. 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 type-check in more than one way. Since interpretations follow the type...

Martin-Lof's type theory is presented in several steps. The kernel is a dependently typed -calculus. Then there are schemata for inductive sets and families of sets and for primitive recursive functions and families of functions. Finally, there are set formers (generic polymorphism) and universes. At each step syntax, inference rules, and set-theoretic semantics are given. 1 Introduction Usually Martin-Lof's type theory is presented as a closed system with rules for a finite collection of set formers. But it is also often pointed out that the system is in principle open to extension: we may introduce new sets when there is a need for them. The principle is that a set is by definition inductively generated - it is defined by its introduction rules, which are rules for generating its elements. The elimination rule is determined by the introduction rules and expresses definition by primitive recursion on the way the elements of the set are generated. (In this paper I shall use the term ...

We investigate semantical normalization proofs for typed combinatory logic and weak -calculus. One builds a model and a function `quote' which inverts the interpretation function. A normalization function is then obtained by composing quote with the interpretation function. Our models are just like the intended model, except that the function space includes a syntactic component as well as a semantic one. We call this a `glued' model because of its similarity with the glueing construction in category theory. Other basic type constructors are interpreted as in the intended model. In this way we can also treat inductively defined types such as natural numbers and Brouwer ordinals. We also discuss how to formalize -terms, and show how one model construction can be used to yield normalization proofs for two different typed -calculi -- one with explicit and one with implicit substitution. The proofs are formalized using Martin-Lof's type theory as a meta language and mechanized using the A...

This thesis contains an investigation of Coquand's Calculus of Constructions, a basic impredicative Type Theory. We review syntactic properties of the calculus, in particular decidability of equality and type-checking, based on the equality-as-judgement presentation. We present a set-theoretic notion of model, CC-structures, and use this to give a new strong normalization proof based on a modification of the realizability interpretation. An extension of the core calculus by inductive types is investigated and we show, using the example of infinite trees, how the realizability semantics and the strong normalization argument can be extended to non-algebraic inductive types. We emphasize that our interpretation is sound for large eliminations, e.g. allows the definition of sets by recursion. Finally we apply the extended calculus to a non-trivial problem: the formalization of the strong normalization argument for Girard's System F. This formal proof has been developed and checked using the...

Linear and other relevant logics have been studied widely in mathematical, philosophical and computational logic. We describe a logical framework, RLF, for defining natural deduction presentations of such logics. RLF consists in a language together, in a manner similar to that of Harper, Honsell and Plotkin's LF, with a representation mechanism: the language of RLF is the lL-calculus; the representation mechanism is judgements-as-types, developed for relevant logics. The lL-calculus type theory is a first-order dependent type theory with two kinds of dependent function spaces: a linear one and an intuitionistic one. We study a natural deduction presentation of the type theory and establish the required proof-theoretic meta-theory. The RLF framework is a conservative extension of LF. We show that RLF uniformly encodes (fragments of) intuitionistic linear logic, Curry's l I -calculus and ML with references. We describe the Curry-Howard-de Bruijn correspondence of the lL-calculus with a s...

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The lL-calculus is a dependent type theory with both linear and intuitionistic dependent function spaces. It can be seen to arise in two ways. Firstly, in logical frameworks, where it is the language of the RLF logical framework and can uniformly represent linear and other relevant logics. Secondly, it is a presentation of the proof-objects of BI, the logic of bunched implications. BI is a logic which directly combines linear and intuitionistic implication and, in its predicate version, has both linear and intuitionistic quantifiers. The lL-calculus is the dependent type theory which generalizes both implications and quantifiers. In this paper, we describe the categorical semantics of the lL-calculus. This is given by Kripke resource models, which are monoid-indexed sets of functorial Kripke models, the monoid giving an account of resource consumption. We describe a class of concrete, set-theoretic models. The models are given by the category of families of sets, parametrized over a small monoidal category, in which the intuitionistic dependent function space is described in the established way, but the linear dependent function space is described using Day's tensor product.

) Eike Ritter and Valeria de Paiva ? School of Computer Science, University of Birmingham Abstract. Calculi with explicit substitutions have found widespread acceptance as a basis for abstract machines for functional languages. In this paper we investigate the relations between variants with de Bruijnnumbers, with variable names, with reduction based on raw expressions and calculi with equational judgements. We show the equivalence between these variants, which is crucial in establishing the correspondence between the semantics of the calculus and its implementations. 1 Introduction Explicit substitution calculi (or oe-calculi for short) first appeared in a seminal paper by Abadi et al. [1]. The basic idea is that instead of having substitutions as a meta-level operation, as in traditional -calculus, we should make them part of the object-level calculus. The advantages of this approach are twofold. Firstly, it makes it possible to design much more efficient abstract machines as we a...

The Calculus of Constructions (CC) ([Coquand 1985]) is a typed lambda calculus for higher order intuitionistic logic: proofs of the higher order logic are interpreted as lambda terms and formulas as types. It is also the union of Girard's system F! ([Girard 1972]), a higher order typed lambda calculus, and a first order dependent typed lambda calculus in the style of de Bruijn's Automath ([de Bruijn 1980]) or Martin-Lof's intuitionistic theory of types ([Martin-Lof 1984]). Using the impredicative coding of data types in F! , the Calculus of Constructions thus becomes a higher order language for the typing of functional programs. We shall introduce and try to explain CC by exploiting especially the first point of view, by introducing a typed lambda calculus that faithfully represent higher order predicate logic (so for this system the Curry-Howard `formulas-as-types isomorphism' is really an isomorphism.) Then we discuss some propositions that are provable in CC but not in the higher or...

Calculi with explicit substitutions have found widespread acceptance as a basis for abstract machines for functional languages. In this paper we investigate the relations between variants with de Bruijn-numbers, with variable names, with reduction based on raw expressions and calculi with equational judgements. We show the equivalence between these variants, which is crucial in establishing the correspondence between the semantics of the calculus and its implementations. 1 Introduction Explicit substitution calculi (or oe-calculi for short) first appeared in a seminal paper by Abadi et al. [1]. The basic idea is that instead of having substitutions as a meta-level operation, as in traditional -calculus, we should make them part of the object-level calculus. The advantages of this approach are twofold. Firstly, it makes it possible to design much more efficient abstract machines as we are allowed to delay substitutions, and secondly it makes it much easier to prove them correct since...