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Consistent disjunctive sequent calculi and Scott domains 一致析取序微积分与Scott域
IF 0.5 4区 计算机科学
Mathematical Structures in Computer Science Pub Date : 2021-08-17 DOI: 10.1017/S0960129521000086
Longchun Wang, Qingguo Li
{"title":"Consistent disjunctive sequent calculi and Scott domains","authors":"Longchun Wang, Qingguo Li","doi":"10.1017/S0960129521000086","DOIUrl":"https://doi.org/10.1017/S0960129521000086","url":null,"abstract":"Abstract Based on the framework of disjunctive propositional logic, we first provide a syntactic representation for Scott domains. Precisely, we establish a category of consistent disjunctive sequent calculi with consequence relations, and show it is equivalent to that of Scott domains with Scott-continuous functions. Furthermore, we illustrate the approach to solving recursive domain equations by introducing some standard domain constructions, such as lifting and sums. The subsystems relation on consistent finitary disjunctive sequent calculi makes these domain constructions continuous. Solutions to recursive domain equations are given by constructing the least fixed point of a continuous function.","PeriodicalId":49855,"journal":{"name":"Mathematical Structures in Computer Science","volume":"32 1","pages":"127 - 150"},"PeriodicalIF":0.5,"publicationDate":"2021-08-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48435907","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}
引用次数: 0
An improved primal-dual approximation algorithm for the k-means problem with penalties 带有惩罚的k-means问题的改进原对偶逼近算法
IF 0.5 4区 计算机科学
Mathematical Structures in Computer Science Pub Date : 2021-08-16 DOI: 10.1017/S0960129521000104
Chunying Ren, Dachuan Xu, D. Du, Min Li
{"title":"An improved primal-dual approximation algorithm for the k-means problem with penalties","authors":"Chunying Ren, Dachuan Xu, D. Du, Min Li","doi":"10.1017/S0960129521000104","DOIUrl":"https://doi.org/10.1017/S0960129521000104","url":null,"abstract":"Abstract In the k-means problem with penalties, we are given a data set $${cal D} subseteq mathbb{R}^ell $$ of n points where each point $$j in {cal D}$$ is associated with a penalty cost pj and an integer k. The goal is to choose a set $${rm{C}}S subseteq {{cal R}^ell }$$ with |CS| ≤ k and a penalized subset $${{cal D}_p} subseteq {cal D}$$ to minimize the sum of the total squared distance from the points in D / Dp to CS and the total penalty cost of points in Dp, namely $$sumnolimits_{j in {cal D}backslash {{cal D}_p}} {d^2}(j,{rm{C}}S) + sumnolimits_{j in {{cal D}_p}} {p_j}$$. We employ the primal-dual technique to give a pseudo-polynomial time algorithm with an approximation ratio of (6.357+ε) for the k-means problem with penalties, improving the previous best approximation ratio 19.849+∊ for this problem given by Feng et al. in Proceedings of FAW (2019).","PeriodicalId":49855,"journal":{"name":"Mathematical Structures in Computer Science","volume":"32 1","pages":"151 - 163"},"PeriodicalIF":0.5,"publicationDate":"2021-08-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48303412","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}
引用次数: 0
Coherence for bicategorical cartesian closed structure 双范畴笛卡尔闭结构的相干性
IF 0.5 4区 计算机科学
Mathematical Structures in Computer Science Pub Date : 2021-08-01 DOI: 10.1017/S0960129521000281
M. Fiore, P. Saville
{"title":"Coherence for bicategorical cartesian closed structure","authors":"M. Fiore, P. Saville","doi":"10.1017/S0960129521000281","DOIUrl":"https://doi.org/10.1017/S0960129521000281","url":null,"abstract":"Abstract We prove a strictification theorem for cartesian closed bicategories. First, we adapt Power’s proof of coherence for bicategories with finite bilimits to show that every bicategory with bicategorical cartesian closed structure is biequivalent to a 2-category with 2-categorical cartesian closed structure. Then we show how to extend this result to a Mac Lane-style “all pasting diagrams commute” coherence theorem: precisely, we show that in the free cartesian closed bicategory on a graph, there is at most one 2-cell between any parallel pair of 1-cells. The argument we employ is reminiscent of that used by Čubrić, Dybjer, and Scott to show normalisation for the simply-typed lambda calculus (Čubrić et al., 1998). The main results first appeared in a conference paper (Fiore and Saville, 2020) but for reasons of space many details are omitted there; here we provide the full development.","PeriodicalId":49855,"journal":{"name":"Mathematical Structures in Computer Science","volume":"31 1","pages":"822 - 849"},"PeriodicalIF":0.5,"publicationDate":"2021-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45689797","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}
引用次数: 2
A special issue on categorical algebras and computation in celebration of John Power’s 60th birthday, part I 为庆祝约翰·鲍尔60岁生日而出版的关于范畴代数和计算的专刊，第一部分
IF 0.5 4区 计算机科学
Mathematical Structures in Computer Science Pub Date : 2021-08-01 DOI: 10.1017/S0960129522000056
Masahito Hasegawa, Stephen Lack, G. McCusker
{"title":"A special issue on categorical algebras and computation in celebration of John Power’s 60th birthday, part I","authors":"Masahito Hasegawa, Stephen Lack, G. McCusker","doi":"10.1017/S0960129522000056","DOIUrl":"https://doi.org/10.1017/S0960129522000056","url":null,"abstract":"John Power, recently retired and now an honorary professor at Macquarie University, turned 60 in December 2019. John hasmade substantial contributions to category theory and its applications to computer science throughout his career. To celebrate John’s achievements, and to inspire further work, two workshops were held in 2019: at Bath on 27 June 2019 (organized by Neil Ghani and Guy McCusker) to mark his retirement, and at Kyoto on 23 December 2019 (organized by Masahito Hasegawa, Ichiro Hasuo, and Makoto Takeyama) to celebrate his 60th birthday. Following the success of these workshops, this Festschrift in honor of John was proposed and a call-for-papers was circulated around April 2020. Among the submissions we received, four are included in this volume. The remaining contributions will appear in the forthcoming second volume of the Festschrift.","PeriodicalId":49855,"journal":{"name":"Mathematical Structures in Computer Science","volume":"31 1","pages":"746 - 747"},"PeriodicalIF":0.5,"publicationDate":"2021-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47335263","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}
引用次数: 0
Doctrines, modalities and comonads 学说、模式和共性
IF 0.5 4区 计算机科学
Mathematical Structures in Computer Science Pub Date : 2021-07-29 DOI: 10.1017/S0960129521000207
Francesco Dagnino, G. Rosolini
{"title":"Doctrines, modalities and comonads","authors":"Francesco Dagnino, G. Rosolini","doi":"10.1017/S0960129521000207","DOIUrl":"https://doi.org/10.1017/S0960129521000207","url":null,"abstract":"Abstract Doctrines are categorical structures very apt to study logics of different nature within a unified environment: the 2-category Dtn of doctrines. Modal interior operators are characterised as particular adjoints in the 2-category Dtn. We show that they can be constructed from comonads in Dtn as well as from adjunctions in it, and we compare the two constructions. Finally we show the amount of information lost in the passage from a comonad, or from an adjunction, to the modal interior operator. The basis for the present work is provided by some seminal work of John Power.","PeriodicalId":49855,"journal":{"name":"Mathematical Structures in Computer Science","volume":"31 1","pages":"769 - 798"},"PeriodicalIF":0.5,"publicationDate":"2021-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46515017","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}
引用次数: 7
A rewriting coherence theorem with applications in homotopy type theory 一个改写相干定理及其在同伦型理论中的应用
IF 0.5 4区 计算机科学
Mathematical Structures in Computer Science Pub Date : 2021-07-04 DOI: 10.1017/S0960129523000026
Nicolai Kraus, Jakob von Raumer
{"title":"A rewriting coherence theorem with applications in homotopy type theory","authors":"Nicolai Kraus, Jakob von Raumer","doi":"10.1017/S0960129523000026","DOIUrl":"https://doi.org/10.1017/S0960129523000026","url":null,"abstract":"Abstract Higher-dimensional rewriting systems are tools to analyse the structure of formally reducing terms to normal forms, as well as comparing the different reduction paths that lead to those normal forms. This higher structure can be captured by finding a homotopy basis for the rewriting system. We show that the basic notions of confluence and wellfoundedness are sufficient to recursively build such a homotopy basis, with a construction reminiscent of an argument by Craig C. Squier. We then go on to translate this construction to the setting of homotopy type theory, where managing equalities between paths is important in order to construct functions which are coherent with respect to higher dimensions. Eventually, we apply the result to approximate a series of open questions in homotopy type theory, such as the characterisation of the homotopy groups of the free group on a set and the pushout of 1-types. This paper expands on our previous conference contribution Coherence via Wellfoundedness by laying out the construction in the language of higher-dimensional rewriting.","PeriodicalId":49855,"journal":{"name":"Mathematical Structures in Computer Science","volume":"32 1","pages":"982 - 1014"},"PeriodicalIF":0.5,"publicationDate":"2021-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45446480","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}
引用次数: 2
Martin Hofmann’s contributions to type theory: Groupoids and univalence 马丁·霍夫曼对类型理论的贡献:类群和一元性
IF 0.5 4区 计算机科学
Mathematical Structures in Computer Science Pub Date : 2021-06-28 DOI: 10.1017/S0960129520000316
Thorsten Altenkirch
{"title":"Martin Hofmann’s contributions to type theory: Groupoids and univalence","authors":"Thorsten Altenkirch","doi":"10.1017/S0960129520000316","DOIUrl":"https://doi.org/10.1017/S0960129520000316","url":null,"abstract":"Abstract My goal is to give an accessible introduction to Martin’s work on the groupoid model and how it is related to the recent notion of univalence in Homotopy Type Theory while sharing some memories of Martin.","PeriodicalId":49855,"journal":{"name":"Mathematical Structures in Computer Science","volume":"31 1","pages":"953 - 957"},"PeriodicalIF":0.5,"publicationDate":"2021-06-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1017/S0960129520000316","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47966794","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}
引用次数: 1
Apartness, sharp elements, and the Scott topology of domains 空间性、尖锐元素和域的Scott拓扑
IF 0.5 4区 计算机科学
Mathematical Structures in Computer Science Pub Date : 2021-06-09 DOI: 10.1017/S0960129523000282
Tom de Jong
{"title":"Apartness, sharp elements, and the Scott topology of domains","authors":"Tom de Jong","doi":"10.1017/S0960129523000282","DOIUrl":"https://doi.org/10.1017/S0960129523000282","url":null,"abstract":"\u0000 Working constructively, we study continuous directed complete posets (dcpos) and the Scott topology. Our two primary novelties are a notion of intrinsic apartness and a notion of sharp elements. Being apart is a positive formulation of being unequal, similar to how inhabitedness is a positive formulation of nonemptiness. To exemplify sharpness, we note that a lower real is sharp if and only if it is located. Our first main result is that for a large class of continuous dcpos, the Bridges–Vîţǎ apartness topology and the Scott topology coincide. Although we cannot expect a tight or cotransitive apartness on nontrivial dcpos, we prove that the intrinsic apartness is both tight and cotransitive when restricted to the sharp elements of a continuous dcpo. These include the strongly maximal elements, as studied by Smyth and Heckmann. We develop the theory of strongly maximal elements highlighting its connection to sharpness and the Lawson topology. Finally, we illustrate the intrinsic apartness, sharpness, and strong maximality by considering several natural examples of continuous dcpos: the Cantor and Baire domains, the partial Dedekind reals, the lower reals and, finally, an embedding of Cantor space into an exponential of lifted sets.","PeriodicalId":49855,"journal":{"name":"Mathematical Structures in Computer Science","volume":"1 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2021-06-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46266401","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}
引用次数: 1
String diagram rewrite theory II: Rewriting with symmetric monoidal structure 弦图改写理论II:对称单轴结构的改写
IF 0.5 4区 计算机科学
Mathematical Structures in Computer Science Pub Date : 2021-04-29 DOI: 10.1017/S0960129522000317
F. Bonchi, F. Gadducci, A. Kissinger, P. Sobocinski, F. Zanasi
{"title":"String diagram rewrite theory II: Rewriting with symmetric monoidal structure","authors":"F. Bonchi, F. Gadducci, A. Kissinger, P. Sobocinski, F. Zanasi","doi":"10.1017/S0960129522000317","DOIUrl":"https://doi.org/10.1017/S0960129522000317","url":null,"abstract":"Abstract Symmetric monoidal theories (SMTs) generalise algebraic theories in a way that make them suitable to express resource-sensitive systems, in which variables cannot be copied or discarded at will. In SMTs, traditional tree-like terms are replaced by string diagrams, topological entities that can be intuitively thought of as diagrams of wires and boxes. Recently, string diagrams have become increasingly popular as a graphical syntax to reason about computational models across diverse fields, including programming language semantics, circuit theory, quantum mechanics, linguistics, and control theory. In applications, it is often convenient to implement the equations appearing in SMTs as rewriting rules. This poses the challenge of extending the traditional theory of term rewriting, which has been developed for algebraic theories, to string diagrams. In this paper, we develop a mathematical theory of string diagram rewriting for SMTs. Our approach exploits the correspondence between string diagram rewriting and double pushout (DPO) rewriting of certain graphs, introduced in the first paper of this series. Such a correspondence is only sound when the SMT includes a Frobenius algebra structure. In the present work, we show how an analogous correspondence may be established for arbitrary SMTs, once an appropriate notion of DPO rewriting (which we call convex) is identified. As proof of concept, we use our approach to show termination of two SMTs of interest: Frobenius semi-algebras and bialgebras.","PeriodicalId":49855,"journal":{"name":"Mathematical Structures in Computer Science","volume":"32 1","pages":"511 - 541"},"PeriodicalIF":0.5,"publicationDate":"2021-04-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45722169","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}
引用次数: 18
On reduction and normalization in the computational core 计算核中的约简与归一化
IF 0.5 4区 计算机科学
Mathematical Structures in Computer Science Pub Date : 2021-04-20 DOI: 10.1017/S0960129522000433
C. Faggian, Giulio Guerrieri, U. De 'liguoro, R. Treglia
{"title":"On reduction and normalization in the computational core","authors":"C. Faggian, Giulio Guerrieri, U. De 'liguoro, R. Treglia","doi":"10.1017/S0960129522000433","DOIUrl":"https://doi.org/10.1017/S0960129522000433","url":null,"abstract":"Abstract We study the reduction in a \u0000$lambda$\u0000 -calculus derived from Moggi’s computational one, which we call the computational core. The reduction relation consists of rules obtained by orienting three monadic laws. Such laws, in particular associativity and identity, introduce intricacies in the operational analysis. We investigate the central notions of returning a value versus having a normal form and address the question of normalizing strategies. Our analysis relies on factorization results.","PeriodicalId":49855,"journal":{"name":"Mathematical Structures in Computer Science","volume":"32 1","pages":"934 - 981"},"PeriodicalIF":0.5,"publicationDate":"2021-04-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43751443","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}
引用次数: 7
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