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Confluence of left-linear higher-order rewrite theories by checking their nested critical pairs 用嵌套临界对检验左线性高阶重写理论的合流性
IF 0.5 4区 计算机科学 Q2 Mathematics Pub Date : 2022-03-17 DOI: 10.1017/S0960129522000044
Gilles Dowek, Gaspard Férey, J. Jouannaud, Jiaxiang Liu
Abstract User-defined higher-order rewrite rules are becoming a standard in proof assistants based on intuitionistic type theory. This raises the question of proving that they preserve the properties of beta-reductions for the corresponding type systems. In a series of papers, we develop techniques based on van Oostrom’s decreasing diagrams that reduce confluence proofs to the checking of various forms of critical pairs for higher-order rewrite rules extending beta-reduction on pure lambda-terms. As shown in a previous paper of the two middle authors, confluence of a terminating set of left-linear rewrite rules is obtained when their critical pairs are joinable, beta-rewrite steps being disallowed. The present paper concentrates on the case where arbitrary beta-rewrite steps are allowed for joining critical pairs. The rewrite relation used for analyzing confluence may rewrite arbitrarily many non-overlapping redexes in a single step. This relation gives rise to critical pairs that overlap both horizontally, as with parallel rewriting, but also vertically, forming chains of successive overlaps. Practical examples of use of this technique are analyzed.
摘要用户定义的高阶重写规则正在成为基于直觉类型理论的证明助手的标准。这就提出了一个问题,即证明它们保留了相应类型系统的贝塔约简的性质。在一系列论文中,我们开发了基于van Oostrom递减图的技术,这些技术将汇流证明简化为检查各种形式的临界对,以用于在纯lambda项上扩展贝塔约简的高阶重写规则。正如两位中间作者之前的一篇论文所示,当左线性重写规则的关键对是可连接的,不允许贝塔重写步骤时,可以获得左线性重写的终止集的汇合。本文集中讨论了允许任意贝塔重写步骤来连接关键对的情况。用于分析合流的重写关系可以在一个步骤中重写任意多个不重叠的redex。这种关系产生了临界对,它们既在水平方向重叠,就像平行重写一样,也在垂直方向重叠,形成连续重叠的链。分析了该技术的应用实例。
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引用次数: 3
Two decades of automatic amortized resource analysis 二十年的自动摊销资源分析
IF 0.5 4区 计算机科学 Q2 Mathematics Pub Date : 2022-03-16 DOI: 10.1017/S0960129521000487
Jan Hoffmann, Steffen Jost
Abstract This article gives an overview of automatic amortized resource analysis (AARA), a technique for inferring symbolic resource bounds for programs at compile time. AARA has been introduced by Hofmann and Jost in 2003 as a type system for deriving linear worst-case bounds on the heap-space consumption of first-order functional programs with eager evaluation strategy. Since then AARA has been the subject of dozens of research articles, which extended the analysis to different resource metrics, other evaluation strategies, non-linear bounds, and additional language features. All these works preserved the defining characteristics of the original paper: local inference rules, which reduce bound inference to numeric (usually linear) optimization; a soundness proof with respect to an operational cost semantics; and the support of amortized analysis with the potential method.
摘要本文概述了自动摊销资源分析(AARA),这是一种在编译时推断程序符号资源边界的技术。Hofmann和Jost在2003年引入了AARA,作为一种类型系统,用于导出具有热切评估策略的一阶函数程序堆空间消耗的线性最坏情况边界。从那时起,AARA已经成为数十篇研究文章的主题,这些文章将分析扩展到不同的资源度量、其他评估策略、非线性边界和其他语言特征。所有这些工作都保留了原始论文的定义特征:局部推理规则,它将边界推理简化为数值(通常是线性)优化;关于操作成本语义的健全性证明;以及潜在法对摊销分析的支持。
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引用次数: 3
On Petrie cycle and Petrie tour partitions of 3- and 4-regular plane graphs 关于3-和4-正则平面图的Petrie环和Petrie tour划分
IF 0.5 4区 计算机科学 Q2 Mathematics Pub Date : 2022-02-01 DOI: 10.1017/S0960129522000238
Xin He, Huaming Zhang, Yijie Han
Abstract Given a plane graph $G=(V,E)$ , a Petrie tour of G is a tour P of G that alternately turns left and right at each step. A Petrie tour partition of G is a collection ${mathscr P}={P_1,ldots,P_q}$ of Petrie tours so that each edge of G is in exactly one tour $P_i in {mathscr P}$ . A Petrie tour P is called a Petrie cycle if all its vertices are distinct. A Petrie cycle partition of G is a collection ${mathscr C}={C_1,ldots,C_p}$ of Petrie cycles so that each vertex of G is in exactly one cycle $C_i in {mathscr C}$ . In this paper, we study the properties of 3-regular plane graphs that have Petrie cycle partitions and 4-regular plane multi-graphs that have Petrie tour partitions. Given a 4-regular plane multi-graph $G=(V,E)$ , a 3-regularization of G is a 3-regular plane graph $G_3$ obtained from G by splitting every vertex $vin V$ into two degree-3 vertices. G is called Petrie partitionable if it has a 3-regularization that has a Petrie cycle partition. The general version of this problem is motivated by a data compression method, tristrip, used in computer graphics. In this paper, we present a simple characterization of Petrie partitionable graphs and show that the problem of determining if G is Petrie partitionable is NP-complete.
摘要给定平面图$G=(V,E)$, G的Petrie巡回是每一步交替向左和向右转的G的巡回P。G的Petrie tour分区是Petrie tour的集合${mathscr P}={P_1,ldots,P_q}$,使得G的每条边都恰好在一个tour $P_i in {mathscr P}$中。如果一个皮特里环的顶点都是不同的,那么它就叫做皮特里环。G的Petrie环划分是一个Petrie环的集合${mathscr C}={C_1,ldots,C_p}$,使得G的每个顶点恰好在一个循环$C_i in {mathscr C}$中。本文研究了具有Petrie循环划分的3正则平面图和具有Petrie循环划分的4正则平面多图的性质。给定一个4-正则平面多图$G=(V,E)$, G的3-正则化是将V$中的每个顶点$ V 拆分为两个3次顶点,得到一个由G得到的3-正则平面图$G_3$。G被称为皮特里可分的如果它有一个3正则化并且有一个皮特里循环划分。这个问题的一般版本是由计算机图形学中使用的数据压缩方法tritrip引起的。本文给出了Petrie可分图的一个简单刻划,并证明了判定G是否Petrie可分的问题是np完全的。
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引用次数: 0
Synchronizing words and monoid factorization, yielding a new parameterized complexity class? 同步单词和monoid因子分解,产生一个新的参数化复杂度类?
IF 0.5 4区 计算机科学 Q2 Mathematics Pub Date : 2022-02-01 DOI: 10.1017/S0960129522000184
H. Fernau, Jens Bruchertseifer
Abstract The concept of a synchronizing word is a very important notion in the theory of finite automata. We consider the associated decision problem to decide if a given DFA possesses a synchronizing word of length at most k, where k is the standard parameter. We show that this problem DFA-SW is equivalent to the problem Monoid Factorization introduced by Cai, Chen, Downey, and Fellows. Apart from the known $textsf{W}[2]$ -hardness results, we show that these problems belong to $textsf{A}[2]$ , $textsf{W}[textsf{P}],$ and $textsf{WNL}$ . This indicates that DFA-SW is not complete for any of these classes, and hence, we suggest a new parameterized complexity class $textsf{W}[textsf{Sync}]$ as a proper home for these (and more) problems. We present quite a number of problems that belong to $textsf{W}[textsf{Sync}]$ or are hard or complete for this new class.
摘要同步字的概念是有限自动机理论中一个非常重要的概念。我们考虑相关的决策问题来决定给定的DFA是否拥有长度最多为k的同步字,其中k是标准参数。我们证明了这个问题DFA-SW等价于蔡、陈、唐尼和费罗斯提出的单调因子分解问题。除了已知的$textsf{W}[2]$硬度结果外,我们还证明了这些问题属于$textsf{A}[2]$、$textsf{W}[textsf{P}]、$和$textsf{WNL}$。这表明DFA-SW对于这些类中的任何一个都不完整,因此,我们建议使用一个新的参数化复杂性类$textsf{W}[textsf{Sync}]$作为这些(以及更多)问题的合适归宿。我们提出了相当多的问题,这些问题属于$textsf{W}[textsf{Sync}]$,或者对于这个新类来说是困难的或完全的。
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引用次数: 1
Rogers semilattices of punctual numberings 准时编号的罗杰斯半格
IF 0.5 4区 计算机科学 Q2 Mathematics Pub Date : 2022-02-01 DOI: 10.1017/S0960129522000093
N. Bazhenov, M. Mustafa, S. Ospichev
Abstract The paper works within the framework of punctual computability, which is focused on eliminating unbounded search from constructions in algebra and infinite combinatorics. We study punctual numberings, that is, uniform computations for families S of primitive recursive functions. The punctual reducibility between numberings is induced by primitive recursive functions. This approach gives rise to upper semilattices of degrees, which are called Rogers pr-semilattices. We show that any infinite, uniformly primitive recursive family S induces an infinite Rogers pr-semilattice R. We prove that the semilattice R does not have minimal elements, and every nontrivial interval inside R contains an infinite antichain. In addition, every non-greatest element from R is a part of an infinite antichain. We show that the $Sigma_1$ -fragment of the theory Th(R) is decidable.
摘要本文在准时可计算性的框架内工作,重点是消除代数和无限组合数学中构造的无界搜索。我们研究了准时数,即原始递归函数族S的一致计算。数之间的准时可约性是由原始递归函数引起的。这种方法产生了度的上半格,称为Rogers-pr半格。我们证明了任何无穷的一致原始递归族S都会诱导一个无穷的Rogers-pr半格R。我们证明了半格R不具有极小元素,并且R内的每个非平凡区间都包含一个无穷反链。此外,来自R的每个非最大元素都是无限反链的一部分。我们证明了Th(R)理论的$ Sigma_1$-片段是可判定的。
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引用次数: 0
Hofmann-Mislove type definitions of non-Hausdorff spaces 非Hausdorff空间的Hofman-Mislove型定义
IF 0.5 4区 计算机科学 Q2 Mathematics Pub Date : 2022-01-01 DOI: 10.1017/S0960129522000196
Chong Shen, Xiaoyong Xi, Xiaoquan Xu, Dongsheng Zhao
Abstract One of the most important results in domain theory is the Hofmann-Mislove Theorem, which reveals a very distinct characterization for the sober spaces via open filters. In this paper, we extend this result to the d-spaces and well-filtered spaces. We do this by introducing the notions of Hofmann-Mislove-system (HM-system for short) and $Psi$ -well-filtered space, which provide a new unified approach to sober spaces, well-filtered spaces, and d-spaces. In addition, a characterization for $Psi$ -well-filtered spaces is provided via $Psi$ -sets. We also discuss the relationship between $Psi$ -well-filtered spaces and H-sober spaces considered by Xu. We show that the category of complete $Psi$ -well-filtered spaces is a full reflective subcategory of the category of $T_0$ spaces with continuous mappings. For each HM-system $Psi$ that has a designated property, we show that a $T_0$ space X is $Psi$ -well-filtered if and only if its Smyth power space $P_s(X)$ is $Psi$ -well-filtered.
摘要域理论中最重要的结果之一是Hofmann Mislove定理,它通过开滤子揭示了清醒空间的一个非常独特的特征。在本文中,我们将这个结果推广到d-空间和良好过滤空间。我们通过引入Hofmann-Mislove系统(简称HM系统)和$Psi$-良好过滤空间的概念来实现这一点,它们为清醒空间、良好过滤空间和d空间提供了一种新的统一方法。此外,通过$Psi$集提供了$Psi$-良好过滤空间的特征化。我们还讨论了Xu所考虑的$Psi$-良好滤波空间与H-ober空间之间的关系。我们证明了完备$Psi$-良好过滤空间的范畴是具有连续映射的$T_0$空间范畴的全反射子范畴。对于每个具有指定属性的HM系统$Psi$,我们证明了$T_0$空间X是$Psi$-良好过滤的当且仅当其Smyth幂空间$P_s(X)$是$Psi$-良好滤波的。
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引用次数: 0
Proof-relevance in Bishop-style constructive mathematics 毕晓普式构造数学中的证明相关性
IF 0.5 4区 计算机科学 Q2 Mathematics Pub Date : 2022-01-01 DOI: 10.1017/S0960129522000159
I. Petrakis
Abstract Bishop’s presentation of his informal system of constructive mathematics BISH was on purpose closer to the proof-irrelevance of classical mathematics, although a form of proof-relevance was evident in the use of several notions of moduli (of convergence, of uniform continuity, of uniform differentiability, etc.). Focusing on membership and equality conditions for sets given by appropriate existential formulas, we define certain families of proof sets that provide a BHK-interpretation of formulas that correspond to the standard atomic formulas of a first-order theory, within Bishop set theory $(mathrm{BST})$ , our minimal extension of Bishop’s theory of sets. With the machinery of the general theory of families of sets, this BHK-interpretation within BST is extended to complex formulas. Consequently, we can associate to many formulas $phi$ of BISH a set ${texttt{Prf}}(phi)$ of “proofs” or witnesses of $phi$ . Abstracting from several examples of totalities in BISH, we define the notion of a set with a proof-relevant equality, and of a Martin-Löf set, a special case of the former, the equality of which corresponds to the identity type of a type in intensional Martin-Löf type theory $(mathrm{MLTT})$ . Through the concepts and results of BST notions and facts of MLTT and its extensions (either with the axiom of function extensionality or with Vooevodsky’s axiom of univalence) can be translated into BISH. While Bishop’s theory of sets is standardly understood through its translation to MLTT, our development of BST offers a partial translation in the converse direction.
毕晓普提出的构造数学BISH的非正式体系有意接近于经典数学的证明无关性,尽管证明相关性的一种形式在模的几个概念(收敛性、一致连续性、一致可微性等)的使用中是明显的,我们在毕晓普集合论$(mathrm{BST})$中定义了某些证明集族,这些证明集族提供了对应于一阶理论的标准原子公式的公式的BHK解释,毕晓普集论$是毕晓普的集合论的最小扩展。利用集合族的一般理论,将BST中的BHK解释推广到复杂公式中。因此,我们可以将$phi$的“证明”或见证的集合${texttt{Prf}}(phi)$关联到BISH的许多公式$phi$。从BISH中的几个总数例子中抽象出,我们定义了具有证明相关等式的集合的概念,以及Martin-Löf集合的概念(前者的特例),其等式对应于内涵Martin-Löf类型理论$(mathrm{MLTT})$中类型的同一类型。通过BST的概念和结果,MLTT及其扩展(无论是用函数可拓性公理还是用Vooevodsky的单价公理)的概念和事实都可以转化为BISH。虽然毕晓普的集合理论通过翻译成MLTT而得到了标准的理解,但我们对BST的发展提供了相反方向的部分翻译。
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引用次数: 4
Implicit computation complexity in higher-order programming languages: A Survey in Memory of Martin Hofmann 高阶编程语言的隐式计算复杂性:马丁·霍夫曼的记忆综述
IF 0.5 4区 计算机科学 Q2 Mathematics Pub Date : 2022-01-01 DOI: 10.1017/S0960129521000505
Ugo Dal Lago
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引用次数: 2
The role of linearity in sharing analysis 线性在共享分析中的作用
IF 0.5 4区 计算机科学 Q2 Mathematics Pub Date : 2022-01-01 DOI: 10.1017/S0960129522000160
G. Amato, M. Meo, F. Scozzari
Abstract Sharing analysis is used to statically discover data structures which may overlap in object-oriented programs. Using the abstract interpretation framework, we show that sharing analysis greatly benefits from linearity information. A variable is linear in a program state when different field paths starting from it always reach different objects. We propose a graph-based abstract domain which can represent aliasing, linearity, and sharing information and define all the necessary abstract operators for the analysis of a Java-like language.
摘要共享分析用于静态地发现面向对象程序中可能重叠的数据结构。使用抽象解释框架,我们表明共享分析极大地受益于线性信息。当从变量开始的不同字段路径总是到达不同的对象时,变量在程序状态下是线性的。我们提出了一个基于图的抽象域,它可以表示混叠、线性和共享信息,并定义了分析类Java语言所需的所有抽象运算符。
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引用次数: 1
Cubical methods in homotopy type theory and univalent foundations 同伦型理论中的三次方法及一元基础
IF 0.5 4区 计算机科学 Q2 Mathematics Pub Date : 2021-12-10 DOI: 10.1017/s0960129521000311
Anders Mörtberg
Cubical methods have played an important role in the development of Homotopy Type Theory and Univalent Foundations (HoTT/UF) in recent years. The original motivation behind these developments was to give constructive meaning to Voevodsky’s univalence axiom, but they have since then led to a range of new results. Among the achievements of these methods is the design of new type theories and proof assistants with native support for notions from HoTT/UF, syntactic and semantic consistency results for HoTT/UF, as well as a variety of independence results and establishing that the univalence axiom does not increase the proof theoretic strength of type theory. This paper is based on lecture notes that were written for the 2019 Homotopy Type Theory Summer School at Carnegie Mellon University. The goal of these lectures was to give an introduction to cubical methods and provide sufficient background in order to make the current research in this very active area of HoTT/UF more accessible to newcomers. The focus of these notes is hence on both the syntactic and semantic aspects of these methods, in particular on cubical type theory and the various cubical set categories that give meaning to these theories.
近年来,三次方法在同伦型理论和一元基础(HoTT/UF)的发展中发挥了重要作用。这些发展背后的最初动机是给Voevodsky的一价公理以建设性的意义,但从那时起,它们导致了一系列新的结果。这些方法的成果包括设计了新的类型理论和原生支持HoTT/UF概念的证明助手,HoTT/UF的句法和语义一致性结果,以及各种独立性结果,并确立了一价公理不会增加类型论的证明理论强度。本文基于卡内基梅隆大学2019年同伦类型理论暑期学校的课堂笔记。这些讲座的目的是介绍立方体方法,并提供足够的背景资料,以便使新人更容易接触到这个非常活跃的HoTT/UF领域的当前研究。因此,这些笔记的重点是这些方法的句法和语义方面,特别是立方体类型理论和赋予这些理论意义的各种立方体集合类别。
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引用次数: 1
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Mathematical Structures in Computer Science
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