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Logics for Temporal Information Systems in Rough Set Theory 粗糙集理论中时态信息系统的逻辑
IF 0.5 4区 数学 Q3 COMPUTER SCIENCE, THEORY & METHODS Pub Date : 2022-07-19 DOI: 10.1145/3549075
Md. Aquil Khan, M. Banerjee, Sibsankar Panda
The article discusses temporal information systems (TISs) that add the dimension of time to complete or incomplete information systems. Through TISs, one can accommodate the possibility of domains or attribute values for objects changing with time or the availability of currently missing information with time. Different patterns of flow of information give different TISs. The corresponding logics with sound and complete axiomatization are presented.
本文讨论了时间信息系统(TIS),它为完整或不完整的信息系统增加了时间维度。通过TIS,可以适应对象的域或属性值随时间变化的可能性,或者当前缺失信息的可用性随时间变化。不同的信息流模式给出了不同的TIS。给出了相应的具有完备公理化的逻辑。
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引用次数: 0
Unifying Operational Weak Memory Verification: An Axiomatic Approach 统一操作弱记忆验证:一种公理化方法
IF 0.5 4区 数学 Q3 COMPUTER SCIENCE, THEORY & METHODS Pub Date : 2022-06-27 DOI: 10.1145/3545117
Simon Doherty, Sadegh Dalvandi, Brijesh Dongol, H. Wehrheim
In this article, we propose an approach to program verification using an abstract characterisation of weak memory models. Our approach is based on a hierarchical axiom scheme that captures the observational properties of a memory model. In particular, we show that it is possible to prove correctness of a program with respect to a particular axiom scheme, and we show this proof to suffice for any memory model that satisfies the axioms. Our axiom scheme is developed using a characterisation of weakest liberal preconditions for weak memory. This characterisation naturally extends to Hoare logic and Owicki-Gries reasoning by lifting weakest liberal preconditions (defined over read/write events) to the level of programs. We study three memory models (SC, TSO, and RC11-RAR) as example instantiations of the axioms, then we demonstrate the applicability of our reasoning technique on a number of litmus tests. The majority of the proofs in this article are supported by mechanisation within Isabelle/HOL.
在这篇文章中,我们提出了一种使用弱记忆模型的抽象特征进行程序验证的方法。我们的方法基于层次公理方案,该方案捕获了记忆模型的观测特性。特别地,我们证明了证明程序相对于特定公理方案的正确性是可能的,并且我们证明了这个证明足以满足任何满足公理的内存模型。我们的公理方案是使用弱记忆的最弱自由先决条件的特征来开发的。通过将最弱的自由先决条件(通过读/写事件定义)提升到程序级别,这种特征自然扩展到霍尔逻辑和Owicki Gries推理。我们研究了三个记忆模型(SC、TSO和RC11-RAR)作为公理的实例,然后我们在一些石蕊测试中证明了我们的推理技术的适用性。本文中的大多数证明都得到了Isabelle/HOL内部机制化的支持。
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引用次数: 6
A Category Theoretic View of Contextual Types: From Simple Types to Dependent Types 语境类型的范畴论视角:从简单类型到依赖类型
IF 0.5 4区 数学 Q3 COMPUTER SCIENCE, THEORY & METHODS Pub Date : 2022-06-06 DOI: 10.1145/3545115
Jason Z. S. Hu, B. Pientka, Ulrich Schöpp
We describe the categorical semantics for a simply typed variant and a simplified dependently typed variant of Cocon, a contextual modal type theory where the box modality mediates between the weak function space that is used to represent higher-order abstract syntax (HOAS) trees and the strong function space that describes (recursive) computations about them. What makes Cocon different from standard type theories is the presence of first-class contexts and contextual objects to describe syntax trees that are closed with respect to a given context of assumptions. Following M. Hofmann’s work, we use a presheaf model to characterise HOAS trees. Surprisingly, this model already provides the necessary structure to also model Cocon. In particular, we can capture the contextual objects of Cocon using a comonad ♭ that restricts presheaves to their closed elements. This gives a simple semantic characterisation of the invariants of contextual types (e.g. substitution invariance) and identifies Cocon as a type-theoretic syntax of presheaf models. We further extend this characterisation to dependent types using categories with families and show that we can model a fragment of Cocon without recursor in the Fitch-style dependent modal type theory presented by Birkedal et al.
我们描述了Cocon的简单类型变体和简化依赖类型变体的分类语义,Cocon是一种上下文模态类型理论,其中盒模态介于用于表示高阶抽象语法(HOAS)树的弱函数空间和描述关于它们的(递归)计算的强函数空间之间。Cocon与标准类型理论的不同之处在于,它存在一级上下文和上下文对象来描述相对于给定假设上下文闭合的语法树。根据M.Hofmann的工作,我们使用预剪切模型来表征HOAS树。令人惊讶的是,该模型已经提供了对Cocon进行建模所需的结构。特别是,我们可以使用comonad捕获Cocon的上下文对象♭ 这将预应力限制在其封闭元件上。这给出了上下文类型不变量(例如替换不变性)的简单语义表征,并将Cocon识别为预学习模型的类型论语法。我们使用带有族的类别将这种表征进一步扩展到依赖类型,并表明我们可以在Birkedal等人提出的Fitch风格依赖模态类型理论中对没有递归的Cocon片段进行建模。
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引用次数: 2
A Generalized Realizability and Intuitionistic Logic 广义可实现性与直觉逻辑
IF 0.5 4区 数学 Q3 COMPUTER SCIENCE, THEORY & METHODS Pub Date : 2022-05-17 DOI: 10.1145/3565367
A. Y. Konovalov
Let V be a set of number-theoretical functions. We define a notion of V-realizability for predicate formulas in such a way that the indices of functions in V are used for interpreting the implication and the universal quantifier. In this article, we prove that Intuitionistic Predicate Calculus is sound with respect to the semantics of V-realizability if and only if some natural conditions for V hold.
设V是一组数论函数。我们为谓词公式定义了V可实现性的概念,使得V中函数的索引用于解释蕴涵和通用量词。在本文中,我们证明了直觉谓词演算关于V可实现性的语义是健全的,当且仅当V的一些自然条件成立。
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引用次数: 1
First-order Logic with Connectivity Operators 带连通算子的一阶逻辑
IF 0.5 4区 数学 Q3 COMPUTER SCIENCE, THEORY & METHODS Pub Date : 2021-07-13 DOI: 10.1145/3595922
Nicole Schirrmacher, S. Siebertz, Alexandre Vigny
First-order logic (FO) can express many algorithmic problems on graphs, such as the independent set and dominating set problem parameterized by solution size. However, FO cannot express the very simple algorithmic question whether two vertices are connected. We enrich FO with connectivity predicates that are tailored to express algorithmic graph problems that are commonly studied in parameterized algorithmics. By adding the atomic predicates connk(x,y,z_1,..., zk) that hold true in a graph if there exists a path between (the valuations of) x and y after (the valuations of) z1,..., zk have been deleted, we obtain separator logic FO + conn. We show that separator logic can express many interesting problems, such as the feedback vertex set problem and elimination distance problems to first-order definable classes. Denote by FO + connk the fragment of separator logic that is restricted to connectivity predicates with at most k + 2 variables (that is, at most k deletions), we show that FO + connk + 1 is strictly more expressive than FO + connk for all k ≥ 0. We then study the limitations of separator logic and prove that it cannot express planarity, and, in particular, not the disjoint paths problem. We obtain the stronger disjoint-paths logic FO + DP by adding the atomic predicates disjoint-pathsk[(x1, y1),..., (xk, yk) that evaluate to true if there are internally vertex-disjoint paths between (the valuations of) xi and yi for all 1 ≤ i ≤ k. Disjoint-paths logic can express the disjoint paths problem, the problem of (topological) minor containment, the problem of hitting (topological) minors, and many more. Again, we show that the fragments FO + DPk that use predicates for at most k disjoint paths form a strict hierarchy of expressiveness. Finally, we compare the expressive power of the new logics with that of transitive-closure logics and monadic second-order logic.
一阶逻辑可以在图上表达许多算法问题,如解大小参数化的独立集和支配集问题。然而,FO不能表达两个顶点是否连通的非常简单的算法问题。我们用连通性谓词来丰富FO,这些谓词专门用于表达通常在参数化算法中研究的算法图问题。通过添加原子谓词connk(x,y,z_1,…), zk)在一个图中成立,如果在z1,…之后x和y的值之间存在一条路径我们得到了分隔符逻辑FO + conn。我们证明了分隔符逻辑可以表达许多有趣的问题,如反馈顶点集问题和一阶可定义类的消去距离问题。用FO + connk表示分隔符逻辑的片段,该片段限制于最多有k + 2个变量的连接谓词(即最多有k个删除),我们表明,对于所有k≥0,FO + connk + 1严格比FO + connk更具表现力。然后,我们研究了分隔逻辑的局限性,并证明了它不能表示平面性,特别是不能表示不相交路径问题。通过添加原子谓词disjoint-pathsk[(x1, y1),…],得到了更强的不相交路径逻辑FO + DP。, (xk, yk),如果对于所有1≤i≤k, xi和yi的(值)之间存在内部顶点不相交的路径,则其值为真。不相交路径逻辑可以表示不相交路径问题,(拓扑)小包含问题,命中(拓扑)小包含问题,以及更多的问题。我们再次证明,片段FO + DPk对最多k个不相交路径使用谓词,形成了严格的表达层次结构。最后,我们将新逻辑的表达能力与传递闭包逻辑和一元二阶逻辑进行了比较。
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引用次数: 8
The Complexity of Quantified Constraints: Collapsibility, Switchability, and the Algebraic Formulation 量化约束的复杂性:可折叠性、可切换性和代数公式
IF 0.5 4区 数学 Q3 COMPUTER SCIENCE, THEORY & METHODS Pub Date : 2021-06-24 DOI: 10.1145/3568397
C. Carvalho, Florent R. Madelaine, B. Martin, Dmitriy Zhuk
Let 𝔸 be an idempotent algebra on a finite domain. By mediating between results of Chen [1] and Zhuk [2], we argue that if 𝔸 satisfies the polynomially generated powers property (PGP) and ℬ is a constraint language invariant under 𝔸 (i.e., in Inv(𝔸)), then QCSP ℬ is in NP. In doing this, we study the special forms of PGP, switchability, and collapsibility, in detail, both algebraically and logically, addressing various questions such as decidability on the way. We then prove a complexity-theoretic converse in the case of infinite constraint languages encoded in propositional logic, that if Inv}(𝔸) satisfies the exponentially generated powers property (EGP), then QCSP (Inv(𝔸)) is co-NP-hard. Since Zhuk proved that only PGP and EGP are possible, we derive a full dichotomy for the QCSP, justifying what we term the Revised Chen Conjecture. This result becomes more significant now that the original Chen Conjecture (see [3]) is known to be false [4]. Switchability was introduced by Chen [1] as a generalization of the already-known collapsibility [5]. There, an algebra 𝔸 :=({ 0,1,2};r) was given that is switchable and not collapsible. We prove that, for all finite subsets Δ of Inv (𝔸 A), Pol (Δ) is collapsible. The significance of this is that, for QCSP on finite structures, it is still possible all QCSP tractability (in NP) explained by switchability is already explained by collapsibility. At least, no counterexample is known to this.
允许𝔸 是有限域上的幂等代数。通过在Chen[1]和Zhuk[2]的结果之间进行中介,我们认为如果𝔸 满足多项式生成功率特性(PGP),并且ℬ 是一个约束语言不变量𝔸 (即,在发票中(𝔸)), 然后QCSPℬ 在NP中。在这样做的过程中,我们从代数和逻辑上详细研究了PGP的特殊形式、可切换性和可折叠性,解决了诸如途中的可判定性等各种问题。然后,我们证明了在命题逻辑中编码的无限约束语言的情况下的复杂性理论逆,即如果Inv}(𝔸) 满足指数生成功率特性(EGP),则QCSP(Inv(𝔸)) 是co-NP困难的。由于Zhuk证明了只有PGP和EGP是可能的,我们导出了QCSP的完全二分法,证明了我们所称的修正Chen猜想的合理性。这个结果变得更加重要,因为原来的陈猜想(见[3])是假的[4]。可切换性由Chen[1]引入,作为已知溃散性[5]的推广。那里有一个代数𝔸 :=({0,1,2};r)是可切换的且不可折叠的。我们证明,对于Inv的所有有限子集Δ(𝔸 A) ,Pol(Δ)是可折叠的。这一点的意义在于,对于有限结构上的QCSP,仍然有可能所有由可切换性解释的QCSP可处理性(在NP中)都已经由溃散性解释了。至少,目前还没有已知的反例。
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引用次数: 0
Expressiveness and Nash Equilibrium in Iterated Boolean Games 迭代布尔对策中的表达性与纳什均衡
IF 0.5 4区 数学 Q3 COMPUTER SCIENCE, THEORY & METHODS Pub Date : 2021-06-08 DOI: 10.1145/3439900
GutiérrezJulián, HarrensteinPaul, PerelliGiuseppe, WooldridgeMichael
We define and investigate a novel notion of expressiveness for temporal logics that is based on game theoretic equilibria of multi-agent systems. We use iterated Boolean games as our abstract model...
基于多智能体系统的博弈论平衡,我们定义并研究了时间逻辑的一个新的表达性概念。我们使用迭代布尔对策作为我们的抽象模型。。。
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引用次数: 3
A Subatomic Proof System for Decision Trees 决策树的亚原子证明系统
IF 0.5 4区 数学 Q3 COMPUTER SCIENCE, THEORY & METHODS Pub Date : 2021-05-04 DOI: 10.1145/3545116
Chris Barrett, Alessio Guglielmi
We design a proof system for propositional classical logic that integrates two languages for Boolean functions: standard conjunction-disjunction-negation and binary decision trees. We give two reasons to do so. The first is proof-theoretical naturalness: The system consists of all and only the inference rules generated by the single, simple, linear scheme of the recently introduced subatomic logic. Thanks to this regularity, cuts are eliminated via a natural construction. The second reason is that the system generates efficient proofs. Indeed, we show that a certain class of tautologies due to Statman, which cannot have better than exponential cut-free proofs in the sequent calculus, have polynomial cut-free proofs in our system. We achieve this by using the same construction that we use for cut elimination. In summary, by expanding the language of propositional logic, we make its proof theory more regular and generate more proofs, some of which are very efficient. That design is made possible by considering atoms as superpositions of their truth values, which are connected by self-dual, non-commutative connectives. A proof can then be projected via each atom into two proofs, one for each truth value, without a need for cuts. Those projections are semantically natural and are at the heart of the constructions in this article. To accommodate self-dual non-commutativity, we compose proofs in deep inference.
我们设计了一个命题经典逻辑的证明系统,该系统集成了布尔函数的两种语言:标准连接-析取-否定和二元决策树。我们给出了两个理由。第一个是证明理论的自然性:该系统由最近引入的亚原子逻辑的单一、简单、线性方案生成的所有推理规则组成。由于这种规律性,通过自然结构消除了切口。第二个原因是该系统生成了有效的证明。事实上,我们证明了一类由Statman引起的重言式在我们的系统中具有多项式无割证明,它在序演算中不可能有比指数无割证明更好的证明。我们通过使用与消除切割相同的结构来实现这一点。总之,通过扩展命题逻辑的语言,我们使其证明理论更加规则,并生成更多的证明,其中一些证明是非常有效的。这种设计是通过将原子视为其真值的叠加而实现的,这些真值由自对偶、非交换连接词连接。然后,一个证明可以通过每个原子投影为两个证明,每个真值一个,而不需要切割。这些投影在语义上是自然的,是本文结构的核心。为了适应自对偶非交换性,我们在深度推理中构造了证明。
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引用次数: 1
Good-for-Game QPTL: An Alternating Hodges Semantics 适合游戏的QPTL:一个交替的Hodges语义
IF 0.5 4区 数学 Q3 COMPUTER SCIENCE, THEORY & METHODS Pub Date : 2021-04-13 DOI: 10.1145/3565365
Dylan Bellier, M. Benerecetti, Dario Della Monica, F. Mogavero
An extension of QPTL is considered where functional dependencies among the quantified variables can be restricted in such a way that their current values are independent of the future values of the other variables. This restriction is tightly connected to the notion of behavioral strategies in game-theory and allows the resulting logic to naturally express game-theoretic concepts. Inspired by the work on logics of dependence and independence, we provide a new compositional semantics for QPTL that allows for expressing such functional dependencies among variables. The fragment where only restricted quantifications are considered, called behavioral quantifications, allows for linear-time properties that are satisfiable if and only if they are realisable in the Pnueli-Rosner sense. This fragment can be decided, for both model checking and satisfiability, in 2Exp Time and is expressively equivalent to QPTL, though significantly less succinct.
考虑QPTL的扩展,其中量化变量之间的函数依赖性可以被限制为使得它们的当前值独立于其他变量的未来值。这种限制与博弈论中的行为策略概念紧密相连,并允许由此产生的逻辑自然地表达博弈论概念。受依赖性和独立性逻辑工作的启发,我们为QPTL提供了一种新的组合语义,允许表达变量之间的函数依赖性。只考虑有限量化的片段,称为行为量化,允许线性时间属性是可满足的,当且仅当它们在Pnueli-Rosner意义上是可实现的。对于模型检查和可满足性,这个片段可以在2Exp-Time中决定,并且在表达上等价于QPTL,尽管明显不那么简洁。
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引用次数: 3
The Intersection of Algorithmically Random Closed Sets and Effective Dimension 算法随机闭集与有效维数的交集
IF 0.5 4区 数学 Q3 COMPUTER SCIENCE, THEORY & METHODS Pub Date : 2021-03-05 DOI: 10.1145/3545114
Adam Case, Christopher P. Porter
In this article, we study several aspects of the intersections of algorithmically random closed sets. First, we answer a question of Cenzer and Weber, showing that the operation of intersecting relatively random closed sets (random with respect to certain underlying measures induced by Bernoulli measures on the space of codes of closed sets), which preserves randomness, can be inverted: a random closed set of the appropriate type can be obtained as the intersection of two relatively random closed sets. We then extend the Cenzer/Weber analysis to the intersection of multiple random closed sets, identifying the Bernoulli measures with respect to which the intersection of relatively random closed sets can be non-empty. We lastly apply our analysis to provide a characterization of the effective Hausdorff dimension of sequences in terms of the degree of intersectability of random closed sets that contain them.
在本文中,我们研究了算法随机闭集的交的几个方面。首先,我们回答了Cenzer和Weber的一个问题,证明了相交相对随机闭集(相对于封闭集码空间上由伯努利测度引起的某些底层测度是随机的)的操作可以被反转,它保留了随机性:两个相对随机闭集的交集可以得到一个适当类型的随机闭集。然后,我们将Cenzer/Weber分析扩展到多个随机闭集的交集,确定了相对随机闭集的交集可以是非空的伯努利测度。最后,我们应用我们的分析,根据包含序列的随机闭集的可相交度,给出了序列的有效豪斯多夫维数的表征。
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引用次数: 0
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ACM Transactions on Computational Logic
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