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19
Toward an objectoriented structure for mathematical text
 MATHEMATICAL KNOWLEDGE MANAGEMENT, 4TH INT’L CONF., PROCEEDINGS. VOLUME 3863 OF LECTURE NOTES IN ARTIFICIAL INTELLIGENCE
, 2006
"... 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 j ..."
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Cited by 17 (11 self)
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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 wordprocessing 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 objectoriented features so that nouns now correspond to classes, and adjectives to mixins.
Assisted proof document authoring
 Mathematical Knowledge Management MKM 2005, LNAI 3863
, 2006
"... 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 machinech ..."
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Cited by 10 (3 self)
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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 machinecheckable proof scripts and the corresponding humanreadable 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 humanreadable 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
Gradual computerisation/formalisation of mathematical texts into Mizar
 From Insight to Proof: Festschrift in Honour of Andrzej Trybulec
"... 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 aspec ..."
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Cited by 9 (4 self)
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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
J.B.: Restoring Natural Language as a Computerised Mathematics Input Method
 Mathematical Knowledge Management, 6th Int’l Conf., Proceedings. Lecture Notes in Artificial Intelligence
, 2007
"... 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 pr ..."
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Cited by 8 (6 self)
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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
Narrative structure of mathematical texts
 In preparation, available at http://www.macs.hw.ac.uk/~mm20
, 2007
"... 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., theore ..."
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Cited by 4 (3 self)
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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 selfconsistent. 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
On Correctness of Mathematical Texts from a Logical and Practical Point of View
 In Proceedings of the 9th AISC international conference, the 15th Calculemas symposium, and the 7th international MKM conference on Intelligent Computer Mathematics
, 2008
"... Abstract. Formalizing mathematical argument is a fascinating activity in itself and (we hope!) also bears important practical applications. While traditional proof theory investigates deducibility of an individual statement from a collection of premises, a mathematical proof, with its structure and ..."
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Cited by 3 (0 self)
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Abstract. Formalizing mathematical argument is a fascinating activity in itself and (we hope!) also bears important practical applications. While traditional proof theory investigates deducibility of an individual statement from a collection of premises, a mathematical proof, with its structure and continuity, can hardly be presented as a single sequent or a set of logical formulas. What is called “mathematical text”, as used in mathematical practice through the ages, seems to be more appropriate. However, no commonly adopted formal notion of mathematical text has emerged so far. In this paper, we propose a formalism which aims to reflect natural (human) style and structure of mathematical argument, yet to be appropriate for automated processing: principally, verification of its correctness (we consciously use the word rather than “soundness ” or “validity”). We consider mathematical texts that are formalized in the ForTheL language (brief description of which is also given) and we formulate a point of view on what a correct mathematical text might be. Logical notion of correctness is formalized with the help of a calculus. Practically, these ideas, methods and algorithms are implemented in a proof assistant called SAD. We give a short description of SAD and a series of examples showing what can be done with it. 1
Abstract Computerizing Mathematical Text with
"... Mathematical texts can be computerized 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 c ..."
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Cited by 1 (0 self)
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Mathematical texts can be computerized 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 computerizing mathematical texts and knowledge which is flexible enough to connect the different approaches to computerization, which allows various degrees of formalization, 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. Three Ph.D. students (Manuel Maarek (2002/2007), Krzysztof Retel (since 2004), and Robert Lamar (since 2006)) and over a dozen master’s degree and undergraduate students have worked on MathLang. The project’s progress and design choices are driven by the needs for computerizing real representative mathematical texts chosen from various
MATHRESS: A MATHEMATICAL RESEARCH SYSTEM Principal Investigator: Arnold Neumaier Funding Period: 5 years
"... 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 s ..."
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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 longterm 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.
(ULTRA group, HeriotWatt University)
"... 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 computerbased for ..."
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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 computerbased formalisation for mathematical authoring habits and we present an overview of our approach for computerising mathematical texts. 1.
A modeling system for mathematics
"... This project aims at the development of a flexible modeling system for the specification of models for largescale 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 ..."
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This project aims at the development of a flexible modeling system for the specification of models for largescale 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