We present a suite of tools for simplifying the creation and maintenance of grammars for the OpenCCG parsing and realization system. The core of our approach relies on a terse but expressive textual format, DotCCG, for declaring CCG grammars. It supports powerful string expansions that allow grammar developers to eliminate redundancy in the declaration of both morphology and category definitions. Grammars written in this format are converted into the XML utilized by OpenCCG using theccg2xml utility, which –like a programming language compiler – provides information regarding errors in the grammar, including the type of error and the line number on which it occurs. DotCCG grammars can be edited with VisCCG, a graphical interface which provides visualization of various components of the grammar and allows local editing of information in a manner inspired by wikis. We also report on resources developed to facilitate wide use of the OpenCCG tool suite presented in this paper and on recent uses of the tools in both academic research and classroom environments. 1

Abstract. When mathematicians present proofs they usually adapt their explanations to their didactic goals and to the (assumed) knowledge of their addressees. Modern automated theorem provers, in contrast, present proofs usually at a fixed level of detail (also called granularity). Often these presentations are neither intended nor suitable for human use. A challenge therefore is to develop user- and goal-adaptive proof presentation techniques that obey common mathematical practice. We present a flexible and adaptive approach to proof presentation based on classification. Expert knowledge for the classification task can be handauthored or extracted from annotated proof examples via machine learning techniques. The obtained models are employed for the automated generation of further proofs at an adapted level of granularity.

The Intelligent Book project aims to improve online education by designing materials that can model the subject matter they teach, in the manner of a Reactive Learning Environment. In this paper, we investigate using an automated proof assistant, particularly Isabelle/HOL, as the model supporting first year undergraduate exercises in which students write proofs in number theory. Automated proof assistants are generally considered to be difficult for novices to learn. We examine whether, by providing a very specialised interface, it is possible to build something that is usable enough to be of educational value. To ensure students cannot “game the system ” the exercise avoids tactic-choosing interaction styles, but asks the student to write out the proof. Proofs are written using MathsTiles: composable tiles that resemble written mathematics. Unlike traditional syntax-directed editors, MathsTiles allow students to keep many answer fragments on the canvas at the same time, and do not constrain the order in which an answer is written. Also, the tile syntax does not need to match the underlying Isar syntax exactly, and different tiles can be used for different questions. The exercises take place within the context of an Intelligent Book. We performed a user study and qualitative analysis of the system. Some users were able to complete proofs with much less training than is usual for the automated proof assistant itself, but there remain significant usability issues to overcome.

Proof tutoring, granularity, machine learning. Even in introductory textbooks on mathematical proof, intermediate proof steps are generally skipped when this seems appropriate. This gives rise to different granularities of proofs, depending on the intended audience and the context in which the proof is presented. We have developed a mechanism to classify whether proof steps of different sizes are appropriate in a tutoring context. The necessary knowledge is learnt from expert tutors via standard machine learning techniques from annotated examples. We discuss the ongoing evaluation of our approach via empirical studies. 1

When mathematicians present proofs they usually adapt their explanations to their didactic goals and to the (assumed) knowledge of their addressees. Modern automated theorem provers, in contrast, present proofs usually at a fixed level of detail (also called granularity). Often these presentations are neither intended nor suitable for human use. A challenge therefore is to develop user- and goal-adaptive proof presentation techniques that obey common mathematical practice. We present a flexible and adaptive approach to proof presentation that exploits machine learning techniques to extract a model of the specific granularity of

Abstract. Mathematical assistance systems and proof assistance systems in general have traditionally been developed as large, monolithic systems which are often hard to maintain and extend. In this article we propose a component network architecture as a means to design and implement such systems. Under this view a mathematical assistance system is an integrated knowledge-based system composed as a network of individual, specialized components. These components manipulate and mutually exchange different kinds of mathematical knowledge encoded within different document formats. Consequently, several units of mathematical knowledge coexist throughout the system within these components and this knowledge changes non-monotonically over time. Our approach has resulted in a lean and maintainable system code and makes the system open for extensions. Moreover, it naturally decomposes the global and complex reasoning and truth maintenance task into local reasoning and truth maintenance tasks inside the system components. The interplay between neighboring components in the network is thereby realized by nonmonotonic updates over agreed interface representations encoding different kinds of mathematical knowledge. 1.

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