Pub Date : 2023-04-01DOI: 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.
{"title":"Divergences on monads for relational program logics","authors":"Tetsuya Sato, Shin-ya Katsumata","doi":"10.1017/s0960129523000245","DOIUrl":"https://doi.org/10.1017/s0960129523000245","url":null,"abstract":"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.","PeriodicalId":49855,"journal":{"name":"Mathematical Structures in Computer Science","volume":"49 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135722102","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-03-30DOI: 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.
{"title":"Behavioural equivalences for continuous-time Markov processes","authors":"Linan Chen, Florence Clerc, P. Panangaden","doi":"10.1017/s0960129523000099","DOIUrl":"https://doi.org/10.1017/s0960129523000099","url":null,"abstract":"\u0000 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.","PeriodicalId":49855,"journal":{"name":"Mathematical Structures in Computer Science","volume":"1 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2023-03-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42827534","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-03-14DOI: 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）$约束违反界和目标误差界。
{"title":"Stochastic linearized generalized alternating direction method of multipliers: Expected convergence rates and large deviation properties","authors":"Jia Hu, T. Guo, Congying Han","doi":"10.1017/s096012952300004x","DOIUrl":"https://doi.org/10.1017/s096012952300004x","url":null,"abstract":"\u0000 Alternating direction method of multipliers (ADMM) receives much attention in the field of optimization and computer science, etc. The generalized ADMM (G-ADMM) proposed by Eckstein and Bertsekas incorporates an acceleration factor and is more efficient than the original ADMM. However, G-ADMM is not applicable in some models where the objective function value (or its gradient) is computationally costly or even impossible to compute. In this paper, we consider the two-block separable convex optimization problem with linear constraints, where only noisy estimations of the gradient of the objective function are accessible. Under this setting, we propose a stochastic linearized generalized ADMM (called SLG-ADMM) where two subproblems are approximated by some linearization strategies. And in theory, we analyze the expected convergence rates and large deviation properties of SLG-ADMM. In particular, we show that the worst-case expected convergence rates of SLG-ADMM are \u0000 \u0000 \u0000 \u0000$mathcal{O}left( {{N}^{-1/2}}right)$\u0000\u0000 \u0000 and \u0000 \u0000 \u0000 \u0000$mathcal{O}left({ln N} cdot {N}^{-1}right)$\u0000\u0000 \u0000 for solving general convex and strongly convex problems, respectively, where N is the iteration number, similarly hereinafter, and with high probability, SLG-ADMM has \u0000 \u0000 \u0000 \u0000$mathcal{O}left ( ln N cdot N^{-1/2} right ) $\u0000\u0000 \u0000 and \u0000 \u0000 \u0000 \u0000$mathcal{O}left ( left ( ln N right )^{2} cdot N^{-1} right ) $\u0000\u0000 \u0000 constraint violation bounds and objective error bounds for general convex and strongly convex problems, respectively.","PeriodicalId":49855,"journal":{"name":"Mathematical Structures in Computer Science","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2023-03-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45770886","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-02-01DOI: 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.
{"title":"A domain-theoretic framework for robustness analysis of neural networks","authors":"Can Zhou, Razin A. Shaikh, Yiran Li, Amin Farjudian","doi":"10.1017/s0960129523000142","DOIUrl":"https://doi.org/10.1017/s0960129523000142","url":null,"abstract":"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.","PeriodicalId":49855,"journal":{"name":"Mathematical Structures in Computer Science","volume":"191 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136019624","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-02-01DOI: 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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Pub Date : 2023-01-12DOI: 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 <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S096012952200041X_inline2.png" /><jats:tex-math> $lambda mu$ </jats:tex-math></jats:alternatives></jats:inline-formula>-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 <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S096012952200041X_inline3.png" /><jats:tex-math> $beta$ </jats:tex-math></jats:alternatives></jats:inline-formula>-, <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S096012952200041X_inline4.png" /><jats:tex-math> $mu$ </jats:tex-math></jats:alternatives></jats:inline-formula>-, and <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S096012952200041X_inline5.png" /><jats:tex-math> $mu '$ </jats:tex-math></jats:alternatives></jats:inline-formula>-reductions, we consider the <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S096012952200041X_inline6.png" /><jats:tex-math> $lambda mu$ </jats:tex-math></jats:alternatives></jats:inline-formula>-calculus augmented with the reduction rules <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S096012952200041X_inline7.png" /><jats:tex-math> $rho$ </jats:tex-math></jats:alternatives></jats:inline-formula>, <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S096012952200041X_inline8.png" /><jats:tex-math> $theta$ </jats:tex-math></jats:alternatives></jats:inline-formula> and <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S096012952200041X_inline9.png" /><jats:tex-math> $varepsilon$ </jats:tex-math></jats:alternatives></jats:inline-formula>. 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 <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S096012952200041X_inline10.png" /><jats:tex-math> $rho$ </jats:tex-math></jats:alternatives></jats:inline-formula>, <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xl
{"title":"Normalization in the simply typed -calculus","authors":"Péter Battyányi, Karim Nour","doi":"10.1017/s096012952200041x","DOIUrl":"https://doi.org/10.1017/s096012952200041x","url":null,"abstract":"In this paper, in connection with the program of extending the Curry–Howard isomorphism to classical logic, we study the <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S096012952200041X_inline2.png\" /><jats:tex-math> $lambda mu$ </jats:tex-math></jats:alternatives></jats:inline-formula>-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 <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S096012952200041X_inline3.png\" /><jats:tex-math> $beta$ </jats:tex-math></jats:alternatives></jats:inline-formula>-, <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S096012952200041X_inline4.png\" /><jats:tex-math> $mu$ </jats:tex-math></jats:alternatives></jats:inline-formula>-, and <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S096012952200041X_inline5.png\" /><jats:tex-math> $mu '$ </jats:tex-math></jats:alternatives></jats:inline-formula>-reductions, we consider the <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S096012952200041X_inline6.png\" /><jats:tex-math> $lambda mu$ </jats:tex-math></jats:alternatives></jats:inline-formula>-calculus augmented with the reduction rules <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S096012952200041X_inline7.png\" /><jats:tex-math> $rho$ </jats:tex-math></jats:alternatives></jats:inline-formula>, <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S096012952200041X_inline8.png\" /><jats:tex-math> $theta$ </jats:tex-math></jats:alternatives></jats:inline-formula> and <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S096012952200041X_inline9.png\" /><jats:tex-math> $varepsilon$ </jats:tex-math></jats:alternatives></jats:inline-formula>. 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 <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S096012952200041X_inline10.png\" /><jats:tex-math> $rho$ </jats:tex-math></jats:alternatives></jats:inline-formula>, <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xl","PeriodicalId":49855,"journal":{"name":"Mathematical Structures in Computer Science","volume":"39 11","pages":""},"PeriodicalIF":0.5,"publicationDate":"2023-01-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138496804","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-01-01DOI: 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.
{"title":"Local Yoneda completions of quasi-metric spaces","authors":"Jing Lu, Bin Zhao","doi":"10.1017/S0960129523000105","DOIUrl":"https://doi.org/10.1017/S0960129523000105","url":null,"abstract":"Abstract In this paper, we study quasi-metric spaces using domain theory. Given a quasi-metric space (X,d), we use \u0000$({bf B}(X,d),leq^{d^{+}}!)$\u0000 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 \u0000$({bf B}(X,d),leq^{d^{+}}!)$\u0000 . 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.","PeriodicalId":49855,"journal":{"name":"Mathematical Structures in Computer Science","volume":"33 1","pages":"33 - 45"},"PeriodicalIF":0.5,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47608027","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-11-22DOI: 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.
{"title":"Semantic analysis of normalisation by evaluation for typed lambda calculus","authors":"Marcelo Fiore","doi":"10.1017/s0960129522000263","DOIUrl":"https://doi.org/10.1017/s0960129522000263","url":null,"abstract":"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.","PeriodicalId":49855,"journal":{"name":"Mathematical Structures in Computer Science","volume":"39 16","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138496803","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-10-18DOI: 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.
{"title":"Preserving consistency in geometric modeling with graph transformations","authors":"A. Arnould, H. Belhaouari, Thomas Bellet, P. L. Gall, R. Pascual","doi":"10.1017/s0960129522000226","DOIUrl":"https://doi.org/10.1017/s0960129522000226","url":null,"abstract":"\u0000 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.","PeriodicalId":49855,"journal":{"name":"Mathematical Structures in Computer Science","volume":"68 1","pages":"300-347"},"PeriodicalIF":0.5,"publicationDate":"2022-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89310297","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-10-09DOI: 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.
{"title":"What should a generic object be?","authors":"Jonathan Sterling","doi":"10.1017/S0960129523000117","DOIUrl":"https://doi.org/10.1017/S0960129523000117","url":null,"abstract":"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, \u0000$lambda2$\u0000 -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.","PeriodicalId":49855,"journal":{"name":"Mathematical Structures in Computer Science","volume":"33 1","pages":"46 - 67"},"PeriodicalIF":0.5,"publicationDate":"2022-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44069563","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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