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The Sequent Calculus Trainer with Automated Reasoning - Helping Students to Find Proofs 序列微积分训练与自动推理-帮助学生找到证明
Pub Date : 2018-03-02 DOI: 10.4204/EPTCS.267.2
Arno Ehle, Norbert Hundeshagen, M. Lange
The sequent calculus is a formalism for proving validity of statements formulated in First-Order Logic. It is routinely used in computer science modules on mathematical logic. Formal proofs in the sequent calculus are finite trees obtained by successively applying proof rules to formulas, thus simplifying them step-by-step. Students often struggle with the mathematical formalities and the level of abstraction that topics like formal logic and formal proofs involve. The difficulties can be categorised as syntactic or semantic. On the syntactic level, students need to understand what a correctly formed proof is, how rules can be applied (on paper for instance) without leaving the mathematical framework of the sequent calculus, and so on. Beyond this, on the semantic level, students need to acquire strategies that let them find the right proof. The Sequent Calculus Trainer is a tool that is designed to aid students in learning the techniques of proving given statements formally. In this paper we describe the didactical motivation behind the tool and the techniques used to address issues on the syntactic as well as on the semantic level.
序贯演算是证明一阶逻辑中表述的有效性的一种形式。它通常用于计算机科学数学逻辑模块。序贯演算中的形式证明是通过对公式相继应用证明规则而得到的有限树,从而逐步简化它们。像形式逻辑和形式证明这样的主题往往涉及数学形式和抽象层次,学生们常常为此而挣扎。这些困难可分为句法上的或语义上的。在句法层面上,学生需要理解正确形成的证明是什么,如何在不离开顺序演算的数学框架的情况下应用规则(例如在纸上),等等。除此之外,在语义层面上,学生需要掌握让他们找到正确证据的策略。序贯微积分训练器是一种工具,旨在帮助学生学习证明给定语句的技术。在本文中,我们描述了该工具背后的教学动机,以及用于解决语法和语义层面问题的技术。
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引用次数: 9
Proceedings 6th International Workshop on Theorem proving components for Educational software, ThEdu@CADE 2017, Gothenburg, Sweden, 6 Aug 2017 第六届教育软件定理证明组件国际研讨会论文集,ThEdu@CADE 2017,瑞典哥德堡,2017年8月6日
Pub Date : 2018-03-02 DOI: 10.4204/EPTCS.267
P. Quaresma, Walther Neuper
The 6th International Workshop on Theorem proving components for Educational software (ThEdu'17) was held in Gothenburg, Sweden, on 6 Aug 2017. It was associated to the conference CADE26. Topics of interest include: methods of automated deduction applied to checking students' input; methods of automated deduction applied to prove post-conditions for particular problem solutions; combinations of deduction and computation enabling systems to propose next steps; automated provers specific for dynamic geometry systems; proof and proving in mathematics education. ThEdu'17 was a vibrant workshop, with one invited talk and eight contributions. It triggered the post-proceedings at hand.
第六届教育软件定理证明组件国际研讨会(ThEdu'17)于2017年8月6日在瑞典哥德堡举行。它与CADE26会议有关。感兴趣的话题包括:用于检查学生输入的自动扣除方法;用于证明特定问题解的后置条件的自动演绎方法;推理和计算的结合,使系统能够提出下一步的步骤;针对动态几何系统的自动证明器;数学教育中的证明与证明。ThEdu'17是一个充满活力的研讨会,有一个邀请演讲和八个贡献。它触发了手头的后续程序。
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引用次数: 3
Carnap: An Open Framework for Formal Reasoning in the Browser 浏览器中形式推理的开放框架
Pub Date : 2018-03-02 DOI: 10.4204/EPTCS.267.5
Graham Leach-Krouse
This paper presents an overview of Carnap, a free and open framework for the development of formal reasoning applications. Carnap's design emphasizes flexibility, extensibility, and rapid prototyping. Carnap-based applications are written in Haskell, but can be compiled to JavaScript to run in standard web browsers. This combination of features makes Carnap ideally suited for educational applications, where ease-of-use is crucial for students and adaptability to different teaching strategies and classroom needs is crucial for instructors. The paper describes Carnap's implementation, along with its current and projected pedagogical applications.
本文介绍了Carnap的概述,它是一个用于开发形式推理应用程序的免费开放框架。Carnap的设计强调灵活性、可扩展性和快速原型。基于快照的应用程序是用Haskell编写的,但可以编译成JavaScript在标准web浏览器中运行。这些功能的结合使Carnap非常适合教育应用程序,其中对学生来说易用性至关重要,对教师来说适应不同的教学策略和课堂需求至关重要。本文描述了Carnap的实现,以及它当前和预计的教学应用。
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引用次数: 11
Exchange of Geometric Information Between Applications 应用程序之间的几何信息交换
Pub Date : 2018-03-02 DOI: 10.4204/EPTCS.267.7
P. Quaresma, Vanda Santos, Nuno Baeta
The Web Geometry Laboratory (WGL) is a collaborative and adaptive e-learning Web platform integrating a well known dynamic geometry system. Thousands of Geometric problems for Geometric Theorem Provers (TGTP) is a Web-based repository of geometric problems to support the testing and evaluation of geometric automated theorem proving systems. The users of these systems should be able to profit from each other. The TGTP corpus must be made available to the WGL user, allowing, in this way, the exploration of TGTP problems and their proofs. On the other direction TGTP could gain by the possibility of a wider users base submitting new problems. Such information exchange between clients (e.g. WGL) and servers (e.g. TGTP) raises many issues: geometric search - someone, working in a geometric problem, must be able to ask for more information regarding that construction; levels of geometric knowledge and interest - the problems in the servers must be classified in such a way that, in response to a client query, only the problems in the user's level and/or interest are returned; different aims of each tool - e.g. WGL is about secondary school geometry, TGTP is about formal proofs in semi-analytic and algebraic proof methods, not a perfect match indeed; localisation issues, e.g. a Portuguese user obliged to make the query and process the answer in English; technical issues-many technical issues need to be addressed to make this exchange of geometric information possible and useful. Instead of a giant (difficult to maintain) tool, trying to cover all, the interconnection of specialised tools seems much more promising. The challenges to make that connection work are many and difficult, but, it is the authors impression, not insurmountable.
网络几何实验室(WGL)是一个协作和自适应的电子学习网络平台,集成了一个众所周知的动态几何系统。几何定理证明者的数千几何问题(TGTP)是一个基于web的几何问题存储库,用于支持几何自动定理证明系统的测试和评估。这些系统的用户应该能够相互获利。TGTP语料库必须提供给WGL用户,以这种方式允许对TGTP问题及其证明进行探索。在另一个方向上,TGTP可以从更广泛的用户群提交新问题的可能性中获益。这种客户端(如WGL)和服务器(如TGTP)之间的信息交换引发了许多问题:几何搜索——处理几何问题的人必须能够询问有关该结构的更多信息;几何知识和兴趣水平——服务器中的问题必须以这样一种方式分类,即在响应客户端查询时,只返回用户级别和/或兴趣的问题;每个工具的目标不同——例如,WGL是关于中学几何的,TGTP是关于半解析和代数证明方法的形式证明,确实不是完美匹配;本地化问题,例如,葡萄牙语用户必须用英语进行查询和处理答案;技术问题—要使这种几何信息交换成为可能和有用,需要解决许多技术问题。而不是一个巨大的(难以维护的)工具,试图覆盖所有,专业工具的互连似乎更有前途。建立这种联系的挑战很多,也很困难,但是,这是作者的印象,不是不可克服的。
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引用次数: 2
Prototyping "Systems that Explain Themselves" for Education 为教育制作“自我解释系统”原型
Pub Date : 2018-03-02 DOI: 10.4204/EPTCS.267.6
Alan Krempler, Walther Neuper
"Systems that Explain Themselves" appears a provocative wording, in particular in the context of mathematics education -- it is as provocative as the idea of building educational software upon technology from computer theorem proving. In spite of recent success stories like the proofs of the Four Colour Theorem or the Kepler Conjecture, mechanised proof is still considered somewhat esoteric by mainstream mathematics. This paper describes the process of prototyping in the ISAC project from a technical perspective. This perspective depends on two moving targets: On the one side the rapidly increasing power and coverage of computer theorem provers and their user interfaces, and on the other side potential users: What can students and teachers request from educational systems based on technology and concepts from computer theorem proving, now and then? By the way of describing the process of prototyping the first comprehensive survey on the state of the ISAC prototype is given as a side effect, made precise by pointers to the code and by citation of all contributing theses.
“解释自己的系统”似乎是一个挑衅性的措辞,特别是在数学教育的背景下——它与基于计算机定理证明的技术构建教育软件的想法一样具有挑衅性。尽管最近出现了一些成功的故事,如四色定理的证明或开普勒猜想的证明,但主流数学仍然认为机械化证明有点深奥。本文从技术角度描述了ISAC项目的原型设计过程。这种观点依赖于两个移动的目标:一方面是计算机定理证明者及其用户界面快速增长的能力和覆盖范围,另一方面是潜在用户:学生和教师可以时不时地从基于计算机定理证明的技术和概念的教育系统中提出什么要求?通过描述原型过程的方式,对ISAC原型的状态进行了第一次全面的调查,作为一个副作用,通过对代码的指针和引用所有贡献论文来精确地进行了调查。
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引用次数: 6
Learning how to Prove: From the Coq Proof Assistant to Textbook Style 学习如何证明:从Coq证明助手到教科书风格
Pub Date : 2018-03-02 DOI: 10.4204/EPTCS.267.1
Sebastian Böhne, C. Kreitz
We have developed an alternative approach to teaching computer science students how to prove. First, students are taught how to prove theorems with the Coq proof assistant. In a second, more difficult, step students will transfer their acquired skills to the area of textbook proofs. In this article we present a realisation of the second step. Proofs in Coq have a high degree of formality while textbook proofs have only a medium one. Therefore our key idea is to reduce the degree of formality from the level of Coq to textbook proofs in several small steps. For that purpose we introduce three proof styles between Coq and textbook proofs, called line by line comments, weakened line by line comments, and structure faithful proofs. While this article is mostly conceptional we also report on experiences with putting our approach into practise.
我们开发了另一种方法来教计算机科学专业的学生如何证明。首先,教授学生如何用Coq证明助手证明定理。在第二步,更困难的是,学生将把他们获得的技能转移到教科书校样领域。在本文中,我们提出了第二步的实现。Coq中的校样具有高度的正式性,而教科书中的校样只有中等的正式性。因此,我们的关键思想是通过几个小步骤来减少从Coq级别到教科书校样的正式程度。为此,我们在Coq和教科书证明之间引入了三种证明风格,称为逐行注释、弱逐行注释和结构忠实证明。虽然这篇文章主要是概念性的,但我们也报告了将我们的方法付诸实践的经验。
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引用次数: 9
On the Structure of Abstract H*-Algebras 关于抽象H*-代数的结构
Pub Date : 2018-02-27 DOI: 10.4204/EPTCS.266.13
Kevin Dunne
Previously we have shown that the topos approach to quantum theory of Doering and Isham can be generalised to a class of categories typically studied within the monoidal approach to quantum theory of Abramsky and Coecke. In the monoidal approach to quantum theory H*-algebras provide an axiomatisation of states and observables. Here we show that H*-algebras naturally correspond with the notions of states and observables in the generalised topos approach to quantum theory. We then combine these results with the dagger-kernel approach to quantumlogic of Heunen and Jacobs, which we use to prove a structure theorem for H*-algebras. This structure theorem is a generalisation of the structure theorem of Ambrose for H*-algebras the category of Hilbert spaces.
以前我们已经证明,Doering和Isham的量子理论的拓扑方法可以推广到一类通常在Abramsky和Coecke的量子理论的一元方法中研究的范畴。在量子理论的一元方法中,H*-代数提供了状态和可观测物的公理化。在这里,我们证明了H*代数自然地与量子理论的广义拓扑方法中的状态和可观测值的概念相对应。然后,我们将这些结果与Heunen和Jacobs的量子逻辑的匕首核方法结合起来,我们使用它来证明H*-代数的结构定理。这个结构定理是Hilbert空间范畴H*-代数的Ambrose结构定理的推广。
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引用次数: 1
Spectral Presheaves, Kochen-Specker Contextuality, and Quantale-Valued Relations 谱预捆,Kochen-Specker上下文性和量子值关系
Pub Date : 2018-02-27 DOI: 10.4204/EPTCS.266.24
Kevin Dunne
In the topos approach to quantum theory of Doering and Isham the Kochen-Specker Theorem, which asserts the contextual nature of quantum theory, can be reformulated in terms of the global sections of a presheaf characterised by the Gelfand spectrum of a commutativeC-Algebra. In previous work we showed how this topos perspective can be generalised to a class of categories typically studied within the monoidal approach to quantum theory of Abramsky and Coecke, and in particular how one can generalise the Gelfand spectrum. Here we study the Gelfand spectrum presheaf for categories of quantale-valued relations, and by considering its global sections we give a non-contextuality result for these categories. We also show that the Gelfand spectrum comes equipped with a topology which has a natural interpretation when thinking of these structures as representing physical theories.
在Doering和Isham的量子理论的拓扑方法中,Kochen-Specker定理,它断言了量子理论的上下文性质,可以用由交换c -代数的Gelfand谱表征的preshef的全局部分来重新表述。在之前的工作中,我们展示了如何将这种topos视角推广到一类通常在Abramsky和Coecke的量子理论的一元方法中研究的类别,特别是如何推广Gelfand谱。本文研究了量子值关系范畴的Gelfand谱预表,并通过考虑其全局部分,给出了这些范畴的非上下文性结果。我们还表明,当将这些结构视为代表物理理论时,Gelfand谱配备了具有自然解释的拓扑结构。
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引用次数: 1
Transforming Proof Tableaux of Hoare Logic into Inference Sequences of Rewriting Induction 将Hoare逻辑的证明表转化为改写归纳的推理序列
Pub Date : 2018-02-16 DOI: 10.4204/EPTCS.265.4
Shinnosuke Mizutani, Naoki Nishida
A proof tableau of Hoare logic is an annotated program with pre- and post-conditions, which corresponds to an inference tree of Hoare logic. In this paper, we show that a proof tableau for partial correctness can be transformed into an inference sequence of rewriting induction for constrained rewriting. We also show that the resulting sequence is a valid proof for an inductive theorem corresponding to the Hoare triple if the constrained rewriting system obtained from the program is terminating. Such a valid proof with termination of the constrained rewriting system implies total correctness of the program w.r.t. the Hoare triple. The transformation enables us to apply techniques for proving termination of constrained rewriting to proving total correctness of programs together with proof tableaux for partial correctness.
霍尔逻辑的证明表是一个带有前置和后置条件的注释程序,它对应于霍尔逻辑的推理树。在本文中,我们证明了部分正确性的证明表可以转化为约束重写的重写归纳推理序列。我们还证明了如果从该程序得到的约束重写系统是终止的,则所得序列是对应于Hoare三元组的一个归纳定理的有效证明。这样一个具有约束重写系统终止的有效证明意味着程序的完全正确性。这种转换使我们能够应用证明约束重写终止的技术来证明程序的全部正确性,以及部分正确性的证明表。
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引用次数: 0
Attributed Hierarchical Port Graphs and Applications 分级端口图和应用程序
Pub Date : 2018-02-16 DOI: 10.4204/EPTCS.265.2
N. Ene, M. Fernández, Bruno Pinaud
We present attributed hierarchical port graphs (AHP) as an extension of port graphs that aims at facilitating the design of modular port graph models for complex systems. AHP consist of a number of interconnected layers, where each layer defines a port graph whose nodes may link to layers further down the hierarchy; attributes are used to store user-defined data as well as visualisation and run-time system parameters. We also generalise the notion of strategic port graph rewriting (a particular kind of graph transformation system, where port graph rewriting rules are controlled by user-defined strategies) to deal with AHP following the Single Push-out approach. We outline examples of application in two areas: functional programming and financial modelling.
我们提出了属性层次端口图(AHP)作为端口图的扩展,旨在促进复杂系统模块化端口图模型的设计。AHP由许多相互连接的层组成,其中每层定义一个端口图,其节点可以链接到层次结构的更低层;属性用于存储用户定义的数据以及可视化和运行时系统参数。我们还概括了策略端口图重写(一种特殊类型的图转换系统,其中端口图重写规则由用户定义的策略控制)的概念,以处理遵循单次推出方法的AHP。我们概述了两个领域的应用示例:函数式编程和财务建模。
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引用次数: 5
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