graph G, the minimum Connected-k-Subgraph Cover problem (MinCkSC) is to find a minimum vertex subset C of G such that every connected subgraph of G on k vertices has at least one vertex in C. If furthermore the subgraph of G induced by C is connected, then the problem is denoted as MinCkSC$_{con}$. In this paper, we first present a PTAS for MinCkSC on an H-minor-free graph, where H is a graph with a constant number of vertices. Then, we design an $O((omega+1)(2(k-1)(omega+2))^{3omega+3})|V|$-time FPT algorithm for MinCkSC$_{con}$ on a graph with treewidth $omega$, based on which we further design an $O(2^{O(sqrt{t}log t)}|V|^{O(1)})$ time subexponential FPT algorithm for MinCkSC$_{con}$ on an H-minor-free graph, where t is an upper bound of solution size.

给定一个图 G，最小连通子图覆盖问题（MinCkSC）就是找到 G 的最小顶点子集 C，使得 G 的 k 个顶点上的每个连通子图都至少有一个顶点在 C 中。在本文中，我们首先提出了在 H-minor-free 图上的 MinCkSC 的 PTAS，其中 H 是具有恒定顶点数的图。然后，我们为具有树宽 $omega$ 的图上的 MinCkSC$_{con}$ 设计了一个 $O((omega+1)(2(k-1)(omega+2))^{3omega+3})|V|$ 时的 FPT 算法、在此基础上，我们进一步为无 H 小数图上的 MinCkSC$_{con}$ 设计了一种 $O(2^{O(sqrt{t}log t)}|V|^{O(1)})$ 时间的亚指数 FPT 算法，其中 t 是解大小的上限。

引用次数: 0

The order-K-ification monads 阶 K 化单子

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

Mathematical Structures in Computer Science

Pub Date : 2023-12-21 DOI: 10.1017/s0960129523000403

Huijun Hou, Hualin Miao, Qingguo Li

Monads prove to be useful mathematical tools in theoretical computer science, notably in denoting different effects of programming languages. In this paper, we investigate a type of monads which arise naturally from Keimel and Lawson’s <img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231220114246647-0443:S0960129523000403:S0960129523000403_inline1.png">$mathbf{K}$</img>-ification.A subcategory of <img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231220114246647-0443:S0960129523000403:S0960129523000403_inline2.png">$mathbf{TOP}_{mathbf{0}}$</img> is called of type <img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231220114246647-0443:S0960129523000403:S0960129523000403_inline3.png">$mathrm{K}^{*}$</img> if it consists of monotone convergence spaces and is of type <img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231220114246647-0443:S0960129523000403:S0960129523000403_inline4.png">$mathrm K$</img> in the sense of Keimel and Lawson. Each such category induces a canonical monad <img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231220114246647-0443:S0960129523000403:S0960129523000403_inline5.png">$mathcal K$</img> on the category <img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231220114246647-0443:S0960129523000403:S0960129523000403_inline6.png">$mathbf{DCPO}$</img> of dcpos and Scott-continuous maps, which is called the order-<img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231220114246647-0443:S0960129523000403:S0960129523000403_inline7.png">$mathbf{K}$</img>-ification monad in this paper. First, for each category of type <img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231220114246647-0443:S0960129523000403:S0960129523000403_inline8.png">$mathrm{K}^{*}$</img>, we characterize the algebras of the corresponding monad <img

在理论计算机科学中，单子被证明是有用的数学工具，特别是在表示编程语言的不同效果时。在本文中，我们将研究一种由 Keimel 和 Lawson 的 $mathbf{K}$-ification 自然产生的单子类型。如果 $mathbf{TOP}_{mathbf{0}}$ 的子类由单调收敛空间组成，并且在 Keimel 和 Lawson 的意义上属于 $mathrm K$ 类型，那么这个子类就叫做 $mathrm{K}^{*}$ 类型。每个这样的范畴都会在dcpos和斯科特连续映射的范畴$mathbf{DCPO}$上诱导出一个典型的单体$mathcal K$，本文称其为order-$mathbf{K}$-ification单体。首先，对于每个$mathrm{K}^{*}$类型的范畴，我们将相应单元$mathcal K$的代数代数同态描述为k-完备的poset，将代数同态描述为k-连续映射，由此我们得到order-$mathbf{K}$-ification单元给出了在poset和Scott-连续映射的范畴$mathbf{POS}_{mathbf{d}$上的自由k-完备poset构造。此外，我们还证明了所有 k-完备正集和斯科特连续映射构成了一个笛卡尔闭合范畴。此外，我们还考虑了阶-K-化一元体的强大性，并以每个阶-K-化一元体总是交换的这一事实得出结论。

引用次数: 0

Visibility and exploitation in social networks 社交网络中的可见度和利用

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

Mathematical Structures in Computer Science

Pub Date : 2023-12-19 DOI: 10.1017/s0960129523000397

Rustam Galimullin, Mina Young Pedersen

Social media is not a neutral channel. How visible information posted online is depends on many factors such as the network structure, the emotional volatility of the content, and the design of the social media platform. In this paper, we use formal methods to study the visibility of agents and information in a social network, as well as how vulnerable the network is to exploitation. We introduce a modal logic to reason about a social network of agents that can follow each other, post, and share information. We show that by imposing some simple rules on the system, a potentially malicious agent can take advantage of the network construction to post an unpopular opinion that may reach many agents. The network is presented both in static and dynamic forms. We prove completeness, expressivity, and model checking problem complexity results for the corresponding logical systems.

社交媒体不是一个中立的渠道。网上发布信息的可见度取决于很多因素，如网络结构、内容的情绪波动性以及社交媒体平台的设计。在本文中，我们使用形式化的方法来研究社交网络中代理和信息的可见性，以及网络被利用的脆弱性。我们引入了一种模态逻辑来推理由代理人组成的社交网络，这些代理人可以互相关注、发布和分享信息。我们的研究表明，通过对系统施加一些简单的规则，潜在的恶意代理可以利用网络的构造来发布不受欢迎的观点，而这些观点可能会传播给许多代理。该网络以静态和动态两种形式呈现。我们证明了相应逻辑系统的完备性、可表达性和模型检查问题复杂性结果。

引用次数: 0

Fixed point logics and definable topological properties 定点逻辑和可定义拓扑特性

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

Mathematical Structures in Computer Science

Pub Date : 2023-12-13 DOI: 10.1017/s0960129523000385

David Fernández-Duque, Quentin Gougeon

Modal logic enjoys topological semantics that may be traced back to McKinsey and Tarski, and the classification of topological spaces via modal axioms is a lively area of research. In the past two decades, there has been interest in extending topological modal logic to the language of the mu-calculus, but previously no class of topological spaces was known to be mu-calculus definable that was not already modally definable. In this paper, we show that the full mu-calculus is indeed more expressive than standard modal logic, in the sense that there are classes of topological spaces (and weakly transitive Kripke frames), which are mu-definable but not modally definable. The classes we exhibit satisfy a modally definable property outside of their perfect core, and thus we dub them imperfect spaces. We show that the mu-calculus is sound and complete for these classes. Our examples are minimal in the sense that they use a single instance of a greatest fixed point, and we show that least fixed points alone do not suffice to define any class of spaces that is not already modally definable.

模态逻辑的拓扑语义可以追溯到麦肯锡和塔尔斯基，通过模态公理对拓扑空间进行分类是一个活跃的研究领域。在过去的二十年里，人们对将拓扑模态逻辑扩展到缪算子语言产生了兴趣，但在此之前，人们还不知道有哪一类拓扑空间是可以用缪算子定义的，而不是已经可以用模态定义的。在本文中，我们证明了完整的mu-calculus确实比标准模态逻辑更具表现力，因为有一些拓扑空间类（以及弱传递克里普克帧）是可以用mu-calculus定义的，但却不能用模态定义。我们展示的这些类在其完美核心之外满足模态可定义的性质，因此我们称它们为不完美空间。我们证明，对于这些类来说，μ算式是健全而完整的。我们的例子是最小的，因为它们只使用了最大定点的一个实例，而且我们证明，仅凭最小定点不足以定义任何一类尚未模态定义的空间。

引用次数: 0

A point-free perspective on lax extensions and predicate liftings 关于松弛扩展和谓词提升的无点视角

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

Mathematical Structures in Computer Science

Pub Date : 2023-12-01 DOI: 10.1017/s096012952300035x

Sergey Goncharov, Dirk Hofmann, Pedro Nora, Lutz Schröder, Paul Wild

Lax extensions of set functors play a key role in various areas, including topology, concurrent systems, and modal logic, while predicate liftings provide a generic semantics of modal operators. We take a fresh look at the connection between lax extensions and predicate liftings from the point of view of quantale-enriched relations. Using this perspective, we show in particular that various fundamental concepts and results arise naturally and their proofs become very elementary. Ultimately, we prove that every lax extension is induced by a class of predicate liftings; we discuss several implications of this result.

集合函子的松散扩展在各个领域发挥关键作用，包括拓扑、并发系统和模态逻辑，而谓词提升提供模态操作符的通用语义。我们从富量子关系的角度重新审视松弛扩展和谓词提升之间的联系。从这个角度来看，我们特别指出，各种基本概念和结果是自然产生的，它们的证明变得非常初级。最后，我们证明了每一个松弛扩展都是由一类谓词提升引起的;我们讨论了这一结果的几个含义。

引用次数: 2

Learning quantum finite automata with queries 用查询学习量子有限自动机

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

Mathematical Structures in Computer Science

Pub Date : 2023-11-30 DOI: 10.1017/s0960129523000373

Daowen Qiu

Learning finite automata (termed as model learning) has become an important field in machine learning and has been useful realistic applications. Quantum finite automata (QFA) are simple models of quantum computers with finite memory. Due to their simplicity, QFA have well physical realizability, but one-way QFA still have essential advantages over classical finite automata with regard to state complexity (two-way QFA are more powerful than classical finite automata in computation ability as well). As a different problem in quantum learning theory and quantum machine learning, in this paper, our purpose is to initiate the study of learning QFA with queries (naturally it may be termed as quantum model learning), and the main results are regarding learning two basic one-way QFA (1QFA): (1) we propose a learning algorithm for measure-once 1QFA (MO-1QFA) with query complexity of polynomial time and (2) we propose a learning algorithm for measure-many 1QFA (MM-1QFA) with query complexity of polynomial time, as well.

有限自动机学习(也称为模型学习)已成为机器学习的一个重要领域，并在现实中得到了广泛的应用。量子有限自动机(QFA)是具有有限内存的量子计算机的简单模型。由于其简单性，QFA具有良好的物理可实现性，但单向QFA在状态复杂性方面仍比经典有限自动机具有本质优势(双向QFA在计算能力方面也比经典有限自动机更强大)。作为量子学习理论和量子机器学习中的一个不同的问题，在本文中，我们的目的是启动使用查询学习QFA的研究(自然它可以被称为量子模型学习)，主要结果是关于学习两个基本的单向QFA (1QFA):(1)提出了一种查询复杂度为多项式时间的测度一次1QFA (MO-1QFA)学习算法;(2)提出了查询复杂度为多项式时间的测度多1QFA (MM-1QFA)学习算法。

引用次数: 0

A concrete model for a typed linear algebraic lambda calculus 一类线性代数λ演算的具体模型

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

Mathematical Structures in Computer Science

Pub Date : 2023-11-21 DOI: 10.1017/s0960129523000361

Alejandro Díaz-Caro, Octavio Malherbe

We give an adequate, concrete, categorical-based model for Lambda-

${mathcal S}$

, which is a typed version of a linear-algebraic lambda calculus, extended with measurements. Lambda-

${mathcal S}$

is an extension to first-order lambda calculus unifying two approaches of non-cloning in quantum lambda-calculi: to forbid duplication of variables and to consider all lambda-terms as algebraic linear functions. The type system of Lambda-

${mathcal S}$

has a superposition constructor S such that a type A is considered as the base of a vector space, while SA is its span. Our model considers S as the composition of two functors in an adjunction relation between the category of sets and the category of vector spaces over

$mathbb C$

. The right adjoint is a forgetful functor U, which is hidden in the language, and plays a central role in the computational reasoning.

我们给出了Lambda- ${mathcal S}$的一个充分的，具体的，基于分类的模型，它是线性代数Lambda演算的一个类型版本，扩展了测量。λ - ${mathcal S}$是对一阶λ演算的扩展，统一了量子λ演算中的两种非克隆方法:禁止变量重复和将所有λ项视为代数线性函数。λ - ${mathcal S}$的类型系统有一个叠加构造函数S，使得类型a被认为是向量空间的基，而SA是它的张成空间。我们的模型认为S是在$mathbb C$上集合范畴与向量空间范畴之间的附加关系中的两个函子的复合。右伴随是一个遗忘的函子U，它隐藏在语言中，在计算推理中起着核心作用。

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引用次数: 8

Material dialogues for first-order logic in constructive type theory: extended version 建构型理论中一阶逻辑的材料对话:扩展版

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

Mathematical Structures in Computer Science

Pub Date : 2023-11-03 DOI: 10.1017/s0960129523000348

Dominik Wehr, Dominik Kirst

Abstract Dialogues are turn-taking games which model debates about the satisfaction of logical formulas. A novel variant played over first-order structures gives rise to a notion of first-order satisfaction. We study the induced notion of validity for classical and intuitionistic first-order logic in the constructive setting of the calculus of inductive constructions. We prove that such material dialogue semantics for classical first-order logic admits constructive soundness and completeness proofs, setting it apart from standard model-theoretic semantics of first-order logic. Furthermore, we prove that completeness with regard to intuitionistic material dialogues fails in both constructive and classical settings. As an alternative, we propose material dialogues played over Kripke structures. These Kripke material dialogues exhibit constructive completeness when restricting to the negative fragment. The results concerning classical material dialogues have been mechanized using the Coq interactive theorem prover.

抽象对话是一种回合制游戏，它模拟了关于逻辑公式满足度的辩论。一阶结构上的一种新变体产生了一阶满足的概念。在归纳构造演算的构造背景下，研究了经典和直觉一阶逻辑的归纳有效性概念。我们证明了经典一阶逻辑的这种物质对话语义具有构造健全性和完备性证明，将其与一阶逻辑的标准模型论语义区分开来。此外，我们证明了关于直觉主义材料对话的完备性在建设性和古典背景下都失败了。作为替代方案，我们建议在Kripke结构上播放材料对话。这些克里普克材料对话在限制于否定片段时表现出建设性的完整性。利用Coq相互作用定理证明，对经典材料对话的结果进行了机械化处理。

引用次数: 0

Dilations and information flow axioms in categorical probability 范畴概率中的扩张和信息流公理

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

Mathematical Structures in Computer Science

Pub Date : 2023-10-25 DOI: 10.1017/s0960129523000324

Tobias Fritz, Tomáš Gonda, Nicholas Gauguin Houghton-Larsen, Antonio Lorenzin, Paolo Perrone, Dario Stein

Abstract We study the positivity and causality axioms for Markov categories as properties of dilations and information flow and also develop variations thereof for arbitrary semicartesian monoidal categories. These help us show that being a positive Markov category is merely an additional property of a symmetric monoidal category (rather than extra structure). We also characterize the positivity of representable Markov categories and prove that causality implies positivity , but not conversely. Finally, we note that positivity fails for quasi-Borel spaces and interpret this failure as a privacy property of probabilistic name generation.

摘要研究了马尔可夫范畴的正性和因果性公理作为膨胀和信息流的性质，并给出了任意半笛卡尔一元范畴的正性和因果性公理的变化。这些帮助我们证明了正马尔可夫范畴仅仅是对称一元范畴的附加性质(而不是额外的结构)。我们还描述了可表征马尔可夫范畴的正性，并证明因果关系意味着正性，而不是相反。最后，我们注意到准borel空间的正性失败，并将这种失败解释为概率名称生成的隐私属性。

引用次数: 6

Bicategorical type theory: semantics and syntax 双范畴类型理论:语义和句法

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

Mathematical Structures in Computer Science

Pub Date : 2023-10-17 DOI: 10.1017/s0960129523000312

Benedikt Ahrens, Paige Randall North, Niels van der Weide

Abstract We develop semantics and syntax for bicategorical type theory. Bicategorical type theory features contexts, types, terms, and directed reductions between terms. This type theory is naturally interpreted in a class of structured bicategories. We start by developing the semantics, in the form of comprehension bicategories . Examples of comprehension bicategories are plentiful; we study both specific examples as well as classes of examples constructed from other data. From the notion of comprehension bicategory, we extract the syntax of bicategorical type theory, that is, judgment forms and structural inference rules. We prove soundness of the rules by giving an interpretation in any comprehension bicategory. The semantic aspects of our work are fully checked in the Coq proof assistant, based on the UniMath library.

摘要:我们发展了双范畴类型理论的语义和句法。双范畴类型理论的特点是语境、类型、术语和术语之间的直接约简。这种类型理论自然地被解释为一类结构化的双范畴。我们从发展语义开始，以理解双范畴的形式。理解分类的例子很多;我们既研究具体的示例，也研究从其他数据构造的示例类。从理解双范畴的概念出发，我们提取了双范畴类型论的句法，即判断形式和结构推理规则。我们通过在任何理解范畴中给出解释来证明规则的合理性。我们工作的语义方面在基于UniMath库的Coq证明助手中进行了全面检查。

引用次数: 3

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Mathematical Structures in Computer Science

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