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Grammatical Framework: A TypeTheoretical Grammar Formalism
, 2003
"... Grammatical Framework (GF) is a specialpurpose functional language for defining grammars. It uses a Logical Framework (LF) for a description of abstract syntax, and adds to this a notation for defining concrete syntax. GF grammars themselves are purely declarative, but can be used both for lineariz ..."
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Cited by 101 (23 self)
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Grammatical Framework (GF) is a specialpurpose functional language for defining grammars. It uses a Logical Framework (LF) for a description of abstract syntax, and adds to this a notation for defining concrete syntax. GF grammars themselves are purely declarative, but can be used both for linearizing syntax trees and parsing strings. GF can describe both formal and natural languages. The key notion of this description is a grammatical object, which is not just a string, but a record that contains all information on inflection and inherent grammatical features such as number and gender in natural languages, or precedence in formal languages. Grammatical objects have a type system, which helps to eliminate runtime errors in language processing. In the same way as an LF, GF uses...
P.rex: An Interactive Proof Explainer
 Automated Reasoning — 1st International Joint Conference, IJCAR 2001, number 2083 in LNAI
, 2001
"... This paper outlines the interactive proof explanation system P.rex, which adapts its explanation to the user and allows him anytime to utter questions or requests, to which it reacts flexibly. As a generic system, it can be connected to different theorem provers. The distribution is available vi ..."
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Cited by 30 (2 self)
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This paper outlines the interactive proof explanation system P.rex, which adapts its explanation to the user and allows him anytime to utter questions or requests, to which it reacts flexibly. As a generic system, it can be connected to different theorem provers. The distribution is available via the P.rex home page at http://www.ags.unisb.de/~prex.
ΩMEGA: Computer supported mathematics
 IN: PROCEEDINGS OF THE 27TH GERMAN CONFERENCE ON ARTIFICIAL INTELLIGENCE (KI 2004)
, 2004
"... The year 2004 marks the fiftieth birthday of the first computer generated proof of a mathematical theorem: “the sum of two even numbers is again an even number” (with Martin Davis’ implementation of Presburger Arithmetic in 1954). While Martin Davis and later the research community of automated dedu ..."
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Cited by 5 (4 self)
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The year 2004 marks the fiftieth birthday of the first computer generated proof of a mathematical theorem: “the sum of two even numbers is again an even number” (with Martin Davis’ implementation of Presburger Arithmetic in 1954). While Martin Davis and later the research community of automated deduction used machine oriented calculi to find the proof for a theorem by automatic means, the Automath project of N.G. de Bruijn – more modest in its aims with respect to automation – showed in the late 1960s and early 70s that a complete mathematical textbook could be coded and proofchecked by a computer. Classical theorem proving procedures of today are based on ingenious search techniques to find a proof for a given theorem in very large search spaces – often in the range of several billion clauses. But in spite of many successful attempts to prove even open mathematical problems automatically, their use in everyday mathematical practice is still limited. The shift
A proofcentric approach to mathematical assistants
 Journal of Applied Logic: Special Issue on Mathematics Assistance Systems
, 2005
"... We present an approach to mathematical assistants which uses readable, executable proof scripts as the central language for interaction. We examine an implementation that combines the Isar language, the Isabelle theorem prover and the IsaPlanner proof planner. We argue that this synergy provides a f ..."
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Cited by 5 (1 self)
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We present an approach to mathematical assistants which uses readable, executable proof scripts as the central language for interaction. We examine an implementation that combines the Isar language, the Isabelle theorem prover and the IsaPlanner proof planner. We argue that this synergy provides a flexible environment for the exploration, certification, and presentation of mathematical proof.
Integrating Proof Assistants as Reasoning and Verification Tools into a Scientific WYSIWYG Editor
, 2005
"... A major problem for the acceptance of mathematical proof assistance systems in mathematical practise is the shortcomings of their user interfaces. Often the interfaces are developed bottomup starting from the mathematical proof assistance system. Therefore they usually focus on the individual syste ..."
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Cited by 3 (1 self)
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A major problem for the acceptance of mathematical proof assistance systems in mathematical practise is the shortcomings of their user interfaces. Often the interfaces are developed bottomup starting from the mathematical proof assistance system. Therefore they usually focus on the individual system and its proof development paradigm and neglect traditional forms to communicate proofs as used by mathematicians. To address this problem we propose a topdown approach where we start from an existing scientific WYSIWYG text editor which supports the preparation of mathematical publications in high quality typesetting and integrate a mathematical proof assistance system to support proof development and validation. Concretely, we extend the document format of the text editor by semantic markup to encode formal mathematical content and to communicate with the formal system. Additionally we provide interaction markup defining contextsensitive means to control the mathematical proof assistance system through the text editor.
Computer Supported Formal Work: Towards a Digital Mathematical Assistant
 STUDIES IN LOGIC, GRAMMAR AND RHETORIC
, 2007
"... The year 2004 marked the fiftieth birthday of the first computer generated proof of a mathematical theorem: “the sum of two even numbers is again an even number ” (with Martin Davis ’ implementation of Presburger Arithmetic in 1954). While Martin Davis and later the research community of automated ..."
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Cited by 1 (1 self)
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The year 2004 marked the fiftieth birthday of the first computer generated proof of a mathematical theorem: “the sum of two even numbers is again an even number ” (with Martin Davis ’ implementation of Presburger Arithmetic in 1954). While Martin Davis and later the research community of automated deduction used machine oriented calculi to find the proof for a theorem by automatic means, the Automath project of N.G. de Bruijn – more modest in its aims with respect to automation – showed in the late 1960s and early 70s that a complete mathematical textbook could be coded and proofchecked by a computer. Roughly at the same time in 1973, the Mizar project started as an attempt to reconstruct mathematics based on computers. Since 1989, the most important activity in the Mizar project has been the development of a database for mathematics. International cooperation resulted in creating a database which includes more than 7000 definitions of mathematical concepts and more than 42000 theorems. The work by
Formal Proof: Reconciling Correctness and Understanding
, 2009
"... Hilbert’s concept of formal proof is an ideal of rigour for mathematics which has important applications in mathematical logic, but seems irrelevant for the practice of mathematics. The advent, in the last twenty years, of proof assistants was followed by an impressive record of deep mathematical th ..."
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Hilbert’s concept of formal proof is an ideal of rigour for mathematics which has important applications in mathematical logic, but seems irrelevant for the practice of mathematics. The advent, in the last twenty years, of proof assistants was followed by an impressive record of deep mathematical theorems formally proved. Formal proof is practically achievable. With formal proof, correctness reaches a standard that no penandpaper proof can match, but an essential component of mathematics — the insight and understanding — seems to be in short supply. So, what makes a proof understandable? To answer this question we first suggest a list of symptoms of understanding. We then propose a vision of an environment in which users can write and check formal proofs as well as query them with reference to the symptoms of understanding. In this way, the environment reconciles the main features of proof: correctness and understanding.
Proof Development with ΩMEGA: √2 Is Irrational
, 2002
"... Freek Wiedijk proposed the wellknown theorem about the irrationality of √2 as a case study and used this theorem for a comparison of fifteen (interactive) theorem proving systems, which were asked to present their solution (see [48]). This represents an important shift of emphasis in the field of a ..."
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Freek Wiedijk proposed the wellknown theorem about the irrationality of √2 as a case study and used this theorem for a comparison of fifteen (interactive) theorem proving systems, which were asked to present their solution (see [48]). This represents an important shift of emphasis in the field of automated deduction away from the somehow artificial problems of the past as represented, for example, in the test set of the TPTP library [45] back to real mathematical challenges. In this paper we present an overview of the Ωmega system as far as it is relevant for the purpose of this paper and show the development of a proof for this theorem.