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17
An Axiomatic Approach to Adequacy
 University of Aarhus
, 1996
"... is permitted for educational or research use on condition that this copyright notice is included in any copy. See back inner page for a list of recent publications in the BRICS Dissertation Series. Copies may be obtained by contacting: BRICS ..."
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Cited by 26 (1 self)
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is permitted for educational or research use on condition that this copyright notice is included in any copy. See back inner page for a list of recent publications in the BRICS Dissertation Series. Copies may be obtained by contacting: BRICS
Arrows, like monads, are monoids
 Proc. of 22nd Ann. Conf. on Mathematical Foundations of Programming Semantics, MFPS XXII, v. 158 of Electron. Notes in Theoret. Comput. Sci
, 2006
"... Monads are by now wellestablished as programming construct in functional languages. Recently, the notion of “Arrow ” was introduced by Hughes as an extension, not with one, but with two type parameters. At first, these Arrows may look somewhat arbitrary. Here we show that they are categorically fai ..."
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Cited by 16 (1 self)
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Monads are by now wellestablished as programming construct in functional languages. Recently, the notion of “Arrow ” was introduced by Hughes as an extension, not with one, but with two type parameters. At first, these Arrows may look somewhat arbitrary. Here we show that they are categorically fairly civilised, by showing that they correspond to monoids in suitable subcategories of bifunctors C op ×C → C. This shows that, at a suitable level of abstraction, arrows are like monads — which are monoids in categories of functors C → C. Freyd categories have been introduced by Power and Robinson to model computational effects, well before Hughes ’ Arrows appeared. It is often claimed (informally) that Arrows are simply Freyd categories. We shall make this claim precise by showing how monoids in categories of bifunctors exactly correspond to Freyd categories.
The Girard Translation Extended with Recursion
 In Proceedings of Computer Science Logic
, 1995
"... This paper extends CurryHoward interpretations of Intuitionistic Logic (IL) and Intuitionistic Linear Logic (ILL) with rules for recursion. The resulting term languages, the rec calculus and the linear rec calculus respectively, are given sound categorical interpretations. The embedding of ..."
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Cited by 11 (0 self)
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This paper extends CurryHoward interpretations of Intuitionistic Logic (IL) and Intuitionistic Linear Logic (ILL) with rules for recursion. The resulting term languages, the rec calculus and the linear rec calculus respectively, are given sound categorical interpretations. The embedding of proofs of IL into proofs of ILL given by the Girard Translation is extended with the rules for recursion, such that an embedding of terms of the rec calculus into terms of the linear rec calculus is induced via the extended CurryHoward isomorphisms. This embedding is shown to be sound with respect to the categorical interpretations. Full version of paper to appear in Proceedings of CSL '94, LNCS 933, 1995. y Basic Research in Computer Science, Centre of the Danish National Research Foundation. Contents 1 Introduction 4 2 The Categorical Picture 6 2.1 Previous Work and Related Results : : : : : : : : : : : : : : : : : : : : : : 6 2.2 How to deal with parameters : : : : : : : ...
Denotational Semantics of Object Specification
 ACTA INFORMATICA
, 1998
"... From an arbitrary temporal logic institution we show how to set up the corresponding institution of objects. The main properties of the resulting institution are studied and used in establishing a categorial, denotational semantics of several basic constructs of object specification, namely aggre ..."
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Cited by 8 (3 self)
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From an arbitrary temporal logic institution we show how to set up the corresponding institution of objects. The main properties of the resulting institution are studied and used in establishing a categorial, denotational semantics of several basic constructs of object specification, namely aggregation (parallel composition), interconnection, abstraction (interfacing) and monotonic specialization. A duality is established between the category of theories and the category of objects, as a corollary of the Galois correspondence between these concrete categories. The special case of linear temporal logic is analysed in detail in order to show that categorial products do reflect interleaving and reducts may lead to internal nondeterminism.
Induction and recursion on the partial real line with applications to Real PCF
 Theoretical Computer Science
, 1997
"... The partial real line is an extension of the Euclidean real line with partial real numbers, which has been used to model exact real number computation in the programming language Real PCF. We introduce induction principles and recursion schemes for the partial unit interval, which allow us to verify ..."
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Cited by 5 (1 self)
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The partial real line is an extension of the Euclidean real line with partial real numbers, which has been used to model exact real number computation in the programming language Real PCF. We introduce induction principles and recursion schemes for the partial unit interval, which allow us to verify that Real PCF programs meet their specification. They resemble the socalled Peano axioms for natural numbers. The theory is based on a domainequationlike presentation of the partial unit interval. The principles are applied to show that Real PCF is universal in the sense that all computable elements of its universe of discourse are definable. These elements include higherorder functions such as integration operators. Keywords: Induction, coinduction, exact real number computation, domain theory, Real PCF, universality. Introduction The partial real line is the domain of compact real intervals ordered by reverse inclusion [28,21]. The idea is that singleton intervals represent total rea...
A General Adequacy Result for a Linear Functional Language
 BRICS Research Series
, 1994
"... A main concern of the paper will be a CurryHoward interpretation of Intuitionistic Linear Logic. It will be extended with recursion, and the resulting functional programming language will be given operational as well as categorical semantics. ..."
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Cited by 4 (1 self)
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A main concern of the paper will be a CurryHoward interpretation of Intuitionistic Linear Logic. It will be extended with recursion, and the resulting functional programming language will be given operational as well as categorical semantics.
Logic of Predicates With Explicit Substitutions
 Mathematical Foundations of Computer Science 1996, 21st Symposium
, 1996
"... This paper aims at supporting the same idea. Our justification of the claim is, however, quite different from the one offered by Girard. The latter, cf. [9], translates every sequent of the usual propositional logic (classical, or intuitionistic) into a sequent of commutative linear logic. Then one ..."
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This paper aims at supporting the same idea. Our justification of the claim is, however, quite different from the one offered by Girard. The latter, cf. [9], translates every sequent of the usual propositional logic (classical, or intuitionistic) into a sequent of commutative linear logic. Then one shows that a sequent can be proved classically, resp., intuitionistically, iff its translation can be proved linearly. By contrast, our embedding only works on the level of predicate logic. We show that every theory of classical logic of predicates with equality lives as a theory within a noncommutative intuitionistic substructural logic: the logic of predicates with equality and explicit substitution. Also, our explanation does not require to call upon so called exponentials  the modalities introduced by Girard just to facilitate his embedding. Our construction is also different from other proposals to move substitutions from the level of metatheory to the theory of logic, cf. [16]. They add substitutions as modal constructions. Here, substitutions are considered new atomic formulae.
Elements of Basic Category Theory
, 1996
"... Category theory provides an elegant and powerful means of expressing relationships across a wide area of mathematics. But further than this it has had a considerable impact on the conceptual basis both of mathematics and many parts of theoretical computer science. Important connections in computer s ..."
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Category theory provides an elegant and powerful means of expressing relationships across a wide area of mathematics. But further than this it has had a considerable impact on the conceptual basis both of mathematics and many parts of theoretical computer science. Important connections in computer science include the design of both functional and imperative programming languages, semantic models of programming languages, semantics of concurrency, specification and development of algorithms, type theory and polymorphism, specification languages, algebraic semantics, constructive logic and automata theory. The purpose of this text is to provide a soft stairway to this infectious and attractive field of mathematics. We provide here a careful and detailed explanation of "basic elements", or more precisely, from the elementary definitions to adjoint situations. The general approach used here is to provide a careful motivation for the majority of constructions as well as a detailed presentat...
The Greedy Algorithms Class: Formalization, Synthesis and Generalization
, 1995
"... On the first hand, this report studies the class of Greedy Algorithms in order to find an as systematic as possible strategy that could be applied to the specification of some problems to lead to a correct program solving that problem. On the other hand, the standard formalisms underlying the G ..."
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On the first hand, this report studies the class of Greedy Algorithms in order to find an as systematic as possible strategy that could be applied to the specification of some problems to lead to a correct program solving that problem. On the other hand, the standard formalisms underlying the Greedy Algorithms (matroid, greedoid and matroid embedding) which are dependent on the particular type set are generalized to a formalism independent of any data type based on an algebraic specification setting.
A Model of Intuitionistic Affine Logic from Stable Domain Theory
 In Proceedings of ICALP '94, LNCS
, 1994
"... Girard worked with the category of coherence spaces and continuous stable maps and observed that the functor that forgets the linearity of linear stable maps has a left adjoint. This fundamental observation gave rise to the discovery of Linear Logic. Since then, the category of coherence spaces and ..."
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Girard worked with the category of coherence spaces and continuous stable maps and observed that the functor that forgets the linearity of linear stable maps has a left adjoint. This fundamental observation gave rise to the discovery of Linear Logic. Since then, the category of coherence spaces and linear stable maps, with the comonad induced by the adjunction, has been considered a canonical model of Linear Logic. Now, the same phenomenon is present if we consider the category of pre dI domains and continuous stable maps, and the category of dI domains and linear stable maps; the functor that forgets the linearity has a left adjoint. This gives an alternative model of Intuitionistic Linear Logic. It turns out that this adjunction can be factored in two adjunctions yielding a model of Intuitionistic Affine Logic; the category of pre dI domains and affine stable functions. It is the goal of this paper to show that this category is actually a model of Intuitionistic Affine Logic, and to ...