Pub Date : 2024-04-18DOI: 10.1017/s0960129524000057
Axel Muller, Metod Saniga, Alain Giorgetti, Henri de Boutray, Frédéric Holweck
We present algorithms and a C code to reveal quantum contextuality and evaluate the contextuality degree (a way to quantify contextuality) for a variety of point-line geometries located in binary symplectic polar spaces of small rank. With this code we were not only able to recover, in a more efficient way, all the results of a recent paper by de Boutray et al. [(2022). Journal of Physics A: Mathematical and Theoretical55 475301], but also arrived at a bunch of new noteworthy results. The paper first describes the algorithms and the C code. Then it illustrates its power on a number of subspaces of symplectic polar spaces whose rank ranges from 2 to 7. The most interesting new results include: (i) non-contextuality of configurations whose contexts are subspaces of dimension 2 and higher, (ii) non-existence of negative subspaces of dimension 3 and higher, (iii) considerably improved bounds for the contextuality degree of both elliptic and hyperbolic quadrics for rank 4, as well as for a particular subgeometry of the three-qubit space whose contexts are the lines of this space, (iv) proof for the non-contextuality of perpsets and, last but not least, (v) contextual nature of a distinguished subgeometry of a multi-qubit doily, called a two-spread, and computation of its contextuality degree. Finally, in the three-qubit polar space we correct and improve the contextuality degree of the full configuration and also describe finite geometric configurations formed by unsatisfiable/invalid constraints for both types of quadrics as well as for the geometry whose contexts are all 315 lines of the space.
我们提出了揭示量子情境性的算法和 C 代码,并评估了位于小秩二元交点极空间中的各种点线几何的情境性程度(量化情境性的一种方法)。有了这套代码,我们不仅能以更高效的方式恢复 de Boutray 等人最近发表的论文[(2022). 物理期刊 A:数学与理论 55 475301]中的所有结果,而且还得出了一系列值得注意的新结果。论文首先介绍了算法和 C 代码。然后说明了它在秩为 2 到 7 的交错极空间的一些子空间上的威力。最有趣的新结果包括(i) 上下文为维数为 2 或更高的子空间的构型的非上下文性,(ii) 维数为 3 或更高的负子空间的不存在性,(iii) 极大地改进了阶为 4 的椭圆和双曲四边形的上下文性程度边界、(iv)证明了周集的非上下文性,最后但并非最不重要的是,(v)证明了多比特多面体的一个杰出子几何(称为双展宽)的上下文性质及其上下文性度的计算。最后,在三比特极坐标空间中,我们修正并改进了完整构型的上下文度,还描述了由两类四面体以及上下文均为空间 315 条线的几何体的不可满足/无效约束所形成的有限几何构型。
{"title":"New and improved bounds on the contextuality degree of multi-qubit configurations","authors":"Axel Muller, Metod Saniga, Alain Giorgetti, Henri de Boutray, Frédéric Holweck","doi":"10.1017/s0960129524000057","DOIUrl":"https://doi.org/10.1017/s0960129524000057","url":null,"abstract":"<p>We present algorithms and a C code to reveal quantum contextuality and evaluate the contextuality degree (a way to quantify contextuality) for a variety of point-line geometries located in binary symplectic polar spaces of small rank. With this code we were not only able to recover, in a more efficient way, all the results of a recent paper by de Boutray et al. [(2022). <span>Journal of Physics A: Mathematical and Theoretical</span> <span>55</span> 475301], but also arrived at a bunch of new noteworthy results. The paper first describes the algorithms and the C code. Then it illustrates its power on a number of subspaces of symplectic polar spaces whose rank ranges from 2 to 7. The most interesting new results include: (i) non-contextuality of configurations whose contexts are subspaces of dimension 2 and higher, (ii) non-existence of negative subspaces of dimension 3 and higher, (iii) considerably improved bounds for the contextuality degree of both elliptic and hyperbolic quadrics for rank 4, as well as for a particular subgeometry of the three-qubit space whose contexts are the lines of this space, (iv) proof for the non-contextuality of perpsets and, last but not least, (v) contextual nature of a distinguished subgeometry of a multi-qubit doily, called a two-spread, and computation of its contextuality degree. Finally, in the three-qubit polar space we correct and improve the contextuality degree of the full configuration and also describe finite geometric configurations formed by unsatisfiable/invalid constraints for both types of quadrics as well as for the geometry whose contexts are all 315 lines of the space.</p>","PeriodicalId":49855,"journal":{"name":"Mathematical Structures in Computer Science","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-04-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140609687","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 : 2024-04-18DOI: 10.1017/s0960129524000112
Dohan Kim
Although reasoning about equations over strings has been extensively studied for several decades, little research has been done for equational reasoning on general clauses over strings. This paper introduces a new superposition calculus with strings and present an equational theorem proving framework for clauses over strings. It provides a saturation procedure for clauses over strings and show that the proposed superposition calculus with contraction rules is refutationally complete. In particular, this paper presents a new decision procedure for solving word problems over strings and provides a new method of solving unification problems over strings w.r.t. a set of conditional equations R over strings if R can be finitely saturated under the proposed inference system with contraction rules.
尽管数十年来人们一直在广泛研究弦上方程的推理,但对弦上一般分句的等式推理却鲜有研究。本文介绍了一种新的字符串叠加微积分,并提出了字符串分句的等式定理证明框架。它为字符串上的子句提供了一个饱和程序,并证明了所提出的带有收缩规则的叠加微积分在反驳上是完备的。特别是,本文提出了一种求解弦上文字问题的新决策程序,并提供了一种求解弦上统一问题的新方法,即如果在所提出的带收缩规则的推理系统下 R 可以有限饱和,则可以求解弦上条件方程组 R。
{"title":"Equational theorem proving for clauses over strings","authors":"Dohan Kim","doi":"10.1017/s0960129524000112","DOIUrl":"https://doi.org/10.1017/s0960129524000112","url":null,"abstract":"<p>Although reasoning about equations over strings has been extensively studied for several decades, little research has been done for equational reasoning on general clauses over strings. This paper introduces a new superposition calculus with strings and present an equational theorem proving framework for clauses over strings. It provides a saturation procedure for clauses over strings and show that the proposed superposition calculus with contraction rules is refutationally complete. In particular, this paper presents a new decision procedure for solving word problems over strings and provides a new method of solving unification problems over strings w.r.t. a set of conditional equations <span>R</span> over strings if <span>R</span> can be finitely saturated under the proposed inference system with contraction rules.</p>","PeriodicalId":49855,"journal":{"name":"Mathematical Structures in Computer Science","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-04-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140609589","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 : 2024-04-17DOI: 10.1017/s0960129524000124
Xiaofei Liu, Weidong Li
Let $T=(V,E)$ be a tree in which each edge is assigned a cost; let $mathcal{P}$ be a set of source–sink pairs of vertices in V in which each source–sink pair produces a profit. Given a lower bound K for the profit, the K-prize-collecting multicut problem in trees with submodular penalties is to determine a partial multicut $Msubseteq E$ such that the total profit of the disconnected pairs after removing M from T is at least K, and the total cost of edges in M plus the penalty of the set of still-connected pairs is minimized, where the penalty is determined by a nondecreasing submodular function. Based on the primal-dual scheme, we present a combinatorial polynomial-time algorithm by carefully increasing the penalty. In the theoretical analysis, we prove that the approximation factor of the proposed algorithm is $(frac{8}{3}+frac{4}{3}kappa+varepsilon)$ , where $kappa$ is the total curvature of the submodular function and $varepsilon$ is any fixed positive number. Experiments reveal that the objective value of the solutions generated by the proposed algorithm is less than 130% compared with that of the optimal value in most cases.
让 $T=(V,E)$ 是一棵树,树中的每条边都有一个成本;让 $mathcal{P}$ 是 V 中顶点的源-汇对集合,其中每个源-汇对都产生一个利润。给定利润的下限 K,具有亚模态惩罚的树中的 K-利润收集多切问题就是确定一个部分多切 $Msubseteq E$,使得从 T 中删除 M 后断开的对的总利润至少为 K,并且 M 中的边的总成本加上仍然连接的对的惩罚最小,其中惩罚由一个非递减的亚模态函数决定。基于初等二元方案,我们提出了一种通过谨慎增加惩罚的组合多项式时间算法。在理论分析中,我们证明了所提算法的近似系数为 $(frac{8}{3}+frac{4}{3}kappa+varepsilon)$ ,其中 $kappa$ 是子模函数的总曲率,$varepsilon$ 是任意固定的正数。实验表明,在大多数情况下,拟议算法生成的解的目标值小于最优值的 130%。
{"title":"An approximation algorithm for the -prize-collecting multicut problem in trees with submodular penalties","authors":"Xiaofei Liu, Weidong Li","doi":"10.1017/s0960129524000124","DOIUrl":"https://doi.org/10.1017/s0960129524000124","url":null,"abstract":"Let <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129524000124_inline3.png\" /> <jats:tex-math> $T=(V,E)$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> be a tree in which each edge is assigned a cost; let <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129524000124_inline4.png\" /> <jats:tex-math> $mathcal{P}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> be a set of source–sink pairs of vertices in <jats:italic>V</jats:italic> in which each source–sink pair produces a profit. Given a lower bound <jats:italic>K</jats:italic> for the profit, the <jats:italic>K</jats:italic>-prize-collecting multicut problem in trees with submodular penalties is to determine a partial multicut <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129524000124_inline5.png\" /> <jats:tex-math> $Msubseteq E$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> such that the total profit of the disconnected pairs after removing <jats:italic>M</jats:italic> from <jats:italic>T</jats:italic> is at least <jats:italic>K</jats:italic>, and the total cost of edges in <jats:italic>M</jats:italic> plus the penalty of the set of still-connected pairs is minimized, where the penalty is determined by a nondecreasing submodular function. Based on the primal-dual scheme, we present a combinatorial polynomial-time algorithm by carefully increasing the penalty. In the theoretical analysis, we prove that the approximation factor of the proposed algorithm is <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129524000124_inline6.png\" /> <jats:tex-math> $(frac{8}{3}+frac{4}{3}kappa+varepsilon)$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, where <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129524000124_inline7.png\" /> <jats:tex-math> $kappa$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> is the total curvature of the submodular function and <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129524000124_inline8.png\" /> <jats:tex-math> $varepsilon$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> is any fixed positive number. Experiments reveal that the objective value of the solutions generated by the proposed algorithm is less than 130% compared with that of the optimal value in most cases.","PeriodicalId":49855,"journal":{"name":"Mathematical Structures in Computer Science","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-04-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140609460","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 : 2024-04-15DOI: 10.1017/s0960129524000094
Sina Hazratpour, Emily Riehl
Consider a locally cartesian closed category with an object $mathbb{I}$ and a class of trivial fibrations, which admit sections and are stable under pushforward and retract as arrows. Define the fibrations to be those maps whose Leibniz exponential with the generic point of $mathbb{I}$ defines a trivial fibration. Then the fibrations are also closed under pushforward.
{"title":"A 2-categorical proof of Frobenius for fibrations defined from a generic point","authors":"Sina Hazratpour, Emily Riehl","doi":"10.1017/s0960129524000094","DOIUrl":"https://doi.org/10.1017/s0960129524000094","url":null,"abstract":"Consider a locally cartesian closed category with an object <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129524000094_inline1.png\" /> <jats:tex-math> $mathbb{I}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> and a class of trivial fibrations, which admit sections and are stable under pushforward and retract as arrows. Define the fibrations to be those maps whose Leibniz exponential with the generic point of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129524000094_inline2.png\" /> <jats:tex-math> $mathbb{I}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> defines a trivial fibration. Then the fibrations are also closed under pushforward.","PeriodicalId":49855,"journal":{"name":"Mathematical Structures in Computer Science","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140563818","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 : 2024-04-11DOI: 10.1017/s0960129524000069
Juan Zou, Yuhan Zhao, Cuixia Miao, Longchun Wang
In this paper, the notion of locally algebraic intersection structure is introduced for algebraic L-domains. Essentially, every locally algebraic intersection structure is a family of sets, which forms an algebraic L-domain ordered by inclusion. It is shown that there is a locally algebraic intersection structure which is order-isomorphic to a given algebraic L-domain. This result extends the classic Stone’s representation theorem for Boolean algebras to the case of algebraic L-domains. In addition, it can be seen that many well-known representations of algebraic L-domains, such as logical algebras, information systems, closure spaces, and formal concept analysis, can be analyzed in the framework of locally algebraic intersection structures. Then, a set-theoretic uniformity across different representations of algebraic L-domains is established.
本文为代数 L 域引入了局部代数交集结构的概念。从本质上讲,每一个局部代数交集结构都是一个集合族,它构成了一个按包含有序排列的代数 L 域。研究表明,有一种局部代数交集结构与给定的代数 L 域是有序同构的。这一结果将布尔代数的经典斯通表示定理扩展到了代数 L 域的情况。此外,我们还可以看到,许多著名的代数 L 域表示,如逻辑代数、信息系统、闭包空间和形式概念分析,都可以在局部代数交集结构的框架内进行分析。然后,建立了代数 L 域不同表示的集合论统一性。
{"title":"A set-theoretic approach to algebraic L-domains","authors":"Juan Zou, Yuhan Zhao, Cuixia Miao, Longchun Wang","doi":"10.1017/s0960129524000069","DOIUrl":"https://doi.org/10.1017/s0960129524000069","url":null,"abstract":"In this paper, the notion of locally algebraic intersection structure is introduced for algebraic L-domains. Essentially, every locally algebraic intersection structure is a family of sets, which forms an algebraic L-domain ordered by inclusion. It is shown that there is a locally algebraic intersection structure which is order-isomorphic to a given algebraic L-domain. This result extends the classic Stone’s representation theorem for Boolean algebras to the case of algebraic L-domains. In addition, it can be seen that many well-known representations of algebraic L-domains, such as logical algebras, information systems, closure spaces, and formal concept analysis, can be analyzed in the framework of locally algebraic intersection structures. Then, a set-theoretic uniformity across different representations of algebraic L-domains is established.","PeriodicalId":49855,"journal":{"name":"Mathematical Structures in Computer Science","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-04-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140563806","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 : 2024-03-12DOI: 10.1017/s0960129524000021
Lauri Hella, Kerkko Luosto, Jouko Väänänen
We introduce three measures of complexity for families of sets. Each of the three measures, which we call dimensions, is defined in terms of the minimal number of convex subfamilies that are needed for covering the given family. For upper dimension, the subfamilies are required to contain a unique maximal set, for dual upper dimension a unique minimal set, and for cylindrical dimension both a unique maximal and a unique minimal set. In addition to considering dimensions of particular families of sets, we study the behavior of dimensions under operators that map families of sets to new families of sets. We identify natural sufficient criteria for such operators to preserve the growth class of the dimensions. We apply the theory of our dimensions for proving new hierarchy results for logics with team semantics. To this end we associate each atom with a natural notion or arity. First, we show that the standard logical operators preserve the growth classes of the families arising from the semantics of formulas in such logics. Second, we show that the upper dimension of $k+1$-ary dependence, inclusion, independence, anonymity, and exclusion atoms is in a strictly higher growth class than that of any k-ary atoms, whence the $k+1$-ary atoms are not definable in terms of any atoms of smaller arity.
{"title":"Dimension in team semantics","authors":"Lauri Hella, Kerkko Luosto, Jouko Väänänen","doi":"10.1017/s0960129524000021","DOIUrl":"https://doi.org/10.1017/s0960129524000021","url":null,"abstract":"<p>We introduce three measures of complexity for families of sets. Each of the three measures, which we call dimensions, is defined in terms of the minimal number of convex subfamilies that are needed for covering the given family. For upper dimension, the subfamilies are required to contain a unique maximal set, for dual upper dimension a unique minimal set, and for cylindrical dimension both a unique maximal and a unique minimal set. In addition to considering dimensions of particular families of sets, we study the behavior of dimensions under operators that map families of sets to new families of sets. We identify natural sufficient criteria for such operators to preserve the growth class of the dimensions. We apply the theory of our dimensions for proving new hierarchy results for logics with team semantics. To this end we associate each atom with a natural notion or arity. First, we show that the standard logical operators preserve the growth classes of the families arising from the semantics of formulas in such logics. Second, we show that the upper dimension of <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240311153531332-0533:S0960129524000021:S0960129524000021_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$k+1$</span></span></img></span></span>-ary dependence, inclusion, independence, anonymity, and exclusion atoms is in a strictly higher growth class than that of any <span>k</span>-ary atoms, whence the <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240311153531332-0533:S0960129524000021:S0960129524000021_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$k+1$</span></span></img></span></span>-ary atoms are not definable in terms of any atoms of smaller arity.</p>","PeriodicalId":49855,"journal":{"name":"Mathematical Structures in Computer Science","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-03-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140107620","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 : 2024-03-04DOI: 10.1017/s0960129524000045
Igor Sedlár
We show that Kozen and Tiuryn’s substructural logic of partial correctness $mathsf{S}$ embeds into the equational theory of Kleene algebra with domain, $mathsf{KAD}$. We provide an implicational formulation of $mathsf{KAD}$ which sets $mathsf{S}$ in the context of implicational extensions of Kleene algebra.
{"title":"Implicational Kleene algebra with domain and the substructural logic of partial correctness","authors":"Igor Sedlár","doi":"10.1017/s0960129524000045","DOIUrl":"https://doi.org/10.1017/s0960129524000045","url":null,"abstract":"<p>We show that Kozen and Tiuryn’s substructural logic of partial correctness <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240301160033202-0894:S0960129524000045:S0960129524000045_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$mathsf{S}$</span></span></img></span></span> embeds into the equational theory of Kleene algebra with domain, <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240301160033202-0894:S0960129524000045:S0960129524000045_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$mathsf{KAD}$</span></span></img></span></span>. We provide an implicational formulation of <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240301160033202-0894:S0960129524000045:S0960129524000045_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$mathsf{KAD}$</span></span></img></span></span> which sets <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240301160033202-0894:S0960129524000045:S0960129524000045_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$mathsf{S}$</span></span></img></span></span> in the context of implicational extensions of Kleene algebra.</p>","PeriodicalId":49855,"journal":{"name":"Mathematical Structures in Computer Science","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140025695","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 : 2024-02-20DOI: 10.1017/s0960129524000033
Juha Kontinen, Yasir Mahmood, Arne Meier, Heribert Vollmer
In this article, we study the complexity of weighted team definability for logics with team semantics. This problem is a natural analog of one of the most studied problems in parameterized complexity, the notion of weighted Fagin-definability, which is formulated in terms of satisfaction of first-order formulas with free relation variables. We focus on the parameterized complexity of weighted team definability for a fixed formula $varphi$ of central team-based logics. Given a first-order structure $mathcal{A}$ and the parameter value $kin mathbb N$ as input, the question is to determine whether $mathcal{A},Tmodels varphi$ for some team T of size k. We show several results on the complexity of this problem for dependence, independence, and inclusion logic formulas. Moreover, we also relate the complexity of weighted team definability to the complexity classes in the well-known W-hierarchy as well as paraNP.
本文研究了具有团队语义的逻辑的加权团队可定义性的复杂性。这个问题是参数化复杂性中研究得最多的问题之一--加权法金可定义性概念--的自然类比,它是用具有自由关系变量的一阶公式的满足度来表述的。我们重点研究基于中心团队逻辑的固定公式 $varphi$ 的加权团队可定义性的参数化复杂度。给定一阶结构 $mathcal{A}$ 和参数值 $kin mathbb N$ 作为输入,问题是确定 $mathcal{A},T 是否为某个大小为 k 的团队 T 的 varphi$ 模型。此外,我们还将加权团队可定义性的复杂性与众所周知的 W-层次结构中的复杂性类别以及 paraNP 联系起来。
{"title":"Parameterized complexity of weighted team definability","authors":"Juha Kontinen, Yasir Mahmood, Arne Meier, Heribert Vollmer","doi":"10.1017/s0960129524000033","DOIUrl":"https://doi.org/10.1017/s0960129524000033","url":null,"abstract":"In this article, we study the complexity of weighted team definability for logics with team semantics. This problem is a natural analog of one of the most studied problems in parameterized complexity, the notion of weighted Fagin-definability, which is formulated in terms of satisfaction of first-order formulas with free relation variables. We focus on the parameterized complexity of weighted team definability for a fixed formula <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129524000033_inline1.png\" /> <jats:tex-math> $varphi$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> of central team-based logics. Given a first-order structure <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129524000033_inline2.png\" /> <jats:tex-math> $mathcal{A}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> and the parameter value <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129524000033_inline3.png\" /> <jats:tex-math> $kin mathbb N$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> as input, the question is to determine whether <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129524000033_inline4.png\" /> <jats:tex-math> $mathcal{A},Tmodels varphi$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> for some team <jats:italic>T</jats:italic> of size <jats:italic>k</jats:italic>. We show several results on the complexity of this problem for dependence, independence, and inclusion logic formulas. Moreover, we also relate the complexity of weighted team definability to the complexity classes in the well-known W-hierarchy as well as paraNP.","PeriodicalId":49855,"journal":{"name":"Mathematical Structures in Computer Science","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-02-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139924735","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 : 2024-01-22DOI: 10.1017/s096012952400001x
J. Rosický
On a locally $lambda$-presentable symmetric monoidal closed category $mathcal {V}$, $lambda$-ary enriched equational theories correspond to enriched monads preserving $lambda$-filtered colimits. We introduce discrete $lambda$-ary enriched equational theories where operations are induced by those having discrete arities (equations are not required to have discrete arities) and show that they correspond to enriched monads preserving preserving $lambda$-filtered colimits and surjections. Using it, we prove enriched Birkhof-type theorems for categories of algebras of discrete theories. This extends known results from metric spaces and posets to general symmetric monoidal closed categories.
{"title":"Discrete equational theories","authors":"J. Rosický","doi":"10.1017/s096012952400001x","DOIUrl":"https://doi.org/10.1017/s096012952400001x","url":null,"abstract":"<p>On a locally <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240119162700723-0670:S096012952400001X:S096012952400001X_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$lambda$</span></span></img></span></span>-presentable symmetric monoidal closed category <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240119162700723-0670:S096012952400001X:S096012952400001X_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$mathcal {V}$</span></span></img></span></span>, <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240119162700723-0670:S096012952400001X:S096012952400001X_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$lambda$</span></span></img></span></span>-ary enriched equational theories correspond to enriched monads preserving <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240119162700723-0670:S096012952400001X:S096012952400001X_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$lambda$</span></span></img></span></span>-filtered colimits. We introduce discrete <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240119162700723-0670:S096012952400001X:S096012952400001X_inline5.png\"><span data-mathjax-type=\"texmath\"><span>$lambda$</span></span></img></span></span>-ary enriched equational theories where operations are induced by those having discrete arities (equations are not required to have discrete arities) and show that they correspond to enriched monads preserving preserving <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240119162700723-0670:S096012952400001X:S096012952400001X_inline6.png\"><span data-mathjax-type=\"texmath\"><span>$lambda$</span></span></img></span></span>-filtered colimits and surjections. Using it, we prove enriched Birkhof-type theorems for categories of algebras of discrete theories. This extends known results from metric spaces and posets to general symmetric monoidal closed categories.</p>","PeriodicalId":49855,"journal":{"name":"Mathematical Structures in Computer Science","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-01-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139516927","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 : 2024-01-10DOI: 10.1017/s0960129523000427
Yuxu Chen, Hui Kou, Zhenchao Lyu, Xiaolin Xie
We give a construction of the free dcpo-cone over any dcpo. There are two steps for getting this result. Firstly, we extend the notion of power domain to directed spaces which are equivalent to $T_0$ monotone-determined spaces introduced by Erné, and we construct the probabilistic powerspace of the monotone determined space, which is defined as a free monotone determined cone. Secondly, we take D-completion of the free monotone determined cone over the dcpo with its Scott topology. In addition, we show that generally the valuation power domain of any dcpo is not the free dcpo-cone.
{"title":"A construction of free dcpo-cones","authors":"Yuxu Chen, Hui Kou, Zhenchao Lyu, Xiaolin Xie","doi":"10.1017/s0960129523000427","DOIUrl":"https://doi.org/10.1017/s0960129523000427","url":null,"abstract":"<p>We give a construction of the free dcpo-cone over any dcpo. There are two steps for getting this result. Firstly, we extend the notion of power domain to directed spaces which are equivalent to <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240109154214644-0603:S0960129523000427:S0960129523000427_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$T_0$</span></span></img></span></span> monotone-determined spaces introduced by Erné, and we construct the probabilistic powerspace of the monotone determined space, which is defined as a free monotone determined cone. Secondly, we take D-completion of the free monotone determined cone over the dcpo with its Scott topology. In addition, we show that generally the valuation power domain of any dcpo is not the free dcpo-cone.</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":"139413455","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}