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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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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
PLATΩ: A mediator between texteditors and proof assistance systems
, 2007
"... We present a generic mediator, called PlatΩ, between texteditors and proof assistants. PlatΩ aims at integrated support for the development, publication, formalization, and verification of mathematical documents in a natural way as possible: The user authors his mathematical documents with a scient ..."
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We present a generic mediator, called PlatΩ, between texteditors and proof assistants. PlatΩ aims at integrated support for the development, publication, formalization, and verification of mathematical documents in a natural way as possible: The user authors his mathematical documents with a scientific WYSIWYG texteditor in the informal language he is used to, that is a mixture of natural language and formulas. These documents are then semantically annotated preserving the textual structure by using the flexible, parameterized proof language which we present. From this informal semantic representation PlatΩ automatically generates the corresponding formal representation for a proof assistant, in our case Ωmega. The primary task of PlatΩ is the maintenance of consistent formal and informal representations during the interactive development of the document.
Example supplement for “Restoring natural language as a computerised mathematics input method”. Available at http://www.macs.hw.ac.uk/ ~rob
, 2007
"... Abstract. Due to a strict page limit for MKM 2007, it was necessary to remove the appendix from the paper [3] and instead provide its contents through an alternative medium. Distributed on the respective web sites of the paper’s authors, the contents are reproduced in this document. The reader will ..."
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Abstract. Due to a strict page limit for MKM 2007, it was necessary to remove the appendix from the paper [3] and instead provide its contents through an alternative medium. Distributed on the respective web sites of the paper’s authors, the contents are reproduced in this document. The reader will find herein an example in which the method of the above mentioned paper is applied to a typical text. The example is given first in a plain text, then as a version annotated according to the method proposed in this paper. A Ring theory example In the following three sections, there are to be found three different views of the same document. This example is a brief selection from [1], and provides a definition of an algebraic ring with some brief corollaries. It was chosen because it concisely exhibits most of our developments in a very accessible text. The observant reader will note that, although the examples throughout the main
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
(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.
◮ Analyzing theories, ◮ Verifying proofs, ◮ Assist tutoring,
, 2006
"... Abstraction with nouns and adjectives ..."
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.