Proof in Context -- Web Editing with Rich, Modeless Contextual Feedback

The Agora system is a prototypical Wiki for formal mathematics: a web-based system for collaborating on formal mathematics, intended to support informal documentation of formal developments. This system requires a reusable proof editor component, both for collaborative editing of documents, and for embedding in the resulting documents. This paper describes the design of Agora's asynchronous editor, that is generic enough to support different tools working on editor content and providing contextual information, with interactive theorem proverss being a special, but important, case described in detail for the Coq theorem prover.Comment: In Proceedings UITP 2012, arXiv:1307.152

Collaborative Verification-Driven Engineering of Hybrid Systems

Hybrid systems with both discrete and continuous dynamics are an important model for real-world cyber-physical systems. The key challenge is to ensure their correct functioning w.r.t. safety requirements. Promising techniques to ensure safety seem to be model-driven engineering to develop hybrid systems in a well-defined and traceable manner, and formal verification to prove their correctness. Their combination forms the vision of verification-driven engineering. Often, hybrid systems are rather complex in that they require expertise from many domains (e.g., robotics, control systems, computer science, software engineering, and mechanical engineering). Moreover, despite the remarkable progress in automating formal verification of hybrid systems, the construction of proofs of complex systems often requires nontrivial human guidance, since hybrid systems verification tools solve undecidable problems. It is, thus, not uncommon for development and verification teams to consist of many players with diverse expertise. This paper introduces a verification-driven engineering toolset that extends our previous work on hybrid and arithmetic verification with tools for (i) graphical (UML) and textual modeling of hybrid systems, (ii) exchanging and comparing models and proofs, and (iii) managing verification tasks. This toolset makes it easier to tackle large-scale verification tasks

Automated Reasoning and Presentation Support for Formalizing Mathematics in Mizar

This paper presents a combination of several automated reasoning and proof presentation tools with the Mizar system for formalization of mathematics. The combination forms an online service called MizAR, similar to the SystemOnTPTP service for first-order automated reasoning. The main differences to SystemOnTPTP are the use of the Mizar language that is oriented towards human mathematicians (rather than the pure first-order logic used in SystemOnTPTP), and setting the service in the context of the large Mizar Mathematical Library of previous theorems,definitions, and proofs (rather than the isolated problems that are solved in SystemOnTPTP). These differences poses new challenges and new opportunities for automated reasoning and for proof presentation tools. This paper describes the overall structure of MizAR, and presents the automated reasoning systems and proof presentation tools that are combined to make MizAR a useful mathematical service.Comment: To appear in 10th International Conference on. Artificial Intelligence and Symbolic Computation AISC 201

Proviola: A Tool for Proof Re-animation

To improve on existing models of interaction with a proof assistant (PA), in particular for storage and replay of proofs, we in- troduce three related concepts, those of: a proof movie, consisting of frames which record both user input and the corresponding PA response; a camera, which films a user's interactive session with a PA as a movie; and a proviola, which replays a movie frame-by-frame to a third party. In this paper we describe the movie data structure and we discuss a proto- type implementation of the camera and proviola based on the ProofWeb system. ProofWeb uncouples the interaction with a PA via a web- interface (the client) from the actual PA that resides on the server. Our camera films a movie by "listening" to the ProofWeb communication. The first reason for developing movies is to uncouple the reviewing of a formal proof from the PA used to develop it: the movie concept enables users to discuss small code fragments without the need to install the PA or to load a whole library into it. Other advantages include the possibility to develop a separate com- mentary track to discuss or explain the PA interaction. We assert that a combined camera+proviola provides a generic layer between a client (user) and a server (PA). Finally we claim that movies are the right type of data to be stored in an encyclopedia of formalized mathematics, based on our experience in filming the Coq standard library.Comment: Accepted for the 9th International Conference on Mathematical Knowledge Management (MKM 2010), 15 page

Towards formal models and languages for verifiable Multi-Robot Systems

Incorrect operations of a Multi-Robot System (MRS) may not only lead to unsatisfactory results, but can also cause economic losses and threats to safety. These threats may not always be apparent, since they may arise as unforeseen consequences of the interactions between elements of the system. This call for tools and techniques that can help in providing guarantees about MRSs behaviour. We think that, whenever possible, these guarantees should be backed up by formal proofs to complement traditional approaches based on testing and simulation. We believe that tailored linguistic support to specify MRSs is a major step towards this goal. In particular, reducing the gap between typical features of an MRS and the level of abstraction of the linguistic primitives would simplify both the specification of these systems and the verification of their properties. In this work, we review different agent-oriented languages and their features; we then consider a selection of case studies of interest and implement them useing the surveyed languages. We also evaluate and compare effectiveness of the proposed solution, considering, in particular, easiness of expressing non-trivial behaviour.Comment: Changed formattin

Open educational resources : conversations in cyberspace

172 p. : ill. ; 25 cm.Libro ElectrónicoEducation systems today face two major challenges: expanding the reach of education and improving its quality. Traditional solutions will not suffice, especially in the context of today's knowledge-intensive societies. The Open Educational Resources movement offers one solution for extending the reach of education and expanding learning opportunities. The goal of the movement is to equalize access to knowledge worldwide through openly and freely available online high-quality content. Over the course of two years, the international community came together in a series of online discussion forums to discuss the concept of Open Educational Resources and its potential. This publication makes the background papers and reports from those discussions available in print.--Publisher's description.A first forum : presenting the open educational resources (OER) movement. Open educational resources : an introductory note / Sally Johnstone -- Providing OER and related issues : an introductory note / Anne Margulies, ... [et al.] -- Using OER and related issues : in introductory note / Mohammed-Nabil Sabry, ... [et al.] -- Discussion highlights / Paul Albright -- Ongoing discussion. A research agenda for OER : discussion highlights / Kim Tucker and Peter Bateman -- A 'do-it-yourself' resource for OER : discussion highlights / Boris Vukovic -- Free and open source software (FOSS) and OER -- A second forum : discussing the OECD study of OER. Mapping procedures and users / Jan Hylén -- Why individuals and institutions share and use OER / Jan Hylén -- Discussion highlights / Alexa Joyce -- Priorities for action. Open educational resources : the way forward / Susan D'Antoni

Interoperability Issues Between Learning Object Repositories and Metadata Harvesters

In this paper we describe an open learning object repository on Statistics based on DSpace which contains true learning objects, that is, exercises, equations, data sets, etc. This repository is part of a large project intended to promote the use of learning object repositories as part of the learning process in virtual learning environments. This involves the creation of a new user interface that provides users with additional services such as resource rating, commenting and so. Both aspects make traditional metadata schemes such as Dublin Core to be inadequate, as there are resources with no title or author, for instance, as those fields are not used by learners to browse and search for learning resources in the repository. Therefore, exporting OAI-PMH compliant records using OAI-DC is not possible, thus limiting the visibility of the learning objects in the repository outside the institution. We propose an architecture based on ontologies and the use of extended metadata records for both storing and refactoring such descriptions

Interoperability Issues Between Learning Object Repositories and Metadata Harvesters

In this paper we describe an open learning object repository on Statistics based on DSpace which contains true learning objects, that is, exercises, equations, data sets, etc. This repository is part of a large project intended to promote the use of learning object repositories as part of the learning process in virtual learning environments. This involves the creation of a new user interface that provides users with additional services such as resource rating, commenting and so. Both aspects make traditional metadata schemes such as Dublin Core to be inadequate, as there are resources with no title or author, for instance, as those fields are not used by learners to browse and search for learning resources in the repository. Therefore, exporting OAI-PMH compliant records using OAI-DC is not possible, thus limiting the visibility of the learning objects in the repository outside the institution. We propose an architecture based on ontologies and the use of extended metadata records for both storing and refactoring such descriptions

Learning-Assisted Automated Reasoning with Flyspeck

The considerable mathematical knowledge encoded by the Flyspeck project is combined with external automated theorem provers (ATPs) and machine-learning premise selection methods trained on the proofs, producing an AI system capable of answering a wide range of mathematical queries automatically. The performance of this architecture is evaluated in a bootstrapping scenario emulating the development of Flyspeck from axioms to the last theorem, each time using only the previous theorems and proofs. It is shown that 39% of the 14185 theorems could be proved in a push-button mode (without any high-level advice and user interaction) in 30 seconds of real time on a fourteen-CPU workstation. The necessary work involves: (i) an implementation of sound translations of the HOL Light logic to ATP formalisms: untyped first-order, polymorphic typed first-order, and typed higher-order, (ii) export of the dependency information from HOL Light and ATP proofs for the machine learners, and (iii) choice of suitable representations and methods for learning from previous proofs, and their integration as advisors with HOL Light. This work is described and discussed here, and an initial analysis of the body of proofs that were found fully automatically is provided

Mathematical Explanation: A Contextual Approach

PurposeIn this article, we aim to present and defend a contextual approach to mathematical explanation.MethodTo do this, we introduce an epistemic reading of mathematical explanation.ResultsThe epistemic reading not only clarifies the link between mathematical explanation and mathematical understanding, but also allows us to explicate some contextual factors governing explanation. We then show how several accounts of mathematical explanation can be read in this approach.ConclusionThe contextual approach defended here clears up the notion of explanation and pushes us towards a pluralist vision on mathematical explanation