Natural language interaction between a student and a tutoring or an assistance system for mathematics is a new multi-disciplinary challenge that requires the interaction of (i) advanced natural language processing, (ii) flexible tutorial dialog strategies including hints, and (iii) mathematical domain reasoning. This paper provides an overview on the current research in the multi-disciplinary research project Dialog, whose goal is to build a prototype dialog-enabled system for teaching to do mathematical proofs. We present the crucial sub-systems in our architecture: the input understanding component and the domain reasoner. We present an interpretation method for mixed-language input consisting of informal and imprecise verbalization of mathematical content, and a proof manager that supports assertion-level automated theorem proving that is a crucial part of our domain reasoning module. Finally, we briefly report on an implementation of a demo system.

Abstract. Literate proving is the analogue for literate programming in the mathematical realm. That is, the goal of literate proving is for humans to produce clear expositions of formal mathematics that could even be enjoyable for people to read whilst remaining faithful representations of the actual proofs. This paper describes maze, a generic literate proving system. Authors markup formal proof files, such as Mizar files, with arbitary XML and use maze to obtain the selected extracts and transform them for presentation, e.g. as L ATEX. To aid its use, maze has built in transformations that include pretty printing and proof sketching for inclusion in L ATEX documents. These transformations challenge the concept of faithfulness in literate proving but it is argued that this should be a distinguishing feature of literate proving from literate programming. 1

Abstract. This paper discusses the semi-formal language of mathematics and presents the Naproche CNL, a controlled natural language for mathematical authoring. Proof Representation Structures, an adaptation of Discourse Representation Structures, are used to represent the semantics of texts written in the Naproche CNL. We discuss how the Naproche CNL can be used in formal mathematics, and present our prototypical Naproche system, a computer program for parsing texts in the Naproche CNL and checking the proofs in them for logical correctness.

Abstract. Natural language interfaces make formal systems accessible in informal language. They have a potential to make systems like theorem provers more widely used by students, mathematicians, and engineers who are not experts in logic. This paper shows that simple but still useful interfaces are easy to build with available technology. They are moreover easy to adapt to different formalisms and natural languages. The language can be made reasonably nice and stylistically varied. However, a fully general translation between logic and natural language also poses difficult, even unsolvable problems. This paper investigates what can be realistically expected and what problems are hard.

This project creates foundations for an automatic system that combines the reliability and speed of a computer with the ability to perform at the level of a good mathematics student. The acronym MATHRESS abbreviating the project title, which may be pronounced “mattress”, indicates that the project serves to provide a good, comfortable foundation for the development of an automatic mathematical research system. The MATHRESS project creates the MATHRESS system that will itself be the foundation on which people will rely for mathematical support. VISION and OBJECTIVES. The ambitious long-term vision for our project is the creation of an expert system that supports mathematicians and scientists dealing with mathematics in: – checking their own work for correctness; – improving the quality of their presentations; – decreasing the time needed for routine work in the preparation of publications; – quickly and reliably reminding them of work done by others; – producing multiple language versions of their manuscripts; – quickly disseminating partially checked results to other users of the system; – intelligently searching a universal database of mathematical knowledge; – learning like a student from the experience accumulated during interaction with the user.

Mathematical documents faithfully computerised: the grammatical and text & symbol aspects of the MathLang framework

In this paper we discuss the importance of good representations in mathematics and relate them to general design issues. Good design makes life easy, bad design difficult. For this reason experienced mathematicians spend a significant amount of their time on the design of their concepts. While many formal systems try to support this by providing a high-level language, we argue that more should be learned from the mathematical practice in order to improve the applicability of formal systems.

Abstract. Automated theorem provers (ATPs) struggle to solve problems with large sets of possibly superfluous axiom. Several algorithms have been developed to reduce the number of axioms, optimally only selecting the necessary axioms. However, most of these algorithms consider only single problems. In this paper, we describe an axiom selection method for series of related problems that is based on logical and textual proximity and tries to mimic a human way of understanding mathematical texts. We present first results that indicate that this approach is indeed useful. Key words: formal mathematics, automated theorem proving, axiom selection 1

This project aims at the development of a flexible modeling system for the specification of models for large-scale numerical work in optimization, data analysis, and partial differential equations. Its input should be provided in a form natural for the working mathematician, while the choice of the numerical solvers and the transformation to the format required by the solvers is done by the interface system. The input format should combine the simplicity of LaTeX source code with the semantic conciseness and modularity of current modeling languages such as AMPL, and it should be as close as possible to the mathematical language people use to explain and communicate their models in publications and lectures. In order that the system is useful for the intended applications, interfaces translating the model formulated in the proposed system into the input required for current state of the art solvers, and into the dominant current modeling languages are needed and shall be provided. Moreover, certain shortcomings of the current generation of modeling languages, such as the lack of support for the correct treatment of uncertainties and rounding errors, shall be overcome. The experience gained in this project will be useful in future work in the more general context

There are already many automatic mathematical assistants that provide expert help in specialized domains. Known classes include computer algebra systems, automated deduction systems, modeling systems, matrix packages, numerical prototyping languages, decision trees for scientific computing software, etc.. Such existing tools already provide partial functionality of the kind to be created in the project but only tied to specific applications, or with a limited scope. This document describes a number of current systems related to the FMathL vision, and some of their limitations when viewed in the light of this vision. The PI’s website (www.mat.univie.ac. at/~neum/FMathL.html) contains a large selection of additional resources and references to existing related systems. L ATEX