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

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

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

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

Call-by-value non-determinism in a linear logic type discipline

We consider the call-by-value lambda-calculus extended with a may-convergent non-deterministic choice and a must-convergent parallel composition. Inspired by recent works on the relational semantics of linear logic and non-idempotent intersection types, we endow this calculus with a type system based on the so-called Girard's second translation of intuitionistic logic into linear logic. We prove that a term is typable if and only if it is converging, and that its typing tree carries enough information to give a bound on the length of its lazy call-by-value reduction. Moreover, when the typing tree is minimal, such a bound becomes the exact length of the reduction

Taylor subsumes Scott, Berry, Kahn and Plotkin

The speculative ambition of replacing the old theory of program approximation based on syntactic continuity with the theory of resource consumption based on Taylor expansion and originating from the differential γ-calculus is nowadays at hand. Using this resource sensitive theory, we provide simple proofs of important results in γ-calculus that are usually demonstrated by exploiting Scott's continuity, Berry's stability or Kahn and Plotkin's sequentiality theory. A paradigmatic example is given by the Perpendicular Lines Lemma for the Böhm tree semantics, which is proved here simply by induction, but relying on the main properties of resource approximants: strong normalization, confluence and linearity

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