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Up-to techniques for behavioural metrics via fibrations 通过振动进行行为度量的最新技术
4区 计算机科学 Q2 Mathematics Pub Date : 2023-04-01 DOI: 10.1017/s0960129523000166
Filippo Bonchi, Barbara König, Daniela Petrisan
Abstract Up-to techniques are a well-known method for enhancing coinductive proofs of behavioural equivalences. We introduce up-to techniques for behavioural metrics between systems modelled as coalgebras, and we provide abstract results to prove their soundness in a compositional way. In order to obtain a general framework, we need a systematic way to lift functors: we show that the Wasserstein lifting of a functor, introduced in a previous work, corresponds to a change of base in a fibrational sense. This observation enables us to reuse existing results about soundness of up-to techniques in a fibrational setting. We focus on the fibrations of predicates and relations valued in a quantale. To illustrate our approach, we provide an example on distances between regular languages.
Up-to技术是一种众所周知的增强行为等价的共归纳证明的方法。我们介绍了行为度量系统之间建模为共代数的最新技术,我们提供了抽象的结果来证明它们在组合方式上的合理性。为了得到一个一般的框架,我们需要一个系统的方法来提升函子:我们证明了在之前的工作中引入的函子的Wasserstein提升,对应于纤维意义上的基的变化。这一观察结果使我们能够重复利用现有的结果,关于在颤振设置的健全技术。我们关注的是量值中谓词和关系的颤动。为了说明我们的方法,我们提供了一个关于常规语言之间距离的示例。
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引用次数: 2
Divergences on monads for relational program logics 关系程序逻辑单子上的分歧
4区 计算机科学 Q2 Mathematics Pub Date : 2023-04-01 DOI: 10.1017/s0960129523000245
Tetsuya Sato, Shin-ya Katsumata
Abstract Several relational program logics have been introduced for integrating reasoning about relational properties of programs and measurement of quantitative difference between computational effects. Toward a general framework for such logics, in this paper, we formalize the concept of quantitative difference between computational effects as divergences on monads , then develop a relational program logic called approximate computational relational logic (acRL for short). It supports generic computational effects and divergences on them. The semantics of the acRL is given by graded strong relational liftings constructed from divergences on monads. We derive two instantiations of the acRL: (1) for the verification of various kinds of differential privacy of higher-order functional probabilistic programs and (2) the other for measuring difference of distributions of cost between higher-order functional probabilistic programs with a cost counting operator.
摘要引入了几种关系程序逻辑,将程序的关系属性推理和计算效果的定量差异度量结合起来。为了建立这种逻辑的一般框架,本文将计算效果之间的数量差异的概念形式化为单子上的散度,然后发展了一种称为近似计算关系逻辑(简称acRL)的关系程序逻辑。它支持通用的计算效果和对它们的发散。acRL的语义是由单体上的散度构造的分级强关系提升给出的。我们推导了acRL的两个实例:(1)用于验证高阶泛函概率规划的各种微分隐私;(2)另一个用于测量高阶泛函概率规划之间的成本分布的差异。
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引用次数: 0
Behavioural equivalences for continuous-time Markov processes 连续时间马尔可夫过程的行为等价
IF 0.5 4区 计算机科学 Q2 Mathematics Pub Date : 2023-03-30 DOI: 10.1017/s0960129523000099
Linan Chen, Florence Clerc, P. Panangaden
Bisimulation is a concept that captures behavioural equivalence of states in a variety of types of transition systems. It has been widely studied in a discrete-time setting. The core of this work is to generalise the discrete-time picture to continuous time by providing a notion of behavioural equivalence for continuous-time Markov processes. In Chen et al. [(2019). Electronic Notes in Theoretical Computer Science347 45–63.], we proposed two equivalent definitions of bisimulation for continuous-time stochastic processes where the evolution is a flow through time: the first one as an equivalence relation and the second one as a cospan of morphisms. In Chen et al. [(2020). Electronic Notes in Theoretical Computer Science.], we developed the theory further: we introduced different concepts that correspond to different behavioural equivalences and compared them to bisimulation. In particular, we studied the relation between bisimulation and symmetry groups of the dynamics. We also provided a game interpretation for two of the behavioural equivalences. The present work unifies the cited conference presentations and gives detailed proofs.
双模拟是一个概念,它捕捉了各种类型的过渡系统中状态的行为等价性。它在离散时间环境中得到了广泛的研究。这项工作的核心是通过为连续时间马尔可夫过程提供行为等价的概念,将离散时间图像推广到连续时间。在Chen等人[(2019).理论计算机科学中的电子注释347 45–63.]中,我们提出了连续时间随机过程的双模拟的两个等价定义,其中进化是通过时间的流动:第一个定义是等价关系,第二个定义是态射的共泛。在Chen等人[(2020).理论计算机科学中的电子笔记。]中,我们进一步发展了这一理论:我们引入了对应于不同行为等价性的不同概念,并将其与互刺激进行了比较。特别地,我们研究了动力学的对称群和互模拟之间的关系。我们还提供了两种行为等价物的博弈解释。本工作将引用的会议报告统一起来,并给出了详细的证明。
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引用次数: 0
Stochastic linearized generalized alternating direction method of multipliers: Expected convergence rates and large deviation properties 乘法器的随机线性化广义交替方向法:预期收敛速率和大偏差特性
IF 0.5 4区 计算机科学 Q2 Mathematics Pub Date : 2023-03-14 DOI: 10.1017/s096012952300004x
Jia Hu, T. Guo, Congying Han
Alternating direction method of multipliers (ADMM) receives much attention in the field of optimization and computer science, etc. The generalized ADMM (G-ADMM) proposed by Eckstein and Bertsekas incorporates an acceleration factor and is more efficient than the original ADMM. However, G-ADMM is not applicable in some models where the objective function value (or its gradient) is computationally costly or even impossible to compute. In this paper, we consider the two-block separable convex optimization problem with linear constraints, where only noisy estimations of the gradient of the objective function are accessible. Under this setting, we propose a stochastic linearized generalized ADMM (called SLG-ADMM) where two subproblems are approximated by some linearization strategies. And in theory, we analyze the expected convergence rates and large deviation properties of SLG-ADMM. In particular, we show that the worst-case expected convergence rates of SLG-ADMM are $mathcal{O}left( {{N}^{-1/2}}right)$ and $mathcal{O}left({ln N} cdot {N}^{-1}right)$ for solving general convex and strongly convex problems, respectively, where N is the iteration number, similarly hereinafter, and with high probability, SLG-ADMM has $mathcal{O}left ( ln N cdot N^{-1/2} right ) $ and $mathcal{O}left ( left ( ln N right )^{2} cdot N^{-1} right ) $ constraint violation bounds and objective error bounds for general convex and strongly convex problems, respectively.
乘法器的交替方向法(ADMM)在优化和计算机科学等领域受到了广泛的关注。Eckstein和Bertsekas提出的广义ADMM(G-ADMM)包含了加速因子,比原来的ADMM更有效。然而,G-ADMM不适用于目标函数值(或其梯度)计算成本高甚至不可能计算的一些模型。在本文中,我们考虑具有线性约束的两块可分离凸优化问题,其中只有目标函数梯度的噪声估计是可访问的。在这种情况下,我们提出了一种随机线性化的广义ADMM(称为SLG-ADMM),其中两个子问题通过一些线性化策略近似。在理论上,我们分析了SLG-ADMM的预期收敛速度和大偏差特性。特别地,我们证明了SLG-ADMM在求解一般凸和强凸问题时,最坏情况下的预期收敛速度分别为$mathcal{O}left({{N}^{-1/2}}right)$和$mathcal{O}left,SLG-ADMM对于一般凸和强凸问题分别具有$mathcal{O}left(ln Ncdot N^{-1/2}right)$和$mathical{O}left(left( ln Nright)^{2}cdot N ^{-1}right)$约束违反界和目标误差界。
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引用次数: 2
A domain-theoretic framework for robustness analysis of neural networks 神经网络鲁棒性分析的领域理论框架
4区 计算机科学 Q2 Mathematics Pub Date : 2023-02-01 DOI: 10.1017/s0960129523000142
Can Zhou, Razin A. Shaikh, Yiran Li, Amin Farjudian
Abstract A domain-theoretic framework is presented for validated robustness analysis of neural networks. First, global robustness of a general class of networks is analyzed. Then, using the fact that Edalat’s domain-theoretic L -derivative coincides with Clarke’s generalized gradient, the framework is extended for attack-agnostic local robustness analysis. The proposed framework is ideal for designing algorithms which are correct by construction. This claim is exemplified by developing a validated algorithm for estimation of Lipschitz constant of feedforward regressors. The completeness of the algorithm is proved over differentiable networks and also over general position ${mathrm{ReLU}}$ networks. Computability results are obtained within the framework of effectively given domains. Using the proposed domain model, differentiable and non-differentiable networks can be analyzed uniformly. The validated algorithm is implemented using arbitrary-precision interval arithmetic, and the results of some experiments are presented. The software implementation is truly validated, as it handles floating-point errors as well.
摘要提出了一种用于神经网络鲁棒性验证分析的领域理论框架。首先,分析了一类网络的全局鲁棒性。然后,利用Edalat的域论L导数与Clarke的广义梯度相吻合的事实,将该框架扩展到攻击不可知的局部鲁棒性分析。所提出的框架对于设计构造正确的算法是理想的。通过开发一种有效的算法来估计前馈回归量的Lipschitz常数,可以证明这一说法。在可微网络和一般位置${ mathm {ReLU}}$网络上证明了算法的完备性。在有效给定域的框架内得到了可计算性结果。利用所提出的领域模型,可以统一地分析可微网络和不可微网络。验证后的算法采用任意精度区间算法实现,并给出了一些实验结果。软件实现得到了真正的验证,因为它也可以处理浮点错误。
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引用次数: 1
Univalent categories of modules 模的单价范畴
IF 0.5 4区 计算机科学 Q2 Mathematics Pub Date : 2023-02-01 DOI: 10.1017/s0960129523000178
J. G. T. Flaten
We show that categories of modules over a ring in homotopy type theory (HoTT) satisfy the internal versions of the AB axioms from homological algebra. The main subtlety lies in proving AB4, which is that coproducts indexed by arbitrary sets are left-exact. To prove this, we replace a set X with the strict category of lists of elements in X. From showing that the latter is filtered, we deduce left-exactness of the coproduct. More generally, we show that exactness of filtered colimits (AB5) implies AB4 for any abelian category in HoTT. Our approach is heavily inspired by Roswitha Harting’s construction of the internal coproduct of abelian groups in an elementary topos with a natural numbers object. To state the AB axioms, we define and study filtered (and sifted) precategories in HoTT. A key result needed is that filtered colimits commute with finite limits of sets. This is a familiar classical result but has not previously been checked in our setting. Finally, we interpret our most central results into an $infty$-topos $ {mathscr{X}} $. Given a ring R in $ {tau_{leq 0}({{mathscr{X}}})} $ – for example, an ordinary sheaf of rings – we show that the internal category of R-modules in $ {mathscr{X}} $ represents the presheaf which sends an object $ X in {mathscr{X}} $ to the category of $ (X{times}R) $-modules in ${mathscr{X}} / X$. In general, our results yield a product-preserving left adjoint to base change of modules over X. When X is 0-truncated, this left adjoint is the internal coproduct. By an internalisation procedure, we deduce left-exactness of the internal coproduct as an ordinary functor from its internal left-exactness coming from HoTT.
证明了同伦类型理论(HoTT)中环上模的范畴满足同伦代数中AB公理的内部版本。主要的微妙之处在于证明AB4,即由任意集合索引的余积是左精确的。为了证明这一点,我们将集合X替换为X中元素列表的严格范畴。通过证明后者是过滤的,我们推导出了余积的左精确性。更一般地,我们证明了对HoTT中任何阿贝尔范畴,滤波边界的精确性(AB5)意味着AB4。我们的方法很大程度上受到Roswitha Harting在具有自然数对象的初等拓扑中构造阿贝群的内副积的启发。为了说明AB公理,我们定义和研究HoTT中的过滤(和筛选)预范畴。需要的一个关键结果是,过滤的边界与有限的集合的极限交换。这是一个熟悉的经典结果,但以前没有在我们的设置中检查过。最后,我们将最核心的结果解释为$infty$ -topos $ {mathscr{X}} $。给定$ {tau_{leq 0}({{mathscr{X}}})} $中的一个环R——例如,一个普通的环束——我们表明,$ {mathscr{X}} $中R-modules的内部类别表示将对象$ X in {mathscr{X}} $发送到${mathscr{X}} / X$中$ (X{times}R) $ -modules的类别的presheaf。一般来说,我们的结果产生了模在X上的基变化的保积左伴随。当X被截断为0时,这个左伴随是内副积。通过内部化过程,我们从HoTT的内左精确性推导出普通函子的内副积的左精确性。
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引用次数: 0
Normalization in the simply typed -calculus 简单类型微积分的归一化
IF 0.5 4区 计算机科学 Q2 Mathematics Pub Date : 2023-01-12 DOI: 10.1017/s096012952200041x
Péter Battyányi, Karim Nour
In this paper, in connection with the program of extending the Curry–Howard isomorphism to classical logic, we study the $lambda mu$ -calculus of Parigot emphasizing the difference between the original version of Parigot and the version of de Groote in terms of normalization properties. In order to talk about a satisfactory representation of the integers, besides the usual $beta$ -, $mu$ -, and $mu '$ -reductions, we consider the $lambda mu$ -calculus augmented with the reduction rules $rho$ , $theta$ and $varepsilon$ . We show that we need all of these rules for this purpose. Then we prove that, with the syntax of Parigot, the calculus enjoys the strong normalization property even when we add the rules $rho$ ,
本文结合将Curry-Howard同构扩展到经典逻辑的方案,研究Parigot的$lambda mu$ -演算,强调Parigot的原始版本与de Groote版本在规格化性质上的区别。为了讨论整数的令人满意的表示,除了通常的$beta$ -, $mu$ -和$mu '$ -约简之外,我们还考虑了$lambda mu$ -演算与约简规则$rho$, $theta$和$varepsilon$的扩充。我们证明了我们需要所有这些规则来达到这个目的。然后我们证明了在Parigot语法下,即使添加了$rho$、$theta$和$epsilon$规则,微积分仍然具有强的规格化性质,而采用更灵活的de groote风格语法的$lambda mu$ -微积分却只有弱的规格化性质。特别地,我们提出了一种归一化算法,用于de Groote-style演算中的$beta mu mu 'rho theta varepsilon$ -约简。
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引用次数: 0
Local Yoneda completions of quasi-metric spaces 拟度量空间的局部Yoneda补全
IF 0.5 4区 计算机科学 Q2 Mathematics Pub Date : 2023-01-01 DOI: 10.1017/S0960129523000105
Jing Lu, Bin Zhao
Abstract In this paper, we study quasi-metric spaces using domain theory. Given a quasi-metric space (X,d), we use $({bf B}(X,d),leq^{d^{+}}!)$ to denote the poset of formal balls of the associated quasi-metric space (X,d). We introduce the notion of local Yoneda-complete quasi-metric spaces in terms of domain-theoretic properties of $({bf B}(X,d),leq^{d^{+}}!)$ . The manner in which this definition is obtained is inspired by Romaguera–Valero theorem and Kostanek–Waszkiewicz theorem. Furthermore, we obtain characterizations of local Yoneda-complete quasi-metric spaces via local nets in quasi-metric spaces. More precisely, we prove that a quasi-metric space is local Yoneda-complete if and only if every local net has a d-limit. Finally, we prove that every quasi-metric space has a local Yoneda completion.
摘要本文利用域理论研究了拟度量空间。给定一个拟度量空间(X,d),我们使用$({bf B}(X,d),leq^{d^{+}!)$表示相关的拟度量空间(X,d)的形式球的偏序集。根据$({bf B}(X,d),leq^{d^{+}}!)$的域理论性质,我们引入了局部Yoneda完备拟度量空间的概念。这个定义的获得方式受到了Romaguera–Valero定理和Kostanek–Waszkiewicz定理的启发。此外,我们通过拟度量空间中的局部网得到了局部Yoneda完备拟度量空间的特征。更确切地说,我们证明了一个拟度量空间是局部Yoneda完备的,当且仅当每个局部网都有一个d-极限。最后,我们证明了每一个拟度量空间都有一个局部Yoneda完备。
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引用次数: 0
Semantic analysis of normalisation by evaluation for typed lambda calculus 类型化λ演算的归一化求值语义分析
IF 0.5 4区 计算机科学 Q2 Mathematics Pub Date : 2022-11-22 DOI: 10.1017/s0960129522000263
Marcelo Fiore
This paper studies normalisation by evaluation for typed lambda calculus from a categorical and algebraic viewpoint. The first part of the paper analyses the lambda definability result of Jung and Tiuryn via Kripke logical relations and shows how it can be adapted to unify definability and normalisation, yielding an extensional normalisation result. In the second part of the paper, the analysis is refined further by considering intensional Kripke relations (in the form of Artin–Wraith glueing) and shown to provide a function for normalising terms, casting normalisation by evaluation in the context of categorical glueing. The technical development includes an algebraic treatment of the syntax and semantics of the typed lambda calculus that allows the definition of the normalisation function to be given within a simply typed metatheory. A normalisation-by-evaluation program in a dependently typed functional programming language is synthesised.
本文从范畴和代数的角度研究了类型化λ演算的归一化。本文第一部分通过Kripke逻辑关系分析了Jung和Tiuryn的lambda可定义性结果,并说明了如何将其用于统一可定义性和规范化,从而得到一个外延规范化结果。在本文的第二部分中,通过考虑内蕴的Kripke关系(以Artin-Wraith粘合的形式)进一步改进了分析,并显示了提供规范化项的函数,在分类粘合的上下文中通过评估来实现规范化。技术发展包括对类型化lambda演算的语法和语义的代数处理,允许在简单类型化元理论中给出规范化函数的定义。合成了一个依赖类型函数式编程语言中的求值归一化程序。
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引用次数: 0
Preserving consistency in geometric modeling with graph transformations 用图形变换保持几何建模的一致性
IF 0.5 4区 计算机科学 Q2 Mathematics Pub Date : 2022-10-18 DOI: 10.1017/s0960129522000226
A. Arnould, H. Belhaouari, Thomas Bellet, P. L. Gall, R. Pascual
Labeled graphs are particularly well adapted to represent objects in the context of topology-based geometric modeling. Thus, graph transformation theory is used to implement modeling operations and check their consistency. This article defines a class of graph transformation rules dedicated to embedding computations. Objects are here defined as a particular subclass of labeled graphs in which arc labels encode their topological structure (i.e., cell subdivision: vertex, edge, face) and node labels encode their embedding (i.e., relevant data: vertex positions, face colors, volume density). Object consistency is defined by labeling constraints which must be preserved by modeling operations that modify topology and/or embedding. Dedicated graph transformation variables allow us to access the existing embedding from the underlying topological structure (e.g., collecting all the points of a face) in order to compute the new embedding using user-provided functions (e.g., compute the barycenter of several points). To ensure the safety of the defined operations, we provide syntactic conditions on rules that preserve the object consistency constraints.
标记图特别适合于在基于拓扑的几何建模环境中表示对象。因此,使用图变换理论来实现建模操作并检查其一致性。本文定义了一类专门用于嵌入计算的图转换规则。对象在这里被定义为标记图的特定子类,其中弧标签编码其拓扑结构(即,细胞细分:顶点,边缘,面),节点标签编码其嵌入(即,相关数据:顶点位置,面颜色,体积密度)。对象一致性是通过标记约束来定义的,这些约束必须通过修改拓扑和/或嵌入的建模操作来保持。专用的图变换变量允许我们从底层拓扑结构(例如,收集人脸的所有点)访问现有的嵌入,以便使用用户提供的函数(例如,计算几个点的重心)计算新的嵌入。为了确保所定义操作的安全性,我们在规则上提供了保留对象一致性约束的语法条件。
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引用次数: 2
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Mathematical Structures in Computer Science
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