Relational Graph Models at Work

We study the relational graph models that constitute a natural subclass of relational models of lambda-calculus. We prove that among the lambda-theories induced by such models there exists a minimal one, and that the corresponding relational graph model is very natural and easy to construct. We then study relational graph models that are fully abstract, in the sense that they capture some observational equivalence between lambda-terms. We focus on the two main observational equivalences in the lambda-calculus, the theory H+ generated by taking as observables the beta-normal forms, and H* generated by considering as observables the head normal forms. On the one hand we introduce a notion of lambda-K\"onig model and prove that a relational graph model is fully abstract for H+ if and only if it is extensional and lambda-K\"onig. On the other hand we show that the dual notion of hyperimmune model, together with extensionality, captures the full abstraction for H*

The Lazy Lambda Calculus : an investigation into the foundations of functional programming

Imperial Users onl

A Set-Theoretical Definition of Application

This paper is in two parts. Part 1 is the previously unpublished 1972 memorandum [41], with editorial changes and some minor corrections. Part 2 presents what happened next, together with some further development of the material. The first part begins with an elementary set-theoretical model of the λβ-calculus. Functions are modelled in a similar way to that normally employed in set theory, by their graphs; difficulties are caused in this enterprise by the axiom of foundation. Next, based on that model, a model of the λβη-calculus is constructed by means of a natural deduction method. Finally, a theorem is proved giving some general properties of those non-trivial models of the λβη-calculus which are continuous complete lattices. In the second part we begin with a brief discussion of models of the λ-calculus in set theories with anti-foundation axioms. Next we review the model of the λβ-calculus of Part 1 and also the closely related- but different!- models of Scott [51, 52] and of Engeler [19, 20]. Then we discuss general frameworks in which elementary constructions of models can be given. Following Longo [36], one can employ certain Scott-Engeler algebras. Following Coppo, Dezani-Ciancaglini, Honsell and Longo [13], one can obtain filter models from their Extended Applicative Type Structures. We give an extended discussion of various ways of constructing models of the λβη-calculus, and the connections between them. Finally we give extensions of the theorem to complete partial orders. Throughout we concentrate on means of constructing models. We hardly consider any analysis of their properties; we do not at all consider their application

A Graph Model for Imperative Computation

Scott's graph model is a lambda-algebra based on the observation that continuous endofunctions on the lattice of sets of natural numbers can be represented via their graphs. A graph is a relation mapping finite sets of input values to output values. We consider a similar model based on relations whose input values are finite sequences rather than sets. This alteration means that we are taking into account the order in which observations are made. This new notion of graph gives rise to a model of affine lambda-calculus that admits an interpretation of imperative constructs including variable assignment, dereferencing and allocation. Extending this untyped model, we construct a category that provides a model of typed higher-order imperative computation with an affine type system. An appropriate language of this kind is Reynolds's Syntactic Control of Interference. Our model turns out to be fully abstract for this language. At a concrete level, it is the same as Reddy's object spaces model, which was the first "state-free" model of a higher-order imperative programming language and an important precursor of games models. The graph model can therefore be seen as a universal domain for Reddy's model

Modelling local variables: possible worlds and object spaces

AbstractLocal variables in imperative languages have been given denotational semantics in at least two fundamentally different ways. One is by use of functor categories, focusing on the idea of possible worlds. The other might be termed event-based, exemplified by Reddy's object spaces and models based on game semantics. O'Hearn and Reddy have related the two approaches by giving functor category models whose worlds are object spaces, then showing that their model is fully abstract for Idealised Algol programs up to order two. But the category of object spaces is not small, and so in order to construct a functor category that is locally small, and hence Cartesian closed, they need to work with a restricted collection of object spaces. This weakens the connection between the object spaces model and the functor-category model: the Yoneda embedding no longer provides a full embedding of the original category of object spaces into the functor-category. Moreoever the choice of the restricted collection of object spaces is ad hoc. In this paper, we refine the approach by proving that the finite objects form a small dense subcategory of a simplified object-spaces model. The functor category over these finite objects is therefore locally small and Cartesian closed, and contains the object-spaces category as a full subcategory. All this work is necessarily enriched in Cpo. We further refine their full abstraction result by showing that full abstraction fails at order three

Universal semantics for the stochastic λ-calculus

We define sound and adequate denotational and operational semantics for the stochastic lambda calculus. These two semantic approaches build on previous work that used an explicit source of randomness to reason about higher-order probabilistic programs

Models and theories of lambda calculus

A quarter of century after Barendregt's main book, a wealth of interesting problems about models and theories of the untyped lambda-calculus are still open. In this thesis we will be mainly interested in the main semantics of lambda-calculus (i.e., the Scott-continuous, the stable, and the strongly stable semantics) but we will also define and study two new kinds of semantics: the relational and the indecomposable semantics. Models of the untyped lambda-calculus may be defined either as reflexive objects in Cartesian closed categories (categorical models) or as combinatory algebras satisfying the five axioms of Curry and the Meyer-Scott axiom (lambda-models). Concerning categorical models we will see that each of them can be presented as a lambda-model, even when the underlying category does not have enough points, and we will provide sufficient conditions for categorical models living in arbitrary cpo-enriched Cartesian closed categories to have H^* as equational theory. We will build a categorical model living in a non-concrete Cartesian closed category of sets and relations (relational semantics) which satisfies these conditions, and we will prove that the associated lambda-model enjoys some algebraic properties which make it suitable for modelling non-deterministic extensions of lambda-calculus. Concerning combinatory algebras, we will prove that they satisfy a generalization of Stone representation theorem stating that every combinatory algebra is isomorphic to a weak Boolean product of directly indecomposable combinatory algebras. We will investigate the semantics of lambda-calculus whose models are directly indecomposable as combinatory algebras (the indecomposable semantics) and we will show that this semantics is large enough to include all the main semantics and all the term models of semi-sensible lambda-theories, and that it is however largely incomplete. Finally, we will investigate the problem of whether there exists a non-syntactical model of lambda-calculus belonging to the main semantics which has an r.e. (recursively enumerable) order or equational theory. This is a natural generalization of Honsell-Ronchi Della Rocca's longstanding open problem about the existence of a Scott-continuous model of lambda-beta or lambda-beta-eta. Then, we introduce an appropriate notion of effective model of lambda-calculus, which covers in particular all the models individually introduced in the literature, and we prove that no order theory of an effective model can be r.e.; from this it follows that its equational theory cannot be lambda-beta or lambda-beta-eta. Then, we show that no effective model living in the stable or strongly stable semantics has an r.e. equational theory. Concerning Scott-continuous semantics, we prove that no order theory of a graph model can be r.e. and that many effective graph models do not have an r.e. equational theory.Dans cette thèse on s'intéresse surtout aux sémantiques principales du λ-calcul (c'est- a-dire la sémantique continue de Scott, la sémantique stable, et la sémantique fortement stable) mais on introduit et étudie aussi deux nouvelles sémantiques : la sémantique relationnelle et la sémantique indécomposable. Les modèles du λ-calcul pur peuvent être définis soit comme des objets réflexifs dans des catégories Cartésiennes fermées (modèles catégoriques) soit comme des algèbres combinatoires satisfaisant les cinq axiomes de Curry et l'axiome de Meyer-Scott ( λ-modèles). En ce qui concerne les modèles catégoriques, on montre que tout modèle catégorique peut être présenté comme un λ-modèle, même si la ccc (catégorie Cartésienne fermée) sous-jacente n'a pas assez de points, et on donne des conditions su santes pour qu'un modèle catégorique vivant dans une ccc \cpo-enriched" arbitraire ait H pour théorie équationnelle. On construit un modèle catégorique qui vit dans une ccc d'ensembles et relations (sémantique relationnelle) et qui satisfait ces conditions. De plus, on montre que le λ-modèle associe possède des propriétés algébriques qui le rendent apte a modéliser des extensions non-déterministes du -calcul. En ce qui concerne les algèbres combinatoires, on montre qu'elles satisfont une généralisation du Théorème de Représentation de Stone qui dit que toute algèbre combinatoire est isomorphe a un produit Booléen faible d'algèbres combinatoires directement indécomposables. On étudie la sémantique du λ-calcul dont les modèles sont directement indécomposable comme algèbres combinatoires (sémantique indécomposable); on prouve en particulier que cette sémantique est assez générale pour inclure d'une part les trois sémantiques principales et d'autre part les modèles de termes de toutes les λ-théories semi-sensibles. Par contre, on montre aussi qu'elle est largement incomplète. Finalement, on étudie la question de l'existence d'un modèle non-syntaxique du λ-calcul appartenant aux sémantiques principales et ayant une théorie équationnelle ou inéquationnelle r.e. (récursivement énumérable). Cette question est une généralisation naturelle du problème de Honsell et Ronchi Della Rocca (ouvert depuis plus que vingt ans) concernant l'existence d'un modèle continu de λβ ou λβη. On introduit une notion adéquate de modèles effectifs du λ-calcul, qui couvre en particulier tous les modèles qui ont été introduits individuellement en littérature, et on prouve que la théorie inéquationnelle d'un modèle effectif n'est jamais r.e. ; en conséquence sa théorie équationnelle ne peut pas être λβ ou λβη. On montre aussi que la théorie équationnelle d'un modèle effectif vivant dans la sémantique stable ou fortement stable n'est jamais r.e. En ce qui concerne la sémantique continue de Scott, on démontre que la théorie in équationnelle d'un modèle de graphe n'est jamais r.e. et qu'il existe beaucoup de modèles de graphes effectifs qui ont une théorie équationnelle qui n'est pas r.e