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*

Lambda theories of effective lambda models

A longstanding open problem is whether there exists a non-syntactical model of untyped lambda-calculus whose theory is exactly the least equational lambda-theory (=Lb). In this paper we make use of the Visser topology for investigating the more general question of whether the equational (resp. order) theory of a non syntactical model M, say Eq(M) (resp. Ord(M)) can be recursively enumerable (= r.e. below). We conjecture that no such model exists and prove the conjecture for several large classes of models. In particular we introduce a notion of effective lambda-model and show that for all effective models M, Eq(M) is different from Lb, and Ord(M) is not r.e. If moreover M belongs to the stable or strongly stable semantics, then Eq(M) is not r.e. Concerning Scott's continuous semantics we explore the class of (all) graph models, show that it satisfies Lowenheim Skolem theorem, that there exists a minimum order/equational graph theory, and that both are the order/equ theories of an effective graph model. We deduce that no graph model can have an r.e. order theory, and also show that for some large subclasses, the same is true for Eq(M).Comment: 15 pages, accepted CSL'0

Effective lambda-models vs recursively enumerable lambda-theories

A longstanding open problem is whether there exists a non syntactical model of the untyped lambda-calculus whose theory is exactly the least lambda-theory (l-beta). In this paper we investigate the more general question of whether the equational/order theory of a model of the (untyped) lambda-calculus can be recursively enumerable (r.e. for brevity). We introduce a notion of effective model of lambda-calculus calculus, which covers in particular all the models individually introduced in the literature. We prove that the order theory of an effective model is never r.e.; from this it follows that its equational theory cannot be l-beta or l-beta-eta. We then show that no effective model living in the stable or strongly stable semantics has an r.e. equational theory. Concerning Scott's semantics, we investigate the class of graph models and prove that no order theory of a graph model can be r.e., and that there exists an effective graph model whose equational/order theory is minimum among all theories of graph models. Finally, we show that the class of graph models enjoys a kind of downwards Lowenheim-Skolem theorem.Comment: 34

Strong Normalization of MLF via a Calculus of Coercions

MLF is a type system extending ML with first-class polymorphism as in system F. The main goal of the present paper is to show that MLF enjoys strong normalization, i.e. it has no infinite reduction paths. The proof of this result is achieved in several steps. We first focus on xMLF, the Church-style version of MLF, and show that it can be translated into a calculus of coercions: terms are mapped into terms and instantiations into coercions. This coercion calculus can be seen as a decorated version of system F, so that the simulation results entails strong normalization of xMLF through the same property of system F. We then transfer the result to all other versions of MLF using the fact that they can be compiled into xMLF and showing there is a bisimulation between the two. We conclude by discussing what results and issues are encountered when using the candidates of reducibility approach to the same problem

Full Abstraction for the Resource Lambda Calculus with Tests, through Taylor Expansion

We study the semantics of a resource-sensitive extension of the lambda calculus in a canonical reflexive object of a category of sets and relations, a relational version of Scott's original model of the pure lambda calculus. This calculus is related to Boudol's resource calculus and is derived from Ehrhard and Regnier's differential extension of Linear Logic and of the lambda calculus. We extend it with new constructions, to be understood as implementing a very simple exception mechanism, and with a "must" parallel composition. These new operations allow to associate a context of this calculus with any point of the model and to prove full abstraction for the finite sub-calculus where ordinary lambda calculus application is not allowed. The result is then extended to the full calculus by means of a Taylor Expansion formula. As an intermediate result we prove that the exception mechanism is not essential in the finite sub-calculus

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

Addressing Machines as models of lambda-calculus

Turing machines and register machines have been used for decades in theoretical computer science as abstract models of computation. Also the $\lambda$-calculus has played a central role in this domain as it allows to focus on the notion of functional computation, based on the substitution mechanism, while abstracting away from implementation details. The present article starts from the observation that the equivalence between these formalisms is based on the Church-Turing Thesis rather than an actual encoding of $\lambda$-terms into Turing (or register) machines. The reason is that these machines are not well-suited for modelling \lam-calculus programs. We study a class of abstract machines that we call \emph{addressing machine} since they are only able to manipulate memory addresses of other machines. The operations performed by these machines are very elementary: load an address in a register, apply a machine to another one via their addresses, and call the address of another machine. We endow addressing machines with an operational semantics based on leftmost reduction and study their behaviour. The set of addresses of these machines can be easily turned into a combinatory algebra. In order to obtain a model of the full untyped $\lambda$-calculus, we need to introduce a rule that bares similarities with the $\omega$-rule and the rule $\zeta_\beta$ from combinatory logic

Degrees of extensionality in the theory of B\"ohm trees and Sall\'e's conjecture

The main observational equivalences of the untyped lambda-calculus have been characterized in terms of extensional equalities between B\"ohm trees. It is well known that the lambda-theory H*, arising by taking as observables the head normal forms, equates two lambda-terms whenever their B\"ohm trees are equal up to countably many possibly infinite eta-expansions. Similarly, two lambda-terms are equal in Morris's original observational theory H+, generated by considering as observable the beta-normal forms, whenever their B\"ohm trees are equal up to countably many finite eta-expansions. The lambda-calculus also possesses a strong notion of extensionality called "the omega-rule", which has been the subject of many investigations. It is a longstanding open problem whether the equivalence B-omega obtained by closing the theory of B\"ohm trees under the omega-rule is strictly included in H+, as conjectured by Sall\'e in the seventies. In this paper we demonstrate that the two aforementioned theories actually coincide, thus disproving Sall\'e's conjecture. The proof technique we develop for proving the latter inclusion is general enough to provide as a byproduct a new characterization, based on bounded eta-expansions, of the least extensional equality between B\"ohm trees. Together, these results provide a taxonomy of the different degrees of extensionality in the theory of B\"ohm trees

Loader and Urzyczyn Are Logically Related

International audienceIn simply typed \lam-calculus with one ground type the following theorem due to Loader holds. $(i)$ Given the full model \cF over a finite set, %with at least seven elements, the question whether some element f\in\cF is \lam-definable is undecidable. In the \lam-calculus with intersection types based on countably many atoms, the following is proved by Urzyczyn. $(ii)$ It is undecidable whether a type is inhabited. Both statements are major results presented in \cite{Bare2}. We show that $(i)$ and $(ii)$ follow from each other in a natural way, by interpreting intersection types as continuous functions logically related to elements of \cF. From this, and a result by Joly on \lam-definability, we get that \Urz's theorem already holds for intersection types with at most two atoms