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Theory Interpretation in Simple Type Theory
 HIGHERORDER ALGEBRA, LOGIC, AND TERM REWRITING, VOLUME 816 OF LECTURE NOTES IN COMPUTER SCIENCE
, 1993
"... Theory interpretation is a logical technique for relating one axiomatic theory to another with important applications in mathematics and computer science as well as in logic itself. This paper presents a method for theory interpretation in a version of simple type theory, called lutins, which admit ..."
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Cited by 37 (17 self)
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Theory interpretation is a logical technique for relating one axiomatic theory to another with important applications in mathematics and computer science as well as in logic itself. This paper presents a method for theory interpretation in a version of simple type theory, called lutins, which admits partial functions and subtypes. The method is patterned on the standard approach to theory interpretation in rstorder logic. Although the method is based on a nonclassical version of simple type theory, it is intended as a guide for theory interpretation in classical simple type theories as well as in predicate logics with partial functions.
A semantic wiki for mathematical knowledge management
 Proceedings of the 1st Workshop on Semantic Wikis, European Semantic Web Conference 2006, Budva, Montenegro, 2006. CEUR Workshop Proceedings. To appear, provisional online version at http://www.eswc2006.org/technologies/ usb/proceedingsworkshops/ eswc200
, 2007
"... SWIM is a semantic wiki for collaboratively building, editing and browsing mathematical knowledge represented in the structural markup language OMDOC. It has been designed to enable groups of scientists to develop new mathematical theories in OMDOC and to enable scholars to browse such a corpus. Aft ..."
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Cited by 37 (15 self)
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SWIM is a semantic wiki for collaboratively building, editing and browsing mathematical knowledge represented in the structural markup language OMDOC. It has been designed to enable groups of scientists to develop new mathematical theories in OMDOC and to enable scholars to browse such a corpus. After a short introduction to semantic wikis and their usefulness for mathematical knowledge, this article presents the architecture and the user interface of the current SWIM prototype and outlines the plans for developing its successor, an ontologybased platform for semantic scientific services that exploit the knowledge and make it accessible to the user. 1
Interpretation of locales in Isabelle: Theories and proof contexts
 MATHEMATICAL KNOWLEDGE MANAGEMENT (MKM 2006), LNAI 4108
, 2006
"... The generic proof assistant Isabelle provides a landscape of specification contexts that is considerably richer than that of most other provers. Theories are the level of specification where objectlogics are axiomatised. Isabelle’s proof language Isar enables local exploration in contexts generated ..."
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Cited by 31 (3 self)
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The generic proof assistant Isabelle provides a landscape of specification contexts that is considerably richer than that of most other provers. Theories are the level of specification where objectlogics are axiomatised. Isabelle’s proof language Isar enables local exploration in contexts generated in the course of natural deduction proofs. Finally, locales, which may be seen as detached proof contexts, offer an intermediate level of specification geared towards reuse. All three kinds of contexts are structured, to different extents. We analyse the “topology ” of Isabelle’s landscape of specification contexts, by means of development graphs, in order to establish what kinds of reuse are possible.
A module calculus for Pure Type Systems
, 1996
"... Several proofassistants rely on the very formal basis of Pure Type Systems. However, some practical issues raised by the development of large proofs lead to add other features to actual implementations for handling namespace management, for developing reusable proof libraries and for separate verif ..."
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Cited by 23 (3 self)
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Several proofassistants rely on the very formal basis of Pure Type Systems. However, some practical issues raised by the development of large proofs lead to add other features to actual implementations for handling namespace management, for developing reusable proof libraries and for separate verification of distincts parts of large proofs. Unfortunately, few theoretical basis are given for these features. In this paper we propose an extension of Pure Type Systems with a module calculus adapted from SMLlike module systems for programming languages. Our module calculus gives a theoretical framework addressing the need for these features. We show that our module extension is conservative, and that type inference in the module extension of a given PTS is decidable under some hypotheses over the considered PTS.
Elements of Mathematical Analysis in PVS
 Ninth international Conference on Theorem Proving in Higher Order Logics TPHOL
, 1996
"... . This paper presents the formalization of some elements of mathematical analysis using the PVS verification system. Our main motivation was to extend the existing PVS libraries and provide means of modelling and reasoning about hybrid systems. The paper focuses on several important aspects of PVS i ..."
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. This paper presents the formalization of some elements of mathematical analysis using the PVS verification system. Our main motivation was to extend the existing PVS libraries and provide means of modelling and reasoning about hybrid systems. The paper focuses on several important aspects of PVS including recent extensions of the type system and discusses their merits and effectiveness. We conclude by a brief comparison with similar developments using other theorem provers. 1 Introduction PVS is a specification and verification system whose ambition is to make formal proofs practical and applicable to large and complex problems. The system is based on a variant of higher order logic which includes complex typing mechanisms such as predicate subtypes or dependent types. It offers an expressive specification language coupled with a theorem prover designed for efficient interactive proof construction. In previous work we have applied PVS to the requirements analysis of a substantially ...
Using LATEX as a semantic markup format
 Mathematics in Computer Science
"... Abstract. One of the great problems of Mathematical Knowledge Management (MKM) systems is to obtain access to a sufficiently large corpus of mathematical knowledge to allow the management/search/navigation techniques developed by the community to display their strength. Such systems usually expect ..."
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Cited by 16 (9 self)
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Abstract. One of the great problems of Mathematical Knowledge Management (MKM) systems is to obtain access to a sufficiently large corpus of mathematical knowledge to allow the management/search/navigation techniques developed by the community to display their strength. Such systems usually expect the mathematical knowledge they operate on in the form of semantically enhanced documents, but mathematicians and publishers in Mathematics have heavily invested into the TEX/LATEX format and workflow. We analyze the current practice of semisemantic markup in LATEX documents and extend it by a markup infrastructure that allows to embed semantic annotations into LATEX documents without changing their visual appearance. This collection of TEX macro packages is called sTEX (semantic TEX) as it allows to markup LATEX documents semantically without leaving the timetried TEX/LATEX workflow, essentially turning LATEX into an MKM format. At the heart of sTEX is a definition mechanism for semantic macros for mathematical objects and a nonstandard scoping construct for them, which is oriented at the semantic dependency relation rather than the document structure. We evaluate the sTEX macro collection on a large case study: the course materials of a twosemester course in Computer Science was annotated semantically and converted to the OMDoc MKM format by Bruce Miller’s LaTeXML system. 1.
Management of Change in Structured Verification
 In Proceedings 15th IEEE International Conference on Automated Software Engineering, number 2000 in ASE
, 2000
"... The use of formal methods in large complex applications implies the need for an evolutionary formal program development in which specification and verification phases are interleaved. But any change of a specification either by adding new parts or by changing erroneous parts affects existing verific ..."
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The use of formal methods in large complex applications implies the need for an evolutionary formal program development in which specification and verification phases are interleaved. But any change of a specification either by adding new parts or by changing erroneous parts affects existing verification work in a subtle way. In this paper we present a truth maintenance system for structured specification and verification. It is based on the simple but powerful notion of a development graph as an underlying datastructure to represent an actual consistent state of a formal development. Based on this notion we try to minimize the consequences of changes of existing verification work. 1. Introduction The application of formal methods in an industrial setting results in an increased complexity of the specification and the corresponding verification. It comprises on the one hand different layers of specifications reflecting the iterated process to refine the requirement specification towa...
IMPS: System Description
 Automated DeductionCADE11, volume 607 of Lecture Notes in Computer Science
, 1992
"... 1 other equally abstract theories. Theory interpretations provide the mechanism for transporting theorems. The little theories style of the axiomatic method is employed extensively in mathematical practice; in [4], we discuss its benets for mechanical theorem provers, and how the approach is used ..."
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Cited by 14 (6 self)
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1 other equally abstract theories. Theory interpretations provide the mechanism for transporting theorems. The little theories style of the axiomatic method is employed extensively in mathematical practice; in [4], we discuss its benets for mechanical theorem provers, and how the approach is used in imps. Logic. Standard mathematical reasoning in many areas focuses on functions and their properties, together with operations on functions. For this reason, imps is based on a version of simple type theory. 1 However, we have adopted a version, called lutins, 2 containing partial functions, because they are ubiquitous in both mathematics and computer science. Although terms, such as 2=0, may be nondenoting, the logic is bivalent and formulas always have a truth value. In particular, an atomic formula is false if any of its constituents is nondenoting. This conventio
An Overview of A Formal Framework For Managing Mathematics
 Annals of Mathematics and Artificial Intelligence
, 2003
"... Mathematics is a process of creating, exploring, and connecting mathematical models. This paper presents an overview of a formal framework for managing the mathematics process as well as the mathematical knowledge produced by the process. The central idea of the framework is the notion of a biform t ..."
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Cited by 13 (6 self)
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Mathematics is a process of creating, exploring, and connecting mathematical models. This paper presents an overview of a formal framework for managing the mathematics process as well as the mathematical knowledge produced by the process. The central idea of the framework is the notion of a biform theory which is simultaneously an axiomatic theory and an algorithmic theory. Representing a collection of mathematical models, a biform theory provides a formal context for both deduction and computation. The framework includes facilities for deriving theorems via a mixture of deduction and computation, constructing sound deduction and computation rules, and developing networks of biform theories linked by interpretations. The framework is not tied to a specific underlying logic; indeed, it is intended to be used with several background logics simultaneously. Many of the ideas and mechanisms used in the framework are inspired by the imps Interactive Mathematical Proof System and the Axiom computer algebra system.
STMM: A Set Theory for Mechanized Mathematics
 JOURNAL OF AUTOMATED REASONING
, 2000
"... Although set theory is the most popular foundation for mathematics, not many mechanized mathematics systems are based on set theory. ZermeloFraenkel (zf) set theory and other traditional set theories are not an adequate foundation for mechanized mathematics. stmm is a version of vonNeumannBerna ..."
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Although set theory is the most popular foundation for mathematics, not many mechanized mathematics systems are based on set theory. ZermeloFraenkel (zf) set theory and other traditional set theories are not an adequate foundation for mechanized mathematics. stmm is a version of vonNeumannBernaysGödel (nbg) set theory that is intended to be a Set Theory for Mechanized Mathematics. stmm allows terms to denote proper classes and to be undened, has a denite description operator, provides a sort system for classifying terms by value, and includes lambdanotation with term constructors for function application and function abstraction. This paper describes stmm and discusses why it is a good foundation for mechanized mathematics.