Computerizing mathematical texts to allow software access to some or all of the texts ’ semantic content is a long and tedious process that currently requires much expertise. We believe it is useful to support computerization that adds some structural and semantic information, but does not require jumping directly from the word-processing level (e.g., L ATEX) to full formalization (e.g., Mizar, Coq, etc.). Although some existing mathematical languages are aimed at this middle ground (e.g., MathML, OpenMath, OMDoc), we believe they miss features needed to capture some important aspects of mathematical texts, especially the portion written with natural language. For this reason, we have been developing MathLang, a language for representing mathematical texts that has weak type checking and support for the special mathematical use of natural language. MathLang is currently aimed at only capturing the essential grammatical and binding structure of mathematical text without requiring full formalization. The development of MathLang is directly driven by experience encoding real mathematical texts. Based on this experience, this paper presents the changes that yield our latest version of MathLang. We have restructured and simplified the core of the language, replaced our old notion of “context” by a new system of blocks and local scoping, and made other changes. Furthermore, we have enhanced our support for the mathematical use of nouns and adjectives with object-oriented features so that nouns now correspond to classes, and adjectives to mixins.

Abstract. Recently, significant advances have been made in formalised mathematical texts for large, demanding proofs. But although such large developments are possible, they still take an inordinate amount of effort and time, and there is a significant gap between the resulting formalised machine-checkable proof scripts and the corresponding human-readable mathematical texts. We present an authoring system for formal proof which addresses these concerns. It is based on a central document format which, in the tradition of literate programming, allows one to extract either a formal proof script or a human-readable document; the two may have differing structure and detail levels, but are developed together in a synchronised way. Additionally, we introduce ways to assist production of the central document, by allowing tools to contribute backflow to update and extend it. Our authoring system builds on the new PG Kit architecture for Proof General, bringing the extra advantage that it works in a uniform interface, generically across various interactive theorem provers. 1

Abstract. We explain in this paper the gradual computerisation process of an ordinary mathematical text into more formal versions ending with a fully formalised Mizar text. The process is part of the MathLang–Mizar project and is divided into a number of steps (called aspects). The first three aspects (CGa, TSa and DRa) are the same for any MathLang–TP project where TP is any proof checker (e.g., Mizar, Coq, Isabelle, etc). These first three aspects are theoretically formalised and implemented and provide the mathematician and/or TP user with useful tools/automation. Using TSa, the mathematician edits his mathematical text just as he would use L ATEX, but at the same time he sees the mathematical text as it appears on his paper. TSa also gives the mathematician easy editing facilities to help assign to parts of the text, grammatical and mathematical roles and to relate different parts through a number of mathematical, rethorical and structural relations. MathLang would then automatically produce CGa and DRa versions of the text, checks

Abstract. Methods for computerised mathematics have found little appeal among mathematicians because they call for additional skills which are not available to the typical mathematician. We herein propose to reconcile computerised mathematics to mathematicians by restoring natural language as the primary medium for mathematical authoring. Our method associates portions of text with grammatical argumentation roles and computerises the informal mathematical style of the mathematician. Typical abbreviations like the aggregation of equations a = b> c, are not usually accepted as input to computerised languages. We propose specific annotations to explicate the morphology of such natural language style, to accept input in this style, and to expand this input in the computer to obtain the intended representation (i.e., a = b and b> c). We have named this method syntax souring in contrast to the usual syntax sugaring. All results have been implemented in a prototype editor developed on top of TEXmacs as a GUI for the core grammatical aspect of MathLang, a framework developed by the ULTRA group to computerise and formalise mathematics. 1

Abstract. There are many styles for the narrative structure of a mathematical document. Each mathematician has its own conventions and traditions about labeling portions of texts (e.g., chapter, section, theorem or proof) and identifying statements according to their logical importance (e.g., theorem is more important than lemma). Such narrative/structuring labels guide the reader’s navigation of the text and form the key components in the reasoning structure of the theory reflected in the text. We present in this paper a method to computerise the narrative structure of a text which includes the relationships between labeled text entities. These labels and relations are input by the user on top of their natural language text. This narrative structure is then automatically analysed to check its consistency. This automatic analysis consists of two phases: (1) checking the correct usage of labels and relations (i.e., that a “proof” justifies a “theorem ” but cannot justify an “axiom”) and (2) checking that the logical precedences in the document are self-consistent. The development of this method was driven by the experience of computerising a number of mathematical documents (covering different authoring styles). We illustrate how such computerised narrative structure could be used for further manipulations, i.e. to build a skeleton of a formal document in a formal system like Mizar, Coq or Isabelle. 1

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.

Abstract. In only few decades, computers have changed the way we approach documents. Throughout history, mathematicians and philosophers had clarified the relationship between mathematical thoughts and their textual and symbolic representations. We discuss here the consequences of computer-based formalisation for mathematical authoring habits and we present an overview of our approach for computerising mathematical texts. 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

MathLang is a language for mathematics on computers. It allows computerisation of existing and new mathematical texts written in the Common Mathematical Language, and checking the grammatical correctness of this computerisation. The framework also allows the user to take incremental steps towards the generation of a fully formalised document in such a way that the result can be checked by a proof checker. This report describes the language itself, its grammar, elements, characteristics and one of the concrete syntaxes: the plain syntax. An encoding of a large example is presented and explained. The main focus of the report lies on the implementation of the heart of the framework: MathLang-Core. While the architecture has changed to a more XML-centred design during the implementation, both the old and proposed new architectures are discussed. Four components can be distinguished in the framework: the parser, the abstract syntax tree, the checker and the printer. The latter has many different instances for many formats.

Mathematical texts can be computerised in many ways that capture differing amounts of the mathematical meaning. At one end, there is document imaging, which captures the arrangement of black marks on paper, while at the other end there are proof assistants (e.g., Mizar, Isabelle, Coq, etc.), which capture the full mathematical meaning and have proofs expressed in a formal foundation of mathematics. In between, there are computer typesetting systems (e.g., LATEX and Presentation MathML) and semantically oriented systems (e.g., Content MathML, OpenMath, OMDoc, etc.). The MathLang project was initiated in 2000 by Fairouz Kamareddine and Joe Wells with the aim of developing an approach for computerising mathematical texts which is flexible enough to connect the different approaches to computerisation, which allows various degrees of formalisation, and which is compatible with different logical frameworks (e.g., set theory, category theory, type theory, etc.) and proof systems. The approach is embodied in a computer representation, which we call MathLang, and associated software tools, which are being developed by ongoing work. Four Ph.D. students (Manuel Maarek (2002/2007), Krzysztof Retel (since 2004), Robert Lamar (since 2006)), and Christoph Zengler (since 2008) and over a dozen master’s degree and undergraduate