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What should a generic object be? 泛型对象应该是什么?
IF 0.5 4区 计算机科学 Q2 Mathematics Pub Date : 2022-10-09 DOI: 10.1017/S0960129523000117
Jonathan Sterling
Abstract Jacobs has proposed definitions for (weak, strong, split) generic objects for a fibered category; building on his definition of (split) generic objects, Jacobs develops a menagerie of important fibrational structures with applications to categorical logic and computer science, including higher order fibrations, polymorphic fibrations, $lambda2$ -fibrations, triposes, and others. We observe that a split generic object need not in particular be a generic object under the given definitions, and that the definitions of polymorphic fibrations, triposes, etc. are strict enough to rule out some fundamental examples: for instance, the fibered preorder induced by a partial combinatory algebra in realizability is not a tripos in this sense. We propose a new alignment of terminology that emphasizes the forms of generic object appearing most commonly in nature, i.e. in the study of internal categories, triposes, and the denotational semantics of polymorphism. In addition, we propose a new class of acyclic generic objects inspired by recent developments in higher category theory and the semantics of homotopy type theory, generalizing the realignment property of universes to the setting of an arbitrary fibration.
摘要Jacobs提出了纤维范畴的(弱、强、分裂)泛型对象的定义;在他对(分裂)泛型对象的定义的基础上,Jacobs开发了一系列重要的fibrational结构,并将其应用于分类逻辑和计算机科学,包括高阶fibrations、多态fibrations,$lambda2$-fibrations和triposes等。我们观察到,在给定的定义下,分裂的泛型对象不一定是泛型对象,多态fibrations、tripoles等的定义足够严格,可以排除一些基本的例子:例如,在可实现性中由部分组合代数诱导的fibed预序在这个意义上不是tripols。我们提出了一种新的术语组合,强调在自然界中最常见的一般对象的形式,即在研究内部类别、三元组和多态性的指称语义时。此外,受高等范畴理论和同源类型理论语义的最新发展的启发,我们提出了一类新的非循环泛型对象,将普遍性的重排性质推广到任意fibration的设置。
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引用次数: 0
Weighted synchronous automata 加权同步自动机
IF 0.5 4区 计算机科学 Q2 Mathematics Pub Date : 2022-10-01 DOI: 10.1017/S0960129522000421
Leandro Gomes, A. Madeira, L. Barbosa
Abstract This paper introduces a class of automata and associated languages, suitable to model a computational paradigm of fuzzy systems, in which both vagueness and simultaneity are taken as first-class citizens. This requires a weighted semantics for transitions and a precise notion of a synchronous product to enforce the simultaneous occurrence of actions. The usual relationships between automata and languages are revisited in this setting, including a specific Kleene theorem.
摘要本文介绍了一类适用于模糊系统计算范式建模的自动机及其相关语言,其中模糊性和同时性均为一等公民。这需要转换的加权语义和同步产品的精确概念,以强制操作同时发生。自动机和语言之间的通常关系在此设置中被重新审视,包括一个特定的Kleene定理。
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引用次数: 0
A linear logic framework for multimodal logics 多模态逻辑的线性逻辑框架
IF 0.5 4区 计算机科学 Q2 Mathematics Pub Date : 2022-10-01 DOI: 10.1017/S0960129522000366
Bruno Xavier, C. Olarte, Elaine Pimentel
Abstract One of the most fundamental properties of a proof system is analyticity, expressing the fact that a proof of a given formula F only uses subformulas of F. In sequent calculus, this property is usually proved by showing that the $mathsf{cut}$ rule is admissible, i.e., the introduction of the auxiliary lemma H in the reasoning “if H follows from G and F follows from H, then F follows from G” can be eliminated. The proof of cut admissibility is usually a tedious, error-prone process through several proof transformations, thus requiring the assistance of (semi-)automatic procedures. In a previous work by Miller and Pimentel, linear logic ( $mathsf{LL}$ ) was used as a logical framework for establishing sufficient conditions for cut admissibility of object logical systems (OL). The OL’s inference rules are specified as an $mathsf{LL}$ theory and an easy-to-verify criterion sufficed to establish the cut-admissibility theorem for the OL at hand. However, there are many logical systems that cannot be adequately encoded in $mathsf{LL}$ , the most symptomatic cases being sequent systems for modal logics. In this paper, we use a linear-nested sequent ( $mathsf{LNS}$ ) presentation of $mathsf{MMLL}$ (a variant of LL with subexponentials), and show that it is possible to establish a cut-admissibility criterion for $mathsf{LNS}$ systems for (classical or substructural) multimodal logics. We show that the same approach is suitable for handling the $mathsf{LNS}$ system for intuitionistic logic.
证明系统的一个最基本的性质是解析性,它表示一个给定公式F的证明只使用F的子公式。在序演学中,通常通过证明$mathsf{cut}$规则是可接受的来证明这一性质,即在“如果H从G引出,F从H引出,那么F从G引出”推理中引入辅助引理H是可以消除的。切割可采性的证明通常是一个冗长且容易出错的过程,需要经过多次证明转换,因此需要(半)自动化程序的帮助。在Miller和Pimentel之前的工作中,线性逻辑($mathsf{LL}$)被用作建立对象逻辑系统(OL)切容许的充分条件的逻辑框架。OL的推理规则被指定为$mathsf{LL}$理论和一个易于验证的准则,足以建立手头OL的可割性定理。然而,有许多逻辑系统不能在$mathsf{LL}$中充分编码,最典型的情况是模态逻辑的顺序系统。在本文中,我们使用$mathsf{MMLL}$的线性嵌套序列($mathsf{LNS}$)表示,并证明了对于(经典或子结构)多模态逻辑,$mathsf{LNS}$系统可以建立一个切容许准则。我们证明了同样的方法也适用于处理直觉逻辑的$mathsf{LNS}$系统。
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引用次数: 1
Preface to Special Issue: LSFA 2019 and 2020 特刊前言:LSFA 2019和2020
IF 0.5 4区 计算机科学 Q2 Mathematics Pub Date : 2022-10-01 DOI: 10.1017/S0960129523000075
A. Felty, Giselle Reis
This special issue collects selected articles from the 14th and 15th editions of the Workshop on Logical and Semantic Frameworks, with Applications (LSFA 2019 and 2020). LSFA 2019 was held on 25–26 August 2019 in Natal as a satellite event of the 27th International Conference on Automated Deduction (CADE 2019). LSFA 2020 was held on 27–28 August as an online workshop, with organizers from Salvador.
本特刊收集了第14和第15届逻辑和语义框架及应用研讨会(LSFA 2019和2020)的精选文章。LSFA 2019于2019年8月25日至26日在纳塔尔举行,作为第27届自动扣款国际会议(CADE 2019)的卫星活动。LSFA 2020于8月27日至28日作为在线研讨会举行,组织者来自萨尔瓦多。
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引用次数: 0
A dual-context sequent calculus for the constructive modal logic S4 构造模态逻辑S4的对偶上下文序列演算
IF 0.5 4区 计算机科学 Q2 Mathematics Pub Date : 2022-10-01 DOI: 10.1017/S0960129522000378
F. E. Miranda-Perea, Lourdes del Carmen González Huesca, Pilar Selene Linares Arévalo
Abstract The proof theory of the constructive modal logic S4 (hereafter $mathsf{CS4}$ ) has been settled since the beginning of this century by means of either standard natural deduction and sequent calculi or by the reconstruction of modal logic through hypothetical and categorical judgments à la Martin-Löf, an approach carried out by using a special kind of sequents, which keeps two separated contexts representing ordinary and enhanced hypotheses, intuitively interpreted as true and valid assumptions. These so-called dual-context sequents, originated in linear logic, are used to define a natural deduction system handling judgments of validity, truth, and possibility, resulting in a formalism equivalent to an axiomatic system for $mathsf{CS4}$ . However, this proof-theoretical study of $mathsf{CS4}$ lacks, to the best of our knowledge, its third fundamental constituent, namely a sequent calculus. In this paper, we define such a dual-context formalism, called ${bf DG_{CS4}}$ , and provide detailed proofs of the admissibility for the ordinary cut rule as well as the elimination of a second cut rule, which manipulates enhanced hypotheses. Furthermore, we make available a formal verification of the equivalence of this proposal with the previously defined axiomatic and dual-context natural deduction systems for $mathsf{CS4}$ , using the Coq proof-assistant.
自本世纪初以来,构造型模态逻辑S4(以下简称$mathsf{CS4}$)的证明理论要么通过标准自然演绎和序列演算,要么通过假设和范畴判断重建模态逻辑(通过使用一类特殊的序列来实现),它保留了代表普通假设和增强假设的两个分离的语境。直观地解释为真实有效的假设。这些所谓的双上下文序列,起源于线性逻辑,被用来定义一个处理有效性、真性和可能性判断的自然演绎系统,从而产生一个等价于$mathsf{CS4}$的公理化系统的形式主义。然而,据我们所知,这个关于$mathsf{CS4}$的证明理论研究缺少它的第三个基本组成部分,即序贯演算。在本文中,我们定义了这样一个双上下文形式,称为${bf DG_{CS4}}$,并提供了普通切割规则的可接受性和第二切割规则的消去性的详细证明,该规则操作增强假设。此外,我们使用Coq证明助手,对该建议与先前定义的$mathsf{CS4}$的公理化和双上下文自然演绎系统的等价性进行了形式化验证。
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引用次数: 0
Indexed and fibered structures for partial and total correctness assertions 用于部分和全部正确性断言的索引结构和纤维结构
IF 0.5 4区 计算机科学 Q2 Mathematics Pub Date : 2022-09-19 DOI: 10.1017/S0960129522000275
Uwe Wolter, A. Martini, E. Haeusler
Abstract Hoare Logic has a long tradition in formal verification and has been continuously developed and used to verify a broad class of programs, including sequential, object-oriented, and concurrent programs. Here we focus on partial and total correctness assertions within the framework of Hoare logic and show that a comprehensive categorical analysis of its axiomatic semantics needs the languages of indexed and fibered category theory. We consider Hoare formulas with local, finite contexts, of program and logical variables. The structural features of Hoare assertions are presented in an indexed setting, while the logical features of deduction are modeled in the fibered one.
摘要Hoare逻辑在形式验证方面有着悠久的传统,并不断被开发和用于验证一系列程序,包括顺序程序、面向对象程序和并发程序。在这里,我们关注霍尔逻辑框架内的部分和完全正确性断言,并表明对其公理语义的全面分类分析需要索引和纤维化范畴理论的语言。我们考虑具有程序和逻辑变量的局部有限上下文的霍尔公式。霍尔断言的结构特征在索引环境中呈现,而推理的逻辑特征在纤维化环境中建模。
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引用次数: 1
Detection and diagnosis of deviations in distributed systems of autonomous agents 分布式自治Agent系统中偏差的检测与诊断
IF 0.5 4区 计算机科学 Q2 Mathematics Pub Date : 2022-09-06 DOI: 10.1017/S0960129522000251
Vivek Nigam, Minyoung Kim, Ian Mason, C. Talcott
Abstract Given the complexity of cyber-physical systems (CPS), such as swarms of drones, often deviations, from a planned mission or protocol, occur which may in some cases lead to harm and losses. To increase the robustness of such systems, it is necessary to detect when deviations happen and diagnose the cause(s) for a deviation. We build on our previous work on soft agents, a formal framework based on using rewriting logic for specifying and reasoning about distributed CPS, to develop methods for diagnosis of CPS at design time. We accomplish this by (1) extending the soft agents framework with Fault Models; (2) proposing a protocol specification language and the definition of protocol deviations; and (3) development of workflows/algorithms for detection and diagnosis of protocol deviations. Our approach is partially inspired by existing work using counterfactual reasoning for fault ascription. We demonstrate our machinery with a collection of experiments.
摘要考虑到网络物理系统(CPS)的复杂性,例如成群的无人机,经常会偏离计划的任务或协议,在某些情况下可能会导致伤害和损失。为了提高此类系统的稳健性,有必要检测偏差何时发生,并诊断偏差的原因。我们在之前关于软代理的工作的基础上,开发了在设计时诊断CPS的方法。软代理是一个基于使用重写逻辑来指定和推理分布式CPS的正式框架。我们通过(1)用故障模型扩展软代理框架来实现这一点;(2) 提出了协议规范语言和协议偏差的定义;以及(3)开发用于检测和诊断协议偏差的工作流程/算法。我们的方法在一定程度上受到了现有工作的启发,这些工作使用反事实推理来进行故障归属。我们用一系列实验来展示我们的机器。
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引用次数: 0
On function spaces equipped with Isbell topology and Scott topology 基于Isbell拓扑和Scott拓扑的功能空间
IF 0.5 4区 计算机科学 Q2 Mathematics Pub Date : 2022-09-01 DOI: 10.1017/S0960129523000014
Xiaoquan Xu, Meng Bao, Xiaoyuan Zhang
Abstract In this paper, we mainly study the function spaces related to H-sober spaces. For an irreducible subset system H and $T_{0}$ spaces X and Y, it is proved that the following three conditions are equivalent: (1) the Scott space $Sigma mathcal O(X)$ of the lattice of all open sets of X is H-sober; (2) for every H-sober space Y, the function space $mathbb{C}(X, Y)$ of all continuous mappings from X to Y equipped with the Isbell topology is H-sober; (3) for every H-sober space Y, the Isbell topology on $mathbb{C}(X, Y)$ has property S with respect to H. One immediate corollary is that for a $T_{0}$ space X, Y is a d-space (resp., well-filtered space) iff the function space $mathbb{C}(X, Y)$ equipped with the Isbell topology is a d-space (resp., well-filtered space). It is shown that for any $T_0$ space X for which the Scott space $Sigma mathcal O(X)$ is non-sober, the function space $mathbb{C}(X, Sigma 2)$ equipped with the Isbell topology is not sober. The function spaces $mathbb{C}(X, Y)$ equipped with the Scott topology, the compact-open topology and the pointwise convergence topology are also discussed. Our study also leads to a number of questions, whose answers will deepen our understanding of the function spaces related to H-sober spaces.
摘要本文主要研究与h -清醒空间相关的函数空间。对于不可约子集系统H和$T_{0}$空间X和Y,证明了下列三个条件是等价的:(1)X的所有开集的格的Scott空间$Sigma mathcal O(X)$是H-清醒的;(2)对于每一个H-sober空间Y,所有具有Isbell拓扑的从X到Y的连续映射的函数空间$mathbb{C}(X, Y)$为H-sober;(3)对于每一个H-sober空间Y, $mathbb{C}(X, Y)$上的Isbell拓扑对于h具有S的性质。一个直接推论是,对于$T_{0}$空间X, Y是一个d-空间(resp。如果具有Isbell拓扑的函数空间$mathbb{C}(X, Y)$是一个d空间(如:(过滤良好的空间)。证明了对于Scott空间$Sigma mathcal O(X)$为非清醒的任意$T_0$空间X,具有Isbell拓扑的函数空间$mathbb{C}(X, Sigma 2)$是不清醒的。讨论了具有Scott拓扑、紧开拓扑和点向收敛拓扑的函数空间$mathbb{C}(X, Y)$。我们的研究还引出了一些问题,这些问题的答案将加深我们对与h -清醒空间相关的函数空间的理解。
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引用次数: 1
Introduction to the special issue: Confluence 特刊简介:合流
IF 0.5 4区 计算机科学 Q2 Mathematics Pub Date : 2022-08-01 DOI: 10.1017/S0960129523000063
M. Ayala-Rincón, S. Mimram
The notion of confluence , which generalizes the one of determinism, is a central and ubiquitous property of computational and deductive systems. Its study is one of the main topics of rewriting theory, where it relates to other properties such as termination, modularity, commutation, and completion. It has been investigated in many formalisms of rewriting, such as conditional and unconditional first- and higher order rewriting, λ -calculi, and constraint rewriting. This special issue presents a selection of novel results and recent computational techniques related to confluence
汇流的概念概括了决定论的概念,是计算和演绎系统的中心和普遍属性。它的研究是改写理论的主要课题之一,它涉及到其他性质,如终止、模块化、交换和补全。它在许多重写形式中被研究,如条件和无条件一阶和高阶重写、λ -演算和约束重写。这期特刊介绍了一些关于合流的新结果和最新的计算技术
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引用次数: 0
Z property for the shuffling calculus 洗牌演算的Z性质
IF 0.5 4区 计算机科学 Q2 Mathematics Pub Date : 2022-08-01 DOI: 10.1017/S0960129522000408
K. Nakazawa, Ken-etsu Fujita, Yuta Imagawa
ABSTRACT This paper gives a new proof of confluence for Carraro and Guerrieri’s call-by-value lambda calculus λ v σ with permutation rules. We adapt the compositional Z theorem to λ v σ .
摘要本文给出了Carraro和Guerrieri用值lambda演算λvσ和置换规则调用的一个新的合流证明。我们将组合Z定理应用于λvσ。
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引用次数: 1
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Mathematical Structures in Computer Science
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