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21
Inheritance As Implicit Coercion
 Information and Computation
, 1991
"... . We present a method for providing semantic interpretations for languages with a type system featuring inheritance polymorphism. Our approach is illustrated on an extension of the language Fun of Cardelli and Wegner, which we interpret via a translation into an extended polymorphic lambda calculus. ..."
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Cited by 120 (3 self)
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. We present a method for providing semantic interpretations for languages with a type system featuring inheritance polymorphism. Our approach is illustrated on an extension of the language Fun of Cardelli and Wegner, which we interpret via a translation into an extended polymorphic lambda calculus. Our goal is to interpret inheritances in Fun via coercion functions which are definable in the target of the translation. Existing techniques in the theory of semantic domains can be then used to interpret the extended polymorphic lambda calculus, thus providing many models for the original language. This technique makes it possible to model a rich type discipline which includes parametric polymorphism and recursive types as well as inheritance. A central difficulty in providing interpretations for explicit type disciplines featuring inheritance in the sense discussed in this paper arises from the fact that programs can typecheck in more than one way. Since interpretations follow the type...
Some Lambda Calculi With Categorical Sums and Products
, 1993
"... . We consider the simply typed calculus with primitive recursion operators and types corresponding to categorical products and coproducts.. The standard equations corresponding to extensionality and to surjectivity of pairing and its dual are oriented as expansion rules. Strong normalization an ..."
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Cited by 20 (1 self)
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. We consider the simply typed calculus with primitive recursion operators and types corresponding to categorical products and coproducts.. The standard equations corresponding to extensionality and to surjectivity of pairing and its dual are oriented as expansion rules. Strong normalization and ground (basetype) confluence is proved for the full calculus; full confluence is proved for the calculus omitting the rule for strong sums. In the latter case, fixedpoint constructors may be added while retaining confluence. 1 Introduction The systems investigated here are simply typed caluli whose types include pairs, unit, sums, an empty type, and a type of natural numbers supporting constructions by primitive recursion. In the core system the types behave as categorical product and coproducts, so the subject at hand is equivalently ([LS86]) the equational theory of the free bicartesian closed category (generated by objects for the base types) with weak natural numbers object. Su...
Universal Profinite Domains
 Information and Computation
, 1987
"... . We introduce a bicartesian closed category of what we call profinite domains. Study of these domains is carried out through the use of an equivalent category of preorders in a manner similar to the information systems approach advocated by Dana Scott and others. A class of universal profinite dom ..."
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Cited by 15 (1 self)
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. We introduce a bicartesian closed category of what we call profinite domains. Study of these domains is carried out through the use of an equivalent category of preorders in a manner similar to the information systems approach advocated by Dana Scott and others. A class of universal profinite domains is defined and used to derive sufficient conditions for the profinite solution of domain equations involving continuous operators. As a special instance of this construction, a universal domain for the category SFP is demonstrated. Necessary conditions for the existence of solutions for domain equations over the profinites are also given and used to derive results about solutions of some equations. A new universal bounded complete domain is also demonstrated using an operator which has bounded complete domains as its fixed points. 1 Introduction. For our purposes a domain equation has the form X ¸ = F (X) where F is an operator on a class of semantic domains (typically, F is an endof...
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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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 : : : : : : : ...
Recursive Types in Games: Axiomatics and Process Representation (Extended Abstract)
 IN PROCEEDINGS O.F LICS'98. IEEE COMPUTER
, 1998
"... This paper presents two basic results on gamebased semantics of FPC, a metalanguage with sums, products, exponentials and recursive types. First we give an axiomatic account of the category of games G introduced in [15], offering a fundamental structural analysis of the category as well as a transp ..."
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Cited by 4 (1 self)
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This paper presents two basic results on gamebased semantics of FPC, a metalanguage with sums, products, exponentials and recursive types. First we give an axiomatic account of the category of games G introduced in [15], offering a fundamental structural analysis of the category as well as a transparent way to prove computational adequacy. As a consequence we obtain an intensional fullabstraction result through a standard definability argument. Next we extend the category G by introducing a category of games G i with optimised strategies; we show that the denotational semantics in G i gives a compilation of FPC terms into core Pict codes (the asynchronous polyadic calculus without summation). The process representation follows a pioneering idea of Hyland and Ong [18]. However, we advance their representation by introducing semantically wellfounded optimisation techniques; we also exte...
A Simple Adequate Categorical Model for PCF
 In Proceedings of Third International Conference on Typed Lambda Calculi and Applications
, 1997
"... Usually types of PCF are interpreted as cpos and terms as continuous functions. It is then the case that nontermination of a closed term of ground type corresponds to the interpretation being bottom; we say that the semantics is adequate. We shall here present an axiomatic approach to adequacy for ..."
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Usually types of PCF are interpreted as cpos and terms as continuous functions. It is then the case that nontermination of a closed term of ground type corresponds to the interpretation being bottom; we say that the semantics is adequate. We shall here present an axiomatic approach to adequacy for PCF in the sense that we will introduce categorical axioms enabling an adequate semantics to be given. We assume the presence of certain "bottom" maps with the role of being the interpretation of nonterminating terms, but the orderstructure is left out. This is different from previous approaches where some kind of ordertheoretic structure has been considered as part of an adequate categorical model for PCF. We take the point of view that partiality is the fundamental notion from which orderstructure should be derived, which is corroborated by the observation that our categorical model induces an ordertheoretic model for PCF in a canonical way.
Metric Spaces in Synthetic Topology
, 2010
"... We investigate the relationship between the synthetic approach to topology, in which every set is equipped with an intrinsic topology, and constructive theory of metric spaces. We relate the synthetic notion of compactness of Cantor space to Brouwer’s Fan Principle. We show that the intrinsic and me ..."
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We investigate the relationship between the synthetic approach to topology, in which every set is equipped with an intrinsic topology, and constructive theory of metric spaces. We relate the synthetic notion of compactness of Cantor space to Brouwer’s Fan Principle. We show that the intrinsic and metric topologies of complete separable metric spaces coincide if they do so for Baire space. In Russian Constructivism the match between synthetic and metric topology breaks down, as even a very simple complete totally bounded space fails to be compact, and its topology is strictly finer than the metric topology. In contrast, in Brouwer’s intuitionism synthetic and metric notions of topology and compactness agree. 1
Abstract CoMeta Project Workshop Preliminary Version Properties of Set Functors ⋆
"... We prove that any endofunctor on a classtheoretic category has a final coalgebra. Moreover, we characterize functors on settheoretic categories which are identical on objects, and functors which are constant on objects. Key words: categories of sets, partially defined endofunctors, identity functo ..."
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We prove that any endofunctor on a classtheoretic category has a final coalgebra. Moreover, we characterize functors on settheoretic categories which are identical on objects, and functors which are constant on objects. Key words: categories of sets, partially defined endofunctors, identity functor, constant functor, final coalgebra.