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Monoidal weak ω-categories as models of a type theory 一元弱范畴作为类型论的模型
IF 0.5 4区 计算机科学 Q2 Mathematics Pub Date : 2022-06-27 DOI: 10.1017/s0960129522000172
Thibaut Benjamin
Weak $omega$ -categories are notoriously difficult to define because of the very intricate nature of their axioms. Various approaches have been explored based on different shapes given to the cells. Interestingly, homotopy type theory encompasses a definition of weak $omega$ -groupoid in a globular setting, since every type carries such a structure. Starting from this remark, Brunerie could extract this definition of globular weak $omega$ -groupoids, formulated as a type theory. By refining its rules, Finster and Mimram have then defined a type theory called $mathsf{CaTT}$ , whose models are weak $omega$ -categories. Here, we generalize this approach to monoidal weak $omega$ -categories. Based on the principle that they should be equivalent to weak $omega$ -categories with only one 0-cell, we are able to derive a type theory $mathsf{MCaTT}$ whose models are monoidal weak $omega$ -categories. This requires changing the rules of the theory in order to encode the information carried by the unique 0-cell. The correctness of the resulting type theory is shown by defining a pair of translations between our type theory $mathsf{MCaTT}$ and the type theory $mathsf{CaTT}$ . Our main contribution is to show that these translations relate the models of our type theory to the models of the type theory $mathsf{CaTT}$ consisting of $omega$ -categories with only one 0-cell by analyzing in details how the notion of models interact with the structural rules of both type theories.
众所周知,弱$omega$类别很难定义,因为它们的公理非常复杂。基于赋予细胞的不同形状,已经探索了各种方法。有趣的是,同伦型理论包含了球状环境中弱$omega$-群胚的定义,因为每个类型都有这样的结构。从这句话开始,Brunerie可以提取出球状弱$omega$-群胚的定义,公式化为一种类型理论。通过完善其规则,Finster和Mimram定义了一个称为$mathsf{CaTT}$的类型理论,其模型是弱$omega$类别。在这里,我们将这种方法推广到单模态弱$omega$-范畴。基于它们应该等价于只有一个0-单元的弱$omega$-类别的原理,我们能够导出类型论$mathsf{MCaTT}$,其模型是单模弱$omega$-类。这需要改变理论的规则,以便对唯一的0单元所携带的信息进行编码。通过定义我们的类型论$mathsf{MCaTT}$和类型论$athsf{CaTT}$之间的一对翻译,表明了所得类型论的正确性。我们的主要贡献是通过详细分析模型的概念如何与两种类型理论的结构规则相互作用,表明这些翻译将我们的类型理论模型与类型理论模型$mathsf{CaTT}$联系起来,该模型由只有一个0单元的$omega$类别组成。
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引用次数: 2
Preface for the special issue in homage to Martin Hofmann Part 2 纪念马丁·霍夫曼特刊序言第二部分
IF 0.5 4区 计算机科学 Q2 Mathematics Pub Date : 2022-06-01 DOI: 10.1017/s0960129522000147
Jan Hoffmann, D. Sannella, Ulrich Schöpp
This is the second part of a two-part special issue dedicated to the memory of our friend and colleague, Martin Hofmann. The first part was published asMathematical Structures in Computer Science (2021), 31(9). On 21 January 2018, Martin Hofmann died in a tragic mountain hiking accident in Japan. He was there to attend a workshop at NII Shonan and arrived early for the workshop in order to spend a day climbing Mount Nikkō-Shirane. On his way down from the 2578 m summit, he was caught in a severe snowstorm and lost his way back to safety. Martin Hofmann studied for a Diplom in Informatics at Universität Erlangen-Nürnberg from November 1984 until August 1991. During an exchange visit at the Université de Nice from October 1987 to June 1988 he obtained in addition the “Maitrise de Mathematiques.” In 1991, he joined the Laboratory for Foundations of Computer Science at the University of Edinburgh. He
这是纪念我们的朋友和同事马丁·霍夫曼的两期特刊的第二部分。第一部分发表在《计算机科学中的数学结构》(2021),31(9)。2018年1月21日,马丁·霍夫曼在日本的一次登山事故中不幸去世。他是为了参加NII Shonan的研讨会而来的,为了花一天时间攀登Nikkō-Shirane山,他提前到了研讨会现场。在从海拔2578米的山顶下山的途中,他遇到了一场严重的暴风雪,失去了返回安全地带的路。从1984年11月到1991年8月,Martin Hofmann在Universität erlangen - n rnberg大学攻读信息学文凭。1987年10月至1988年6月在尼斯大学交换访问期间,他还获得了“Maitrise de Mathematiques”学位。1991年,他加入了爱丁堡大学计算机科学基础实验室。他
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引用次数: 0
An algebraic representation of the fixed-point closure of *-continuous Kleene algebras – A categorical Chomsky–Schützenberger theorem *-连续Kleene代数不动点闭包的代数表示——范畴Chomsky–Schützenberger定理
IF 0.5 4区 计算机科学 Q2 Mathematics Pub Date : 2022-06-01 DOI: 10.1017/S0960129522000329
Hans Leiß
Abstract The family ${mathcal{R}} X^*$ of regular subsets of the free monoid $X^*$ generated by a finite set X is the standard example of a ${}^*$ -continuous Kleene algebra. Likewise, the family ${mathcal{C}} X^*$ of context-free subsets of $X^*$ is the standard example of a $mu$ -continuous Chomsky algebra, i.e. an idempotent semiring that is closed under a well-behaved least fixed-point operator $mu$ . For arbitrary monoids M, ${mathcal{C}} M$ is the closure of ${mathcal{R}}M$ as a $mu$ -continuous Chomsky algebra, more briefly, the fixed-point closure of ${mathcal{R}} M$ . We provide an algebraic representation of ${mathcal{C}} M$ in a suitable product of ${mathcal{R}} M$ with $C_2'$ , a quotient of the regular sets over an alphabet $Delta_2$ of two pairs of bracket symbols. Namely, ${mathcal{C}}M$ is isomorphic to the centralizer of $C_2'$ in the product of ${mathcal{R}} M$ with $C_2'$ , i.e. the set of those elements that commute with all elements of $C_2'$ . This generalizes a well-known result of Chomsky and Schützenberger (1963, Computer Programming and Formal Systems, 118–161) and admits us to denote all context-free languages over finite sets $Xsubseteq M$ by regular expressions over $XcupDelta_2$ interpreted in the product of ${mathcal{R}} M$ and $C_2'$ . More generally, for any ${}^*$ -continuous Kleene algebra K the fixed-point closure of K can be represented algebraically as the centralizer of $C_2'$ in the product of K with $C_2'$ .
由有限集X生成的自由幺半群$X^*$的正则子集的族${mathcal{R}}X ^*$是连续Kleene代数的标准例子。同样,$X^*$的上下文无关子集的族${mathcal{C}}X ^*$是$mu$连续Chomsky代数的标准例子,即在表现良好的最不定点算子$mu$$下闭合的幂等半环。对于任意monoids M,${mathcal{C}}M$是作为$mu$连续的Chomsky代数的${ mathcal{R}M$$的闭包,更简单地说,是${mathcal}M$1的定点闭包。我们在${mathcal{R}}M$与$C_2'$的适当乘积中提供了${ mathcal{C}}M$的代数表示,$C_2'$是两对括号符号的字母表$Delta_2$上的正则集的商。也就是说,${mathcal{C}}M$同构于${ mathcal{R}M$C_2'$与$C_2'$$的乘积中的$C_2'$的中心化子,即与$C_2'$的所有元素交换的那些元素的集合。这推广了Chomsky和Schützenberger(1963,Computer Programming and Formal Systems,118-161)的一个著名结果,并允许我们用$XcupDelta_2$上的正则表达式来表示有限集$XsubsteqM$上的所有上下文无关语言,这些正则表达式被解释为${mathcal{R}}M$和$C_2'$的乘积。更一般地,对于任何${}^*$-连续的Kleene代数K,K的不动点闭包可以代数地表示为K与$C_2'$的乘积中的$C_2'$。
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引用次数: 0
Efficient representation of piecewise linear functions into Łukasiewicz logic modulo satisfiability 分段线性函数的有效表示成Łukasiewicz逻辑模可满足性
IF 0.5 4区 计算机科学 Q2 Mathematics Pub Date : 2022-05-17 DOI: 10.1017/S096012952200010X
Sandro Preto, M. Finger
Abstract This work concerns the representation of a class of continuous functions into Logic, so that one may automatically reason about properties of these functions using logical tools. Rational McNaughton functions may be implicitly represented by logical formulas in Łukasiewicz Infinitely-valued Logic by constraining the set of allowed valuations; such a restriction contemplates only those valuations that satisfy specific formulas. This work investigates two approaches to such depiction, called representation modulo satisfiability. Furthermore, a polynomial-time algorithm that builds this representation is presented, producing a pair of formulas consisting of the representative formula and the constraining one, given as input a rational McNaughton function in a suitable encoding. An implementation of the algorithm is discussed.
摘要这项工作涉及一类连续函数在逻辑中的表示,以便使用逻辑工具自动推理这些函数的性质。Rational McNaughton函数可以通过约束允许估值的集合,由Łukasiewicz无穷值逻辑中的逻辑公式隐式表示;这种限制只考虑那些满足特定公式的估值。这项工作研究了两种描述方法,称为表示模可满足性。此外,还提出了一种建立这种表示的多项式时间算法,产生了一对由代表公式和约束公式组成的公式,并以适当的编码将有理McNaughton函数作为输入。讨论了该算法的实现。
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引用次数: 1
Some representations of real numbers using integer sequences 实数的一些整数序列表示
IF 0.5 4区 计算机科学 Q2 Mathematics Pub Date : 2022-05-01 DOI: 10.1017/S0960129522000342
L. Mazo, Marie-Andrée Da Col-Jacob, Laurent Fuchs, Nicolas Magaud, Gaëlle Skapin
Abstract The paper describes three models of the real field based on subsets of the integer sequences. The three models are compared to the Harthong–Reeb line. Two of the new models, contrary to the Harthong–Reeb line, provide accurate integer “views” on real numbers at a sequence of growing scales $B^n$ ( $Bge2$ ).
摘要本文描述了基于整数序列子集的实域的三个模型。这三款车型与Harthong-Reeb系列进行了比较。与Harthong–Reeb线相反,其中两个新模型在一系列增长尺度$B^n$($Bge2$)上提供了实数的精确整数“视图”。
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引用次数: 0
A special issue on categorical algebras and computation in celebration of John Power’s 60th birthday, part II 庆祝约翰·鲍尔60岁生日的范畴代数与计算特刊,第二部分
IF 0.5 4区 计算机科学 Q2 Mathematics Pub Date : 2022-04-01 DOI: 10.1017/s0960129522000391
Masahito Hasegawa, Stephen Lack, G. McCusker
This volume is the second part of the Festschrift in honour of John Power, who turned 60 in December 2019. For a brief history of John and his work, we refer to the preface of the first volume of the Festschrift (Mathematical Structures in Computer Science (2021), 31(7)). Reflecting John’s influence on category theory and its applications to computer science throughout his career, we received a number of submissions on a wide range of topics. All the submitted papers have been fully peer-reviewed to the usual standards of the journal, and we finally accepted 11 papers. Among them, four papers already appeared in the first part. This volume consists of the remaining seven papers. We are very grateful to the authors and reviewers whose hard work was essential in preparing this special issue. We would like to thank all of them for their help and patience.
本卷是纪念2019年12月年满60岁的约翰·鲍尔的Festschrift的第二部分。关于约翰及其工作的简史,我们可以参考Festschrift第一卷的序言(计算机科学中的数学结构(2021),31(7))。在约翰的整个职业生涯中,他对范畴理论及其在计算机科学中的应用产生了影响,我们收到了许多关于广泛主题的意见书。所有提交的论文都按照期刊的常规标准进行了全面的同行评审,我们最终接受了11篇论文。其中,第一部分已经出现了四篇论文。本卷由其余七篇论文组成。我们非常感谢作者和审稿人,他们的辛勤工作对编写本特刊至关重要。我们要感谢他们所有人的帮助和耐心。
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引用次数: 0
Complete algebraic semantics for second-order rewriting systems based on abstract syntax with variable binding 基于变量绑定抽象语法的二阶重写系统的完全代数语义
IF 0.5 4区 计算机科学 Q2 Mathematics Pub Date : 2022-04-01 DOI: 10.1017/S0960129522000287
M. Hamana
Abstract By using algebraic structures in a presheaf category over finite sets, following Fiore, Plotkin and Turi, we develop sound and complete models of second-order rewriting systems called second-order computation systems (CSs). Restricting the algebraic structures to those equipped with well-founded relations, we obtain a complete characterisation of terminating CSs. We also extend the characterisation to rewriting on meta-terms using the notion of $Sigma$ -monoid.
摘要继Fiore、Plotkin和Turi之后,我们利用有限集上预剪切范畴中的代数结构,建立了二阶重写系统的健全完整模型,称为二阶计算系统(CS)。将代数结构限制为那些具有良好基础关系的代数结构,我们获得了终止CS的完整刻画。我们还使用$Sigma$-monoid的概念将特征化扩展到元项上的重写。
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引用次数: 0
Hom weak ω-categories of a weak ω-category 弱ω-类别的弱ω-类别
IF 0.5 4区 计算机科学 Q2 Mathematics Pub Date : 2022-04-01 DOI: 10.1017/S0960129522000111
Thomas Cottrell, Soichiro Fujii
Abstract Classical definitions of weak higher-dimensional categories are given inductively, for example, a bicategory has a set of objects and hom categories, and a tricategory has a set of objects and hom bicategories. However, more recent definitions of weak n-categories for all natural numbers n, or of weak $omega$ -categories, take more sophisticated approaches, and the nature of the ‘hom is often not immediate from the definitions’. In this paper, we focus on Leinster’s definition of weak $omega$ -category based on an earlier definition by Batanin and construct, for each weak $omega$ -category $mathcal{A}$ , an underlying (weak $omega$ -category)-enriched graph consisting of the same objects and for each pair of objects x and y, a hom weak $omega$ -category $mathcal{A}(x,y)$ . We also show that our construction is functorial with respect to weak $omega$ -functors introduced by Garner.
摘要归纳地给出了弱高维范畴的经典定义,如双范畴有一组对象和若干范畴,三范畴有一组对象和若干范畴。然而,最近对所有自然数n的弱n-类别的定义,或弱$ $ -类别的定义,采用了更复杂的方法,并且“家”的性质通常不是直接从定义中得到的。本文在Batanin先前的定义基础上,重点讨论了Leinster对弱$omega$ -范畴的定义,并构造了对于每一个弱$omega$ -范畴$mathcal{A}$,一个由相同对象组成的底层(弱$omega$ -范畴)富图,对于每一对对象x和y,一个弱$omega$ -范畴$mathcal{A}(x,y)$。我们也证明了我们的构造对于加纳引入的弱函子是泛函的。
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引用次数: 1
E-Unification based on Generalized Embedding 基于广义嵌入的e -统一
IF 0.5 4区 计算机科学 Q2 Mathematics Pub Date : 2022-03-24 DOI: 10.1017/s0960129522000019
P. Szabó, J. Siekmann
Ordering is a well-established concept in mathematics and also plays an important role in many areas of computer science, where quasi-orderings, most notably well-founded quasi-orderings and well-quasi-orderings, are of particular interest. This paper deals with quasi-orderings on first-order terms and introduces a new notion of unification based on a special quasi-order, known as homeomorphic tree embedding. Historically, the development of unification theory began with the central notion of a most general unifier based on the subsumption order. A unifier $sigma$ is most general, if it subsumes any other unifier $tau$ , that is, if there is a substitution $lambda$ with $tau=_{E}sigmalambda$ , where E is an equational theory and $=_{E}$ denotes equality under E. Since there is in general more than one most general unifier for unification problems under equational theories E, called E-Unification, we have the notion of a complete and minimal set of unifiers under E for a unification problem $varGamma$ , denoted as $mumathcal{U}Sigma_{E}(Gamma)$ . This set is still the basic notion in unification theory today. But, unfortunately, the subsumption quasi-order is not a well-founded quasi-order, which is the reason why for certain equational theories there are solvable E-unification problems, but the set $mumathcal{U}Sigma_{E}(Gamma)$ does not exist. They are called type nullary in the unification hierarchy. In order to overcome this problem and also to substantially reduce the number of most general unifiers, we extended the well-known encompassment order on terms to an encompassment order on substitutions (modulo E). Unification under the encompassment order is called essential unification and if $mumathcal{U}Sigma_{E}(Gamma)$ exists, then the complete set of essential unifiers $emathcal{U}Sigma_{E}(Gamma)$ is a subset of $mumathcal{U}Sigma_{E}(Gamma)$ . An interesting effect is that many E-unification problems with an infinite set of most general unifiers (under the subsumption order) reduce to a problem with only finitely many essential unifiers. Moreover, there are cases of an equational theory E, for which the complete set of most general unifiers does not exist, the minimal and complete set of essential unifiers however does exist. Unfortunately again, the encompassment order is not a well-founded quasi-ordering either, that is, there are still theories with a solvable unification problem, for which a minimal and complete set of essential unifiers does not exist. This paper deals with a third approach, namely the extension of the well-known homeomorphic embedding of terms to a homeomorphic embedding of substitutions (modulo E). We examine the set of most general, minimal, and complete E-unifiers under the quasi-order of homeomorphic embedment modulo an equational theory E, called $varphi USigma_{E}(Gamma)$ , and p
排序在数学中是一个公认的概念,在计算机科学的许多领域也起着重要的作用,其中准排序,特别是有充分根据的准排序和良好的准排序,是特别感兴趣的。本文讨论了一阶项上的拟序,并引入了一种新的基于一种特殊的拟序的统一概念,即同胚树嵌入。历史上,统一理论的发展始于一个基于包容秩序的最普遍统一者的中心概念。一个统一子$sigma$是最一般的,如果它包含任何其他统一子$tau$,也就是说,如果用$tau=_{E}sigmalambda$替换$lambda$,其中E是一个方程理论,$=_{E}$表示E下的相等性,因为在方程理论E下的统一问题通常有不止一个最一般的统一子,称为E-统一,对于统一问题$varGamma$,我们有E下统一器的完备最小集的概念,记为$mumathcal{U}Sigma_{E}(Gamma)$。这个集合在今天仍然是统一理论的基本概念。但是,不幸的是,包含拟序并不是一个建立良好的拟序,这就是为什么某些方程理论存在可解的e -统一问题,而集合$mumathcal{U}Sigma_{E}(Gamma)$不存在的原因。在统一层次结构中,它们被称为nullary类型。为了克服这一问题,也为了大大减少大多数一般统一子的数量,我们将众所周知的项上的包涵顺序推广到替换上的包涵顺序(模E)。包涵顺序下的统一称为本质统一,如果$mumathcal{U}Sigma_{E}(Gamma)$存在,则本质统一子的完备集$emathcal{U}Sigma_{E}(Gamma)$是$mumathcal{U}Sigma_{E}(Gamma)$的一个子集。一个有趣的结果是,许多具有无限个最一般统一子集的e -统一问题(在包含阶下)简化为只有有限个基本统一子的问题。此外,在等式理论E的某些情况下,大多数一般统一子的完备集不存在,但是存在基本统一子的极小和完备集。不幸的是,包涵序也不是一个有充分根据的拟序,也就是说,仍然存在具有可解的统一问题的理论,对于这些理论,基本统一子的最小和完全集合是不存在的。本文讨论了第三种方法,即将众所周知的同胚嵌入项推广到替换的同胚嵌入(模E)。我们研究了同胚嵌入模E的准阶下的最一般、最小和完全E统一子集合,称为$varphi USigma_{E}(Gamma)$。并提出了一个合适的定义框架,该框架基于统一理论的标准概念,由树嵌入定理或Kruskal定理的概念扩展。主要结果是:对于正则理论,极小集和完备集$varphimathcal{U}Sigma_{E}(Gamma)$总是存在的。如果我们将E-嵌入顺序限制为纯E-嵌入,这是一种众所周知的逻辑编程和项重写技术,其中变量之间的差异被忽略,集合$varphi_{pi}mathcal{U}Sigma_{E}(Gamma)$总是存在的,甚至对任何理论E都是有限的。
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引用次数: 1
Monoidal reverse differential categories 一元反微分范畴
IF 0.5 4区 计算机科学 Q2 Mathematics Pub Date : 2022-03-23 DOI: 10.1017/S096012952200038X
G. Cruttwell, J. Gallagher, J. Lemay, D. Pronk
Abstract Cartesian reverse differential categories (CRDCs) are a recently defined structure which categorically model the reverse differentiation operations used in supervised learning. Here, we define a related structure called a monoidal reverse differential category, prove important results about its relationship to CRDCs, and provide examples of both structures, including examples coming from models of quantum computation.
笛卡尔反微分范畴(CRDCs)是最近定义的一种结构,它对监督学习中使用的反微分操作进行了明确的建模。在这里,我们定义了一种称为单体反微分范畴的相关结构,证明了它与CRDC关系的重要结果,并提供了这两种结构的例子,包括来自量子计算模型的例子。
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引用次数: 1
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Mathematical Structures in Computer Science
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