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Universes for Generic Programs and Proofs in Dependent Type Theory
 Nordic Journal of Computing
, 2003
"... We show how to write generic programs and proofs in MartinL of type theory. To this end we consider several extensions of MartinL of's logical framework for dependent types. Each extension has a universes of codes (signatures) for inductively defined sets with generic formation, introductio ..."
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Cited by 52 (2 self)
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We show how to write generic programs and proofs in MartinL of type theory. To this end we consider several extensions of MartinL of's logical framework for dependent types. Each extension has a universes of codes (signatures) for inductively defined sets with generic formation, introduction, elimination, and equality rules. These extensions are modeled on Dybjer and Setzer's finitely axiomatized theories of inductiverecursive definitions, which also have a universe of codes for sets, and generic formation, introduction, elimination, and equality rules.
Setoids in Type Theory
, 2000
"... Formalising mathematics in dependent type theory often requires to use setoids, i.e. types with an explicit equality relation, as a representation of sets. This paper surveys some possible denitions of setoids and assesses their suitability as a basis for developing mathematics. In particular, we ..."
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Cited by 43 (3 self)
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Formalising mathematics in dependent type theory often requires to use setoids, i.e. types with an explicit equality relation, as a representation of sets. This paper surveys some possible denitions of setoids and assesses their suitability as a basis for developing mathematics. In particular, we argue that a commonly advocated approach to partial setoids is unsuitable, and more generally that total setoids seem better suited for formalising mathematics. 1
General recursion via coinductive types
 Logical Methods in Computer Science
"... Vol. 1 (2:1) 2005, pp. 1–28 ..."
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Program Extraction from Large Proof Developments
, 2003
"... It is well known that mathematical proofs often contain (abstract) algorithms, but although these algorithms can be understood by a human, it still takes a lot of time and effort to implement this algorithm on a computer; moreover, one runs the risk of making mistakes in the process. From a fully... ..."
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Cited by 9 (5 self)
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It is well known that mathematical proofs often contain (abstract) algorithms, but although these algorithms can be understood by a human, it still takes a lot of time and effort to implement this algorithm on a computer; moreover, one runs the risk of making mistakes in the process. From a fully...
Higher Order Abstract Syntax in Type Theory
"... We develop a general tool to formalize higherorder languages and reason about them in a prooftool based on type theory (Coq). A language is specified by its signature, which consists of sets of sort and operation names and typing rules. These rules prescribe the sorts and bindings of each operat ..."
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Cited by 2 (1 self)
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We develop a general tool to formalize higherorder languages and reason about them in a prooftool based on type theory (Coq). A language is specified by its signature, which consists of sets of sort and operation names and typing rules. These rules prescribe the sorts and bindings of each operation. An algebra of terms is associated to a signature, using de Bruijn notation. Then a higherorder notation is built on top of the de Bruijn level, so that the user can work with metavariables instead of de Bruijn indices. We also provide recursion and induction principles formulated directly on the higherorder syntax. This generalizes work on the Hybrid approach to higherorder syntax in Isabelle and our earlier work on a constructive extension to Hybrid formalized in Coq. In particular, a large class of theorems that must be repeated for each object language in Hybrid is done once in our new approach and can be applied directly to each object language.
Coinductive Field of Exact Real Numbers and General Corecursion
, 2006
"... In this article we present a method to define algebraic structure (field operations) on a representation of real numbers by coinductive streams. The field operations will be given in two algorithms (homographic and quadratic algorithm) that operate on streams of Möbius maps. The algorithms can be se ..."
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Cited by 2 (0 self)
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In this article we present a method to define algebraic structure (field operations) on a representation of real numbers by coinductive streams. The field operations will be given in two algorithms (homographic and quadratic algorithm) that operate on streams of Möbius maps. The algorithms can be seen as coalgebra maps on the coalgebra of streams and hence they will be formalised as general corecursive functions. We use the machinery of Coq proof assistant for coinductive types to present the formalisation.
PROGRAMMING LANGUAGE by
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Use policy The fulltext may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or notforprofit purposes provided that: • a full bibliographic reference is made to the original source • a link is made to the metadata record in Durham ETheses • the fulltext is not changed in any way The fulltext must not be sold in any format or medium without the formal permission of the copyright holders. Please consult the full Durham ETheses policy for further details.
Privacy in Data Mining Using Formal Methods
"... Abstract. There is growing public concern about personal data collected by both private and public sectors. People have very little control over what kinds of data are stored and how such data is used. Moreover, the ability to infer new knowledge from existing data is increasing rapidly with advance ..."
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Abstract. There is growing public concern about personal data collected by both private and public sectors. People have very little control over what kinds of data are stored and how such data is used. Moreover, the ability to infer new knowledge from existing data is increasing rapidly with advances in database and data mining technologies. We describe a solution which allows people to take control by specifying constraints on the ways in which their data can be used. User constraints are represented in formal logic, and organizations that want to use this data provide formal proofs that the software they use to process data meets these constraints. Checking the proof by an independent verifier demonstrates that user constraints are (or are not) respected by this software. Our notion of “privacy correctness” differs from general software correctness in two ways. First, properties of interest are simpler and thus their proofs should be easier to automate. Second, this kind of correctness is stricter; in addition to showing a certain relation between input and output is realized, we must also show that only operations that respect privacy constraints are applied during execution. We have therefore an intensional notion of correctness, rather that the usual extensional one. We discuss how our mechanism can be put into practice, and we present the technical aspects via an example. Our example shows how users can exercise control when their data is to be used as input to a decision tree learning algorithm. We have formalized the example and the proof of preservation of privacy constraints in Coq. 1
DOI: 10.1017/S0956796802004501 Printed in the United Kingdom Setoids in type theory
"... Formalising mathematics in dependent type theory often requires to represent sets as setoids, i.e. types with an explicit equality relation. This paper surveys some possible definitions of setoids and assesses their suitability as a basis for developing mathematics. According to whether the equality ..."
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Formalising mathematics in dependent type theory often requires to represent sets as setoids, i.e. types with an explicit equality relation. This paper surveys some possible definitions of setoids and assesses their suitability as a basis for developing mathematics. According to whether the equality relation is required to be reflexive or not we have total or partial setoid, respectively. There is only one definition of total setoid, but four different definitions of partial setoid, depending on four different notions of setoid function. We prove that one approach to partial setoids in unsuitable, and that the other approaches can be divided in two classes of equivalence. One class contains definitions of partial setoids that are equivalent to total setoids; the other class contains an inherently different definition, that has been useful in the modeling of type systems. We also provide some elements of discussion on the merits of each approach from the viewpoint of formalizing mathematics. In particular, we exhibit a difficulty with the common definition of subsetoids in the partial setoid approach. 1