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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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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.
Computerizing Mathematical Text with MathLang
"... 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 17 (4 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
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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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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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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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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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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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
Computerising Mathematical Text with MathLang
"... 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 c ..."
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Cited by 1 (0 self)
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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
A Partial Translation Path from MathLang to Isabelle
, 2011
"... thesis or use of any of the information contained in it must acknowledge this thesis as the source of the quotation or information. This dissertation describes certain developments in computer techniques for managingmathematical knowledge. Computers currently assist mathematicians in presenting and ..."
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thesis or use of any of the information contained in it must acknowledge this thesis as the source of the quotation or information. This dissertation describes certain developments in computer techniques for managingmathematical knowledge. Computers currently assist mathematicians in presenting and archiving mathematics, as well as performing calculation and verification tasks. MathLang is a framework for computerising mathematical documents which features new approaches to these issues. In this dissertation, several extensions to MathLang are described: a system and notation for annotating text; improved methods for annotating complex mathematical expressions; and a method for creating rules to translate document annotations. A typical MathLang work flow for document annotation and computerisation is demonstrated, showing how writing style can complicate the annotation process and how these may be resolved. This workflow is compared with the standard process for producing formal computer theories in a computer proof assistant (Isabelle is the system we choose). The rules for translation are further discussed as a way of producing text in the syntax of Isabelle (without a deep knowledge of the system), with possible use cases of providing a text which can be used either as an aid to learning Isabelle, or as a skeleton framework to be used as a starting point for a formal document. i ii For my family both present and future
(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.