Advanced Proof Viewing in ProofTool

Sequent calculus is widely used for formalizing proofs. However, due to the proliferation of data, understanding the proofs of even simple mathematical arguments soon becomes impossible. Graphical user interfaces help in this matter, but since they normally utilize Gentzen's original notation, some of the problems persist. In this paper, we introduce a number of criteria for proof visualization which we have found out to be crucial for analyzing proofs. We then evaluate recent developments in tree visualization with regard to these criteria and propose the Sunburst Tree layout as a complement to the traditional tree structure. This layout constructs inferences as concentric circle arcs around the root inference, allowing the user to focus on the proof's structural content. Finally, we describe its integration into ProofTool and explain how it interacts with the Gentzen layout.Comment: In Proceedings UITP 2014, arXiv:1410.785

Mixing the reactive with the personal: Opportunities for end-user programming in personal information management

The transition of personal information management (PIM) tools off the desktop to the Web presents an opportunity to augment these tools with capabilities provided by the wealth of real-time information readily available. In this chapter, we describe a personal information assistance engine that lets end-users delegate to it various simple context- and activity-reactive tasks and reminders. Our system, Atomate, treats RSS/ATOM feeds from social networking and life-tracking sites as sensor streams, integrating information from such feeds into a simple unified RDF world model representing people, places and things and their time-varying states and activities. Combined with other information sources on the web, including the user's online calendar, web-based e-mail client, news feeds and messaging services, Atomate can be made to automatically carry out a variety of simple tasks for the user, ranging from context-aware filtering and messaging, to sharing and social coordination actions. Atomate's open architecture and world model easily accommodate new information sources and actions via the addition of feeds and web services. To make routine use of the system easy for non-programmers, Atomate provides a constrained-input natural language interface (CNLI) for behavior specification, and a direct-manipulation interface for inspecting and updating its world model

Combining SysML and AADL for the design, validation and implementation of critical systems

The realization of critical systems goes through multiple phases of specification, design, integration, validation, and testing. It starts from high-level sketches down to the final product. Model-Based Design has been acknowledged as a good conveyor to capture these steps. Yet, there is no universal solution to represent all activities. Two candidates are the OMG-based SysML to perform high-level modeling tasks, and the SAE AADL to perform lower-level ones, down to the implementation. The paper shares an experience on the seamless use of SysML and the AADL to model, validate/verify and implement a flight management system

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

Rotate-and-Press: A Non-Visual Alternative to Point-and-Click

Most computer applications manifest visually rich and dense graphical user interfaces (GUIs) that are primarily tailored for an easy-and-efficient sighted interaction using a combination of two default input modalities, namely the keyboard and the mouse/touchpad. However, blind screen-reader users predominantly rely only on keyboard, and therefore struggle to interact with these applications, since it is both arduous and tedious to perform the visual \u27point-and-click\u27 tasks such as accessing the various application commands/features using just keyboard shortcuts supported by screen readers. In this paper, we investigate the suitability of a \u27rotate-and-press\u27 input modality as an effective non-visual substitute for the visual mouse to easily interact with computer applications, with specific focus on word processing applications serving as the representative case study. In this regard, we designed and developed bTunes, an add-on for Microsoft Word that customizes an off-the-shelf Dial input device such that it serves as a surrogate mouse for blind screen-reader users to quickly access various application commands and features using a set of simple rotate and press gestures supported by the Dial. Therefore, with bTunes, blind users too can now enjoy the benefits of two input modalities, as their sighted counterparts. A user study with 15 blind participants revealed that bTunes significantly reduced both the time and number of user actions for doing representative tasks in a word processing application, by as much as 65.1% and 36.09% respectively. The participants also stated that they did not face any issues switching between keyboard and Dial, and furthermore gave a high usability rating (84.66 avg. SUS score) for bTunes

A Tool for Producing Verified, Explainable Proofs

Mathematicians are reluctant to use interactive theorem provers. In this thesis I argue that this is because proof assistants don't emphasise explanations of proofs; and that in order to produce good explanations, the system must create proofs in a manner that mimics how humans would create proofs. My research goals are to determine what constitutes a human-like proof and to represent human-like reasoning within an interactive theorem prover to create formalised, understandable proofs. Another goal is to produce a framework to visualise the goal states of this system. To demonstrate this, I present HumanProof: a piece of software built for the Lean 3 theorem prover. It is used for interactively creating proofs that resemble how human mathematicians reason. The system provides a visual, hierarchical representation of the goal and a system for suggesting available inference rules. The system produces output in the form of both natural language and formal proof terms which are checked by Lean's kernel. This is made possible with the use of a structured goal state system which interfaces with Lean's tactic system which is detailed in Chapter 3. In Chapter 4, I present the subtasks automation planning subsystem, which is used to produce equality proofs in a human-like fashion. The basic strategy of the subtasks system is break a given equality problem in to a hierarchy of tasks and then maintain a stack of these tasks in order to determine the order in which to apply equational rewriting moves. This process produces equality chains for simple problems without having to resort to brute force or specialised procedures such as normalisation. This makes proofs more human-like by breaking the problem into a hierarchical set of tasks in the same way that a human would. To produce the interface for this software, I also created the ProofWidgets system for Lean 3. This system is detailed in Chapter 5. The ProofWidgets system uses Lean's metaprogramming framework to allow users to write their own interactive, web-based user interfaces to display within the VSCode editor and in an online web-editor. The entire tactic state is available to the rendering engine, and hence expression structure and types of subexpressions can be explored interactively. The ProofWidgets system also allows the user interface to interactively edit the proof document, enabling a truly interactive modality for creating proofs; human-like or not. In Chapter 6, the system is evaluated by asking real mathematicians about the output of the system, and what it means for a proof to be understandable to them. The user group study asks participants to rank and comment on proofs created by HumanProof alongside natural language and pure Lean proofs. The study finds that participants generally prefer the HumanProof format over the Lean format. The verbal responses collected during the study indicate that providing intuition and signposting are the most important properties of a proof that aid understanding.EPSR

Extracting proofs from documents

Often, theorem checkers like PVS are used to check an existing proof, which is part of some document. Since there is a large difference between the notations used in the documents and the notations used in the theorem checkers, it is usually a laborious task to convert an existing proof into a format which can be checked by a machine. In the system that we propose, the author is assisted in the process of converting an existing proof into the PVS language and having it checked by PVS. 1 Introduction The now-classic ALGOL 60 report [5] recognized three different levels of language: a reference language, a publication language and several hardware representations, whereby the publication language was intended to admit variations on the reference language and was to be used for stating and communicating processes. The importance of publication language ---often referred to nowadays as "pseudo-code"--- is difficult to exaggerate since a publication language is the most effective way..