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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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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 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
A DocumentOriented Coq Plugin for TEXmacs
, 2006
"... This article discusses the integration of the authoring of a mathematical document with the formalisation of the mathematics contained in that document. To achieve this we have started the development of a Coq plugin for the TEXmacs scientific editor, called tmEgg. TEXmacs allows the wysiwyg editing ..."
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Cited by 5 (4 self)
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This article discusses the integration of the authoring of a mathematical document with the formalisation of the mathematics contained in that document. To achieve this we have started the development of a Coq plugin for the TEXmacs scientific editor, called tmEgg. TEXmacs allows the wysiwyg editing of mathematical documents, much in the style of LATEX. Our plugin allows to integrate into a TEXmacs document mathematics formalised in the Coq proof assistant: formal definitions, lemmas and proofs. The plugin is still under development. Its main current hallmark is a documentconsistent interaction model, instead of the calculatorlike approach usual for TEXmacs plugins. This means that the Coq code in the TEXmacs document is interpreted as one (consistent) Coq file: executing a Coq command in the document means to execute it in the context (state) of all the Coq commands before it. 1
Grammars as software libraries
, 2008
"... Grammars of natural languages are needed in programs like natural language interfaces and dialogue systems, but also more generally, in software localization. Writing grammar implementations is a highly specialized task. For various reasons, no libraries have been available to ease this task. This p ..."
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Cited by 4 (1 self)
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Grammars of natural languages are needed in programs like natural language interfaces and dialogue systems, but also more generally, in software localization. Writing grammar implementations is a highly specialized task. For various reasons, no libraries have been available to ease this task. This paper shows how grammar libraries can be written in GF (Grammatical Framework), focusing on the software engineering aspects rather than the linguistic aspects. As an implementation of the approach, the GF Resource Grammar Library currently comprises ten languages. As an application, a translation system from formalized mathematics to text in three languages is outlined. 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.
Proof Assistants: history, ideas and future
"... In this paper we will discuss the fundamental ideas behind proof assistants: What are they and what is a proof anyway? We give a short history of the main ideas, emphasizing the way they ensure the correctness of the mathematics formalized. We will also briefly discuss the places where proof assista ..."
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In this paper we will discuss the fundamental ideas behind proof assistants: What are they and what is a proof anyway? We give a short history of the main ideas, emphasizing the way they ensure the correctness of the mathematics formalized. We will also briefly discuss the places where proof assistants are used and how we envision their extended use in the future. While being an introduction into the world of proof assistants and the main issues behind them, this paper is also a position paper that pushes the further use of proof assistants. We believe that these systems will become the future of mathematics, where definitions, statements, computations and proofs are all available in a computerized form. An important application is and will be in computer supported modelling and verification of systems. But their is still along road ahead and we will indicate what we believe is needed for the further proliferation of proof assistants.
Formalizing Arrow’s theorem
"... Abstract. We present a small project in which we encoded a proof of Arrow’s theorem – probably the most famous results in the economics field of social choice theory – in the computer using the Mizar system. We both discuss the details of this specific project, as well as describe the process of for ..."
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Abstract. We present a small project in which we encoded a proof of Arrow’s theorem – probably the most famous results in the economics field of social choice theory – in the computer using the Mizar system. We both discuss the details of this specific project, as well as describe the process of formalization (encoding proofs in the computer) in general. Keywords: formalization of mathematics, Mizar, social choice theory, Arrow’s theorem, GibbardSatterthwaite theorem, proof errors.
Comparing two userfriendly formal languages for mathematics: Weak Type Theory and Mizar
, 2004
"... syntax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.2.3 Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.2.4 Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.2.5 Binders . . . . . . . . . . . . . . . . . . . . . ..."
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Cited by 2 (0 self)
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syntax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.2.3 Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.2.4 Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.2.5 Binders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.2.6 Phrases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.2.7 Statements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.2.8 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.2.9 Contexts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.2.10 Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.2.11 Books . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.3 A derivation system for WTT . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.3.1 Weak types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.3.2 The preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.3.3 Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.3.4 Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.3.5 Binders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.3.6 Phrases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.3.7 Statements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.3.8 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.3.9 Contexts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.3.10 Books . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ....
Student Proof Exercises using MathsTiles and Isabelle/HOL in an Intelligent Book
"... The Intelligent Book project aims to improve online education by designing materials that can model the subject matter they teach, in the manner of a Reactive Learning Environment. In this paper, we investigate using an automated proof assistant, particularly Isabelle/HOL, as the model supporting fi ..."
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The Intelligent Book project aims to improve online education by designing materials that can model the subject matter they teach, in the manner of a Reactive Learning Environment. In this paper, we investigate using an automated proof assistant, particularly Isabelle/HOL, as the model supporting first year undergraduate exercises in which students write proofs in number theory. Automated proof assistants are generally considered to be difficult for novices to learn. We examine whether, by providing a very specialised interface, it is possible to build something that is usable enough to be of educational value. To ensure students cannot “game the system ” the exercise avoids tacticchoosing interaction styles, but asks the student to write out the proof. Proofs are written using MathsTiles: composable tiles that resemble written mathematics. Unlike traditional syntaxdirected editors, MathsTiles allow students to keep many answer fragments on the canvas at the same time, and do not constrain the order in which an answer is written. Also, the tile syntax does not need to match the underlying Isar syntax exactly, and different tiles can be used for different questions. The exercises take place within the context of an Intelligent Book. We performed a user study and qualitative analysis of the system. Some users were able to complete proofs with much less training than is usual for the automated proof assistant itself, but there remain significant usability issues to overcome.