Pub Date : 2024-01-10DOI: 10.1017/s0960129523000415
Lauri Hella, Kerkko Luosto
A game that characterizes equivalence of structures with respect to all first-order sentences containing a given number of quantifiers was introduced by Immerman in 1981. We define three other games and prove that they are all equivalent to the Immerman game, and hence also give a characterization for the number of quantifiers needed for separating structures. In the Immerman game, Duplicator has a canonical optimal strategy, and hence Duplicator can be completely removed from the game by replacing her moves with default moves given by this optimal strategy. On the other hand, in the last two of our games there is no such optimal strategy for Duplicator. Thus, the Immerman game can be regarded as a one-player game, but two of our games are genuine two-player games.
{"title":"Game characterizations for the number of quantifiers","authors":"Lauri Hella, Kerkko Luosto","doi":"10.1017/s0960129523000415","DOIUrl":"https://doi.org/10.1017/s0960129523000415","url":null,"abstract":"<p>A game that characterizes equivalence of structures with respect to all first-order sentences containing a given number of quantifiers was introduced by Immerman in 1981. We define three other games and prove that they are all equivalent to the Immerman game, and hence also give a characterization for the number of quantifiers needed for separating structures. In the Immerman game, Duplicator has a canonical optimal strategy, and hence Duplicator can be completely removed from the game by replacing her moves with default moves given by this optimal strategy. On the other hand, in the last two of our games there is no such optimal strategy for Duplicator. Thus, the Immerman game can be regarded as a one-player game, but two of our games are genuine two-player games.</p>","PeriodicalId":49855,"journal":{"name":"Mathematical Structures in Computer Science","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-01-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139413500","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}
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。
{"title":"Approximation Algorithm and FPT Algorithm for Connected-k-Subgraph Cover on Minor-Free Graphs","authors":"Pengcheng Liu, Zhao Zhang, Yingli Ran, Xiaohui Huang","doi":"10.1017/s0960129523000439","DOIUrl":"https://doi.org/10.1017/s0960129523000439","url":null,"abstract":"<p>Given a graph G, the minimum Connected-<span>k</span>-Subgraph Cover problem (MinC<span>k</span>SC) is to find a minimum vertex subset <span>C</span> of <span>G</span> such that every connected subgraph of <span>G</span> on <span>k</span> vertices has at least one vertex in <span>C</span>. If furthermore the subgraph of <span>G</span> induced by <span>C</span> is connected, then the problem is denoted as MinC<span>k</span>SC<span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240109155214832-0728:S0960129523000439:S0960129523000439_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$_{con}$</span></span></img></span></span>. In this paper, we first present a PTAS for MinC<span>k</span>SC on an <span>H</span>-minor-free graph, where <span>H</span> is a graph with a constant number of vertices. Then, we design an <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240109155214832-0728:S0960129523000439:S0960129523000439_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$O((omega+1)(2(k-1)(omega+2))^{3omega+3})|V|$</span></span></img></span></span>-time FPT algorithm for MinC<span>k</span>SC<span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240109155214832-0728:S0960129523000439:S0960129523000439_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$_{con}$</span></span></img></span></span> on a graph with treewidth <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240109155214832-0728:S0960129523000439:S0960129523000439_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$omega$</span></span></img></span></span>, based on which we further design an <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240109155214832-0728:S0960129523000439:S0960129523000439_inline5.png\"><span data-mathjax-type=\"texmath\"><span>$O(2^{O(sqrt{t}log t)}|V|^{O(1)})$</span></span></img></span></span> time subexponential FPT algorithm for MinC<span>k</span>SC<span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240109155214832-0728:S0960129523000439:S0960129523000439_inline6.png\"><span data-mathjax-type=\"texmath\"><span>$_{con}$</span></span></img></span></span> on an <span>H</span>-minor-free graph, where <span>t</span> is an upper bound of solution size.</p>","PeriodicalId":49855,"journal":{"name":"Mathematical Structures in Computer Science","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-01-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139413501","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 : 2023-12-21DOI: 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 $mathbf{K}$-ification.
A subcategory of $mathbf{TOP}_{mathbf{0}}$ is called of type $mathrm{K}^{*}$ if it consists of monotone convergence spaces and is of type $mathrm K$ in the sense of Keimel and Lawson. Each such category induces a canonical monad $mathcal K$ on the category $mathbf{DCPO}$ of dcpos and Scott-continuous maps, which is called the order-$mathbf{K}$-ification monad in this paper. First, for each category of type $mathrm{K}^{*}$, we characterize the algebras of the corresponding monad
{"title":"The order-K-ification monads","authors":"Huijun Hou, Hualin Miao, Qingguo Li","doi":"10.1017/s0960129523000403","DOIUrl":"https://doi.org/10.1017/s0960129523000403","url":null,"abstract":"<p>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 <span><span><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\"><span data-mathjax-type=\"texmath\"><span>$mathbf{K}$</span></span></img></span></span>-ification.</p><p>A subcategory of <span><span><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\"><span data-mathjax-type=\"texmath\"><span>$mathbf{TOP}_{mathbf{0}}$</span></span></img></span></span> is called of type <span><span><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\"><span data-mathjax-type=\"texmath\"><span>$mathrm{K}^{*}$</span></span></img></span></span> if it consists of monotone convergence spaces and is of type <span><span><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\"><span data-mathjax-type=\"texmath\"><span>$mathrm K$</span></span></img></span></span> in the sense of Keimel and Lawson. Each such category induces a canonical monad <span><span><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\"><span data-mathjax-type=\"texmath\"><span>$mathcal K$</span></span></img></span></span> on the category <span><span><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\"><span data-mathjax-type=\"texmath\"><span>$mathbf{DCPO}$</span></span></img></span></span> of dcpos and Scott-continuous maps, which is called the order-<span><span><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\"><span data-mathjax-type=\"texmath\"><span>$mathbf{K}$</span></span></img></span></span>-ification monad in this paper. First, for each category of type <span><span><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\"><span data-mathjax-type=\"texmath\"><span>$mathrm{K}^{*}$</span></span></img></span></span>, we characterize the algebras of the corresponding monad <span><span><img ","PeriodicalId":49855,"journal":{"name":"Mathematical Structures in Computer Science","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-12-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138824025","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 : 2023-12-19DOI: 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.
{"title":"Visibility and exploitation in social networks","authors":"Rustam Galimullin, Mina Young Pedersen","doi":"10.1017/s0960129523000397","DOIUrl":"https://doi.org/10.1017/s0960129523000397","url":null,"abstract":"<p>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.</p>","PeriodicalId":49855,"journal":{"name":"Mathematical Structures in Computer Science","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138744582","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 : 2023-12-13DOI: 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.
{"title":"Fixed point logics and definable topological properties","authors":"David Fernández-Duque, Quentin Gougeon","doi":"10.1017/s0960129523000385","DOIUrl":"https://doi.org/10.1017/s0960129523000385","url":null,"abstract":"<p>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 <span>imperfect spaces.</span> 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.</p>","PeriodicalId":49855,"journal":{"name":"Mathematical Structures in Computer Science","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-12-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138579044","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 : 2023-12-01DOI: 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.
{"title":"A point-free perspective on lax extensions and predicate liftings","authors":"Sergey Goncharov, Dirk Hofmann, Pedro Nora, Lutz Schröder, Paul Wild","doi":"10.1017/s096012952300035x","DOIUrl":"https://doi.org/10.1017/s096012952300035x","url":null,"abstract":"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.","PeriodicalId":49855,"journal":{"name":"Mathematical Structures in Computer Science","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138496806","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 : 2023-11-30DOI: 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.
{"title":"Learning quantum finite automata with queries","authors":"Daowen Qiu","doi":"10.1017/s0960129523000373","DOIUrl":"https://doi.org/10.1017/s0960129523000373","url":null,"abstract":"<jats:italic>Learning finite automata</jats:italic> (termed as <jats:italic>model learning</jats:italic>) 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 <jats:italic>quantum learning theory</jats:italic> and <jats:italic>quantum machine learning</jats:italic>, in this paper, our purpose is to initiate the study of <jats:italic>learning QFA with queries</jats:italic> (naturally it may be termed as <jats:italic>quantum model learning</jats:italic>), 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.","PeriodicalId":49855,"journal":{"name":"Mathematical Structures in Computer Science","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-11-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138517247","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 : 2023-11-21DOI: 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.
{"title":"A concrete model for a typed linear algebraic lambda calculus","authors":"Alejandro Díaz-Caro, Octavio Malherbe","doi":"10.1017/s0960129523000361","DOIUrl":"https://doi.org/10.1017/s0960129523000361","url":null,"abstract":"We give an adequate, concrete, categorical-based model for Lambda-<jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129523000361_inline1.png\" /> <jats:tex-math> ${mathcal S}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, which is a typed version of a linear-algebraic lambda calculus, extended with measurements. Lambda-<jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129523000361_inline2.png\" /> <jats:tex-math> ${mathcal S}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> 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-<jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129523000361_inline3.png\" /> <jats:tex-math> ${mathcal S}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> has a superposition constructor <jats:italic>S</jats:italic> such that a type <jats:italic>A</jats:italic> is considered as the base of a vector space, while <jats:italic>SA</jats:italic> is its span. Our model considers <jats:italic>S</jats:italic> as the composition of two functors in an adjunction relation between the category of sets and the category of vector spaces over <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129523000361_inline4.png\" /> <jats:tex-math> $mathbb C$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. The right adjoint is a forgetful functor <jats:italic>U</jats:italic>, which is hidden in the language, and plays a central role in the computational reasoning.","PeriodicalId":49855,"journal":{"name":"Mathematical Structures in Computer Science","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-11-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138496805","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 : 2023-11-03DOI: 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.
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Pub Date : 2023-10-25DOI: 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.
{"title":"Dilations and information flow axioms in categorical probability","authors":"Tobias Fritz, Tomáš Gonda, Nicholas Gauguin Houghton-Larsen, Antonio Lorenzin, Paolo Perrone, Dario Stein","doi":"10.1017/s0960129523000324","DOIUrl":"https://doi.org/10.1017/s0960129523000324","url":null,"abstract":"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.","PeriodicalId":49855,"journal":{"name":"Mathematical Structures in Computer Science","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135113929","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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