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Proving and Computing: a certified version of the Buchberger's algorithm
, 1997
"... This paper shows on a nontrivial example that it is possible to mix proving and computing using current technologies. We present a proof of the Buchberger's algorithm that has been developed in the Coq proof assistant. The formulation of the algorithm in Coq can then be efficiently compiled an ..."
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This paper shows on a nontrivial example that it is possible to mix proving and computing using current technologies. We present a proof of the Buchberger's algorithm that has been developed in the Coq proof assistant. The formulation of the algorithm in Coq can then be efficiently compiled and used to do computation.
Interactive Theorem Proving with Temporal Logic
 JOURNAL OF SYMBOLIC COMPUTATION
, 1996
"... In this paper, we present a theorem prover for linear temporal logic. Our goal is to extend the capabilities of existing interactive and automatic systems for verifying temporal properties of software and hardware systems. We focus on increasing the eoeectiveness of user interaction in such systems. ..."
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In this paper, we present a theorem prover for linear temporal logic. Our goal is to extend the capabilities of existing interactive and automatic systems for verifying temporal properties of software and hardware systems. We focus on increasing the eoeectiveness of user interaction in such systems. In particular, we extend the techniques of proof by pointing and point and shoot for mousedriven proof construction in ørstorder logic to temporal logic. In addition, we show how to generate text from proofs by extending a previously given translation for ørstorder logic to the temporal operators. Our theorem prover implements an inference system for temporal logic that we have deøned. The inference rules of this system are more intuitive than the rules commonly given for temporal logics and thus they are better suited to our goals. We present this inference system and prove that it is sound and complete with respect to a known system.
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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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
Presenting Proofs Using Logicographic Symbols
 In Proc. of the Workshop on Proof Transformation and Presentation and Proof Complexities (PTP01
, 2001
"... Abstract. Mathematics has a rich tradition in creating symbols and notation that is soundly integrated into the syntax of the underlying formal language and, at the same time, conveys the intuition behind the concepts described by the symbols and notation. Continuing this idea, in the Theorema syste ..."
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Abstract. Mathematics has a rich tradition in creating symbols and notation that is soundly integrated into the syntax of the underlying formal language and, at the same time, conveys the intuition behind the concepts described by the symbols and notation. Continuing this idea, in the Theorema system, with the new feature of logicographic symbols, we now provide a means to invent arbitrary new symbols and notation. In this paper we describe how logicographic symbols can be created, declared, and afterwards used as a part of the formal language of Theorema with an example. Also with logicographic symbols, formal proof text automatically generated by Theorema provers can become easy to read in a way that resembles telling a pictorial story about the mathematical concepts involved. 1
Expressing References to Rules in Proof Presentations
"... Presenting machinegenerated proofs in natural language has deserved considerable attention over the past decade. Despite respectable progress achieved, the produced texts appear stilted in several places and are less coherent than comparable textbook presentations. One of the reasons for these shor ..."
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Presenting machinegenerated proofs in natural language has deserved considerable attention over the past decade. Despite respectable progress achieved, the produced texts appear stilted in several places and are less coherent than comparable textbook presentations. One of the reasons for these shortcomings is the low degree of variation in which references to the pieces of knowledge underlying a proof, theorems and axioms, are made. Addressing this deficit, we present a repertoire of descriptions of inference rules in proof presentations that indicate accurately how the underlying piece of generic knowledge is used in an involved inference. Generated expressions refer to conceptualizations of a rule's components, including partial instantiations, the role of assertions in inferencing, and focus emphasizing structural hints. Quite interestingly, some proof justifications are altered opportunistcally to enable the use of referring expressions that make the text fluent. The produced texts make reasoning lines much easier to follow, which is particularly valuable for texts serving didactic purposes.
A Multilingual NaturalLanguage Interface to Regular Expressions
 Proceedings of the International Workshop on Finite State Methods in Natural Language Processing
, 1998
"... This report explains a naturallanguage interface to the frmilsm of XFST (Xerox Finite State Tool), which is a rich language used for specifying finite state automata and transducers. By using the interface, it is possible to give input to XFST in English aad French, as well as to translate forma ..."
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This report explains a naturallanguage interface to the frmilsm of XFST (Xerox Finite State Tool), which is a rich language used for specifying finite state automata and transducers. By using the interface, it is possible to give input to XFST in English aad French, as well as to translate formal XFST code into these languages. It is also possible to edit XFST source files and their naturallaaguage equivalents interac tively, in parallel.
Formal and informal software specifications
, 2005
"... The topic of this thesis is to bridge the gap between formal and informal software specifications. Formal specifications are required for the use of formal methods to verify the correctness of software. If we expect formal methods to be used in realistic software development projects, we need to ena ..."
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The topic of this thesis is to bridge the gap between formal and informal software specifications. Formal specifications are required for the use of formal methods to verify the correctness of software. If we expect formal methods to be used in realistic software development projects, we need to enable people with varying levels of familiarity with formal specification languages to understand, maintain and create formal specifications. To address these problems, we provide a tool for translating specifications written in the formal language OCL, a substandard of UML, to natural language. We also provide a multilingual, syntaxdirected editor where OCL and natural language specifications can be edited in parallel. The implementation of our work is to a large extent based on the Grammatical Framework (GF). GF is a grammar formalism based on type theory, which provides a special purpose language for defining grammars, and a compiler for this language. We have developed a GF grammar for specifications
Certified and Portable Mathematical Documents from Formal Contexts
, 2001
"... This paper deals with the problem of generating interactive natural language documents based on formal mathematics. It describes how formal mathematical developments, carried out in the type theoretical theorem prover Coq, can be transformed to readable and interactive documents viewable using s ..."
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This paper deals with the problem of generating interactive natural language documents based on formal mathematics. It describes how formal mathematical developments, carried out in the type theoretical theorem prover Coq, can be transformed to readable and interactive documents viewable using standard Web browser technology. The transformation process produces documents encoded in an xml application called omdoc, suited for describing mathematical documents.
XML, Stylesheets and the Remathematization of Formal Content
, 2001
"... An important part of the descriptive power of mathematics derives from its ability to represent formal concepts in a highly evolved, twodimensional system of symbolic notations. Tools for the mechanisation of mathematics and the automation of formal reasoning must eventually face the problem of re ..."
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An important part of the descriptive power of mathematics derives from its ability to represent formal concepts in a highly evolved, twodimensional system of symbolic notations. Tools for the mechanisation of mathematics and the automation of formal reasoning must eventually face the problem of remathematization of the logical, symbolic content of the information, especially in view of their integration with the World Wide Web. In a different work [APSS00c], we already discussed the pivotal role that XML [eXtensible Markup Language] technology [XML] is likely to play in such an integration. In this paper, we focus on the problem of (Web) publishing, advocating the use of XSL [eXtensible Stylesheet Language] Transformations, in conjunction with MathML [Mathematical Markup Language], as a standard, application independent and modular way for associating notation to formal mathematical content.
The Proofassistant Yarrow
, 1998
"... Yarrow is an interactive proof assistant based on the theory of Pure Type Systems, a family of typed lambda calculi. Yarrow has been designed as a flexible environment for experimentation with various typed lambda calculi. It offers both graphical and textual interfaces. It has been coded entirel ..."
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Yarrow is an interactive proof assistant based on the theory of Pure Type Systems, a family of typed lambda calculi. Yarrow has been designed as a flexible environment for experimentation with various typed lambda calculi. It offers both graphical and textual interfaces. It has been coded entirely in Haskell, making use of the Fudget library for the graphical interface. In this paper we concentrate on the software architecture of Yarrow, in particular the use of monads, the coupling of user interface and proof engine, polymorphic output routines, and flexible representations of lambda terms. We also treat the presentation of proofs in the flagstyle format. 1 1 Introduction In this paper we describe the system Yarrow, an interactive proof assistant based on the theory of Pure Type Systems, a family of typed lambda calculi. In typed lambda calculi, theorems and proofs can be represented as welltyped terms, proof checking amounts to type checking, and proof construction to the...