Bifibrational Functorial Semantics For Parametric Polymorphism

Reynolds’ theory of parametric polymorphism captures the invariance of polymorphically typed programs under change of data representation. Semantically, reflexive graph categories and fibrations are both known to give a categorical understanding of parametric polymorphism. This paper contributes further to this categorical perspective by showing the relevance of bifibrations. We develop a bifibrational framework for models of System F that are parametric, in that they verify the Identity Extension Lemma and Reynolds’ Abstraction Theorem. We also prove that our models satisfy expected properties, such as the existence of initial algebras and final coalgebras, and that parametricity implies dinaturality

Bifibrational functorial semantics of parametric polymorphism

Reynolds' theory of parametric polymorphism captures the invariance of polymorphically typed programs under change of data representation. Semantically, reflexive graph categories and fibrations are both known to give a categorical understanding of parametric polymorphism. This paper contributes further to this categorical perspective by showing the relevance of bifibrations. We develop a bifibrational framework for models of System F that are parametric, in that they verify the Identity Extension Lemma and Reynolds' Abstraction Theorem. We also prove that our models satisfy expected properties, such as the existence of initial algebras and final coalgebras, and that parametricity implies dinaturality

A computable expression of closure to efficient causation

International audienceIn this paper, we propose a mathematical expression of closure to efficient causation in terms of lambda-calculus; we argue that this opens up the perspective of developing principled computer simulations of systems closed to efficient causation in an appropriate programming language. An important implication of our formulation is that, by exhibiting an expression in lambda-calculus, which is a paradigmatic formalism for computability and programming, we show that there are no conceptual or principled problems in realizing a computer simulation or model of closure to efficient causation. We conclude with a brief discussion of the question whether closure to efficient causation captures all relevant properties of living systems. We suggest that it might not be the case, and that more complex definitions could indeed create crucial some obstacles to computability

Internal models of system F for decompilation

AbstractThis paper considers Girard’s internal coding of each term of System F by some term of a code type. This coding is the type-erasing coding definable already in the simply typed lambda-calculus using only abstraction on term variables. It is shown that there does not exist any decompiler for System F in System F, where the decompiler maps a term of System F to its code. An internal model of F is given by interpreting each type of F by some type equipped with maps between the type and the code type. This paper gives a decompiler–normalizer for this internal model in F, where the decompiler–normalizer maps any term of the internal model to the code of its normal form. It is also shown that for any model of F the composition of this internal model and the model produces another model of F whose equational theory is below untyped beta–eta-equality

Cercles vicieux, mathématiques et formalisations logiques

Some forms of circularity in Logic and Mathematics (self-membership, self-application, impredicativity, …) are analyzed as closure properties of suitable mathematical structures since they can be considered as solutions of some systems of equations. At the same time, from a philosophical point of view, we stress the contribution of these circularities to the power of mathematics in making the world intelligible;Certaines formes de circularité logiques et mathématiques (auto-appartenance, auto-implication, imprédicativité) sont analysées comme des propriétés de fermeture de certaines structures mathématiques puisqu'on peut les interpréter comme des solutions de certains systèmes d'équations. Parallèlement, du point de vue philosophique, on met en évidence la contribution de ces circularités au pouvoir des mathématiques à rendre le monde intelligible

Cercles vicieux, mathématiques et formalisations logiques

Some forms of circularity in Logic and Mathematics (self-membership, self-application, impredicativity, …) are analyzed as closure properties of suitable mathematical structures since they can be considered as solutions of some systems of equations. At the same time, from a philosophical point of view, we stress the contribution of these circularities to the power of mathematics in making the world intelligible;Certaines formes de circularité logiques et mathématiques (auto-appartenance, auto-implication, imprédicativité) sont analysées comme des propriétés de fermeture de certaines structures mathématiques puisqu'on peut les interpréter comme des solutions de certains systèmes d'équations. Parallèlement, du point de vue philosophique, on met en évidence la contribution de ces circularités au pouvoir des mathématiques à rendre le monde intelligible

Constructive Natural Deduction and its `Omega-Set' Interpretation

International audienceVarious Theories of Types are introduced, by stressing the analogy ‘propositions-as-types’: from propositional to higher order types (and Logic). In accordance with this, proofs are described as terms of various calculi, in particular of polymorphic (second order) λ-calculus. A semantic explanation is then given by interpreting individual types and the collection of all types in two simple categories built out of the natural numbers (the modest sets and the universe of ω-sets). The first part of this paper (syntax) may be viewed as a short tutorial with a constructive understanding of the deduction theorem and some work on the expressive power of first and second order quantification. Also in the second part (semantics, §§6–7) the presentation is meant to be elementary, even though we introduce some new facts on types as quotient sets in order to interpret ‘explicit polymorphism’. (The experienced reader in Type Theory may directly go, at first reading, to §§6–8)

On the Semantics of Intensionality and Intensional Recursion

Intensionality is a phenomenon that occurs in logic and computation. In the most general sense, a function is intensional if it operates at a level finer than (extensional) equality. This is a familiar setting for computer scientists, who often study different programs or processes that are interchangeable, i.e. extensionally equal, even though they are not implemented in the same way, so intensionally distinct. Concomitant with intensionality is the phenomenon of intensional recursion, which refers to the ability of a program to have access to its own code. In computability theory, intensional recursion is enabled by Kleene's Second Recursion Theorem. This thesis is concerned with the crafting of a logical toolkit through which these phenomena can be studied. Our main contribution is a framework in which mathematical and computational constructions can be considered either extensionally, i.e. as abstract values, or intensionally, i.e. as fine-grained descriptions of their construction. Once this is achieved, it may be used to analyse intensional recursion.Comment: DPhil thesis, Department of Computer Science & St John's College, University of Oxfor