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Flexible encoding of mathematics on the computer
 In MKM 2004, volume 3119 of LNCS
, 2004
"... Abstract. This paper reports on refinements and extensions to the MathLang framework that add substantial support for natural language text. We show how the extended framework supports multiple views of mathematical texts, including natural language views using the exact text that the mathematician ..."
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Abstract. This paper reports on refinements and extensions to the MathLang framework that add substantial support for natural language text. We show how the extended framework supports multiple views of mathematical texts, including natural language views using the exact text that the mathematician wants to use. Thus, MathLang now supports the ability to capture the essential mathematical structure of mathematics written using natural language text. We show examples of how arbitrary mathematical text can be encoded in MathLang without needing to change any of the words or symbols of the texts or their order. In particular, we show the encoding of a theorem and its proof that has been used by Wiedijk for comparing many theorem prover representations of mathematics, namely the irrationality of √ 2 (originally due to Pythagoras). We encode a 1960 version by Hardy and Wright, and a more recent version by Barendregt. 1 On the way to a mathematical vernacular for computers Mathematicians now use computer software for a variety of tasks: typing mathematical texts, performing calculation, analyzing theories, verifying proofs. Software tools like
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 18 (12 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.
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 10 (5 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
An experimental grammar for German mathematical text
, 2009
"... This is a report on work in progress, for the FMathL project. It describes a preliminary grammar designed for ultimately parsing the German language lecture notes on analysis and linear algebra by the second author. The report will be updated from time to time. ..."
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Cited by 4 (3 self)
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This is a report on work in progress, for the FMathL project. It describes a preliminary grammar designed for ultimately parsing the German language lecture notes on analysis and linear algebra by the second author. The report will be updated from time to time.
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
Under supervision of:
, 2006
"... 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 th ..."
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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: MathLangCore. While the architecture has changed to a more XMLcentred 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.
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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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
(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.
Digitised Mathematics:
, 2006
"... 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 ..."
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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
rst chapter of Foundations of Analysis
, 2004
"... y 1 = y; of Theorem 28, 1 y = y; 1 y = y 1; so that 1 belongs to M. xy = yx; xy + y = yx + y = yx y = xy + y; y = yx belongs to M. The assertion therefore holds for all x. Theorem 30 (Distributive Law) x(y + z) = xy + xz: Preliminary Remark The Formula (y + z)x = ..."
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y 1 = y; of Theorem 28, 1 y = y; 1 y = y 1; so that 1 belongs to M. xy = yx; xy + y = yx + y = yx y = xy + y; y = yx belongs to M. The assertion therefore holds for all x. Theorem 30 (Distributive Law) x(y + z) = xy + xz: Preliminary Remark The Formula (y + z)x = yx + zx which results from Theorem 30 and Theorem 29, and similar analogues later on, need not be speci ned for all y and be such that a 1 = x, b 1 = x, a y 0 = ay + x, b y 0 = by + x for every y. a1 = x = b1 ; hence 1 belongs to M. hence, a y 0 = ay + x = by + x = b y 0 ; belongs to M. to de ning xy for all y in such a way that most one possibility of de rst show that for each nition 6 To every pair of numbers x; y, we may assign in exactly one way a natural number, called x y ( to be read \times"; however, the dot is usualy omitted), such that x 1 = x for every x; (8) x y = (x y) + x for every x and every y: (9) x y is called the prod