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ECC, an Extended Calculus of Constructions
, 1989
"... We present a higherorder calculus ECC which can be seen as an extension of the calculus of constructions [CH88] by adding strong sum types and a fully cumulative type hierarchy. ECC turns out to be rather expressive so that mathematical theories can be abstractly described and abstract mathematics ..."
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We present a higherorder calculus ECC which can be seen as an extension of the calculus of constructions [CH88] by adding strong sum types and a fully cumulative type hierarchy. ECC turns out to be rather expressive so that mathematical theories can be abstractly described and abstract mathematics may be adequately formalized. It is shown that ECC is strongly normalizing and has other nice prooftheoretic properties. An !\GammaSet (realizability) model is described to show how the essential properties of the calculus can be captured settheoretically.
Syntactical properties of an extension of Girard's System F where types can be taken as "generic" inputs
, 1995
"... We investigate from a syntactical point of view the theory FC introduced by Longo, Milsted and Soloviev. This theory is an extension of Girard's System F with axiom C which implies that types are "generic" inputs for polymorphic functions (i.e. if a polymorphic term M gives outputs inhabiting in the ..."
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We investigate from a syntactical point of view the theory FC introduced by Longo, Milsted and Soloviev. This theory is an extension of Girard's System F with axiom C which implies that types are "generic" inputs for polymorphic functions (i.e. if a polymorphic term M gives outputs inhabiting in the same type then it must be constant). This theory is interesting because it sheds more light on the nature of parametric polymorphism. In this paper we define the theory FC by means of a confluent and normalizing reduction system !C . It follows that the theory FC is decidable. 1 Introduction The starting point of this work is the dependency of the polymorphic calculus terms on types in the paper by Longo, Milsted and Soloviev [5]. The main characteristic of polymorphic lambda calculus, introduced in logic by Girard [2] and in computer science by Reynolds [6], is the possibility of abstracting a term with respect to a type variable. By this it allows for a natural representation of polymo...
Adjectives in a Modern TypeTheoretical Setting ⋆
"... Abstract. In this paper we discuss the semantics of adjectives from the perspective of a Modern Type Theory (MTT) with an adequate subtyping mechanism. In an MTT, common nouns (CNs) can be interpreted as types and, in particular, CNs modified by intersective and subsective adjectives can be given se ..."
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Abstract. In this paper we discuss the semantics of adjectives from the perspective of a Modern Type Theory (MTT) with an adequate subtyping mechanism. In an MTT, common nouns (CNs) can be interpreted as types and, in particular, CNs modified by intersective and subsective adjectives can be given semantics by means of Σtypes. However, an interpretation of CNs as types would not be viable without a proper notion of subtyping which, as we explain, is given by coercive subtyping, an adequate notion of subtyping for MTTs. It is also shown that suitable uses of universes are one of the key ingredients that have made such an analysis adequate. Privative and noncommittal adjectives require different treatments than the use of Σtypes. We propose to deal with privative adjectives using the disjoint union type while noncommittal adjectives by making use of the typetheoretical notion of context, as used by Ranta [27] to approximate the modeltheoretic notion of a possible world. Our approach to adjectives has a number of advantages over those proposed within the Montagovian setting, one of which is that the inferences related with the adjectives arise via typing and not by some kind of extra semantic meaning in the form of a meaning postulate. 1