On the characterization of models of H*: The semantical aspect

We give a characterization, with respect to a large class of models of untyped lambda-calculus, of those models that are fully abstract for head-normalization, i.e., whose equational theory is H* (observations for head normalization). An extensional K-model $D$ is fully abstract if and only if it is hyperimmune, {\em i.e.}, not well founded chains of elements of D cannot be captured by any recursive function. This article, together with its companion paper, form the long version of [Bre14]. It is a standalone paper that presents a purely semantical proof of the result as opposed to its companion paper that presents an independent and purely syntactical proof of the same result

On the discriminating power of tests in resource lambda-calculus

Since its discovery, differential linear logic (DLL) inspired numerous domains. In denotational semantics, categorical models of DLL are now commune, and the simplest one is Rel, the category of sets and relations. In proof theory this naturally gave birth to differential proof nets that are full and complete for DLL. In turn, these tools can naturally be translated to their intuitionistic counterpart. By taking the co-Kleisly category associated to the ! comonad, Rel becomes MRel, a model of the \Lcalcul that contains a notion of differentiation. Proof nets can be used naturally to extend the \Lcalcul into the lambda calculus with resources, a calculus that contains notions of linearity and differentiations. Of course MRel is a model of the \Lcalcul with resources, and it has been proved adequate, but is it fully abstract? That was a strong conjecture of Bucciarelli, Carraro, Ehrhard and Manzonetto. However, in this paper we exhibit a counter-example. Moreover, to give more intuition on the essence of the counter-example and to look for more generality, we will use an extension of the resource \Lcalcul also introduced by Bucciarelli et al for which \Minf is fully abstract, the tests

Modelling Coeffects in the Relational Semantics of Linear Logic

Various typing system have been recently introduced giving a parametric version of the exponential modality of linear logic. The parameters are taken from a semi-ring, and allow to express coeffects - i.e. specific requirements of a program with respect to the environment (availability of a resource, some prerequisite of the input, etc.). We show that all these systems can be interpreted in the relational category (Rel) of sets and relations. This is possible because of the notion of multiplicity semi-ring and allowing a great variety of exponential comonads in Rel. The interpretation of a particular typing system corresponds then to give a suitable notion of stratification of the exponential comonad associated with the semi-ring parametrising the exponential modality

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*

On Higher-Order Probabilistic Subrecursion

We study the expressive power of subrecursive probabilistic higher-order calculi. More specifically, we show that endowing a very expressive deterministic calculus like Godel's T with various forms of probabilistic choice operators may result in calculi which are not equivalent as for the class of distributions they give rise to, although they all guarantee almost-sure termination. Along the way, we introduce a probabilistic variation of the classic reducibility technique, and we prove that the simplest form of probabilistic choice leaves the expressive power of T essentially unaltered. The paper ends with some observations about the functional expressive power: expectedly, all the considered calculi capture the functions which T itself represents, at least when standard notions of observations are considered

Combining Effects and Coeffects via Grading

This is the author accepted manuscript. It is currently under an indefinite embargo pending publication by the Association for Computing Machinery.$\textit{Effects}$ and $\textit{coeffects}$ are two general, complementary aspects of program behaviour. They roughly correspond to computations which change the execution context (effects) versus computations which make demands on the context (coeffects). Effectful features include partiality, non-determinism, input-output, state, and exceptions. Coeffectful features include resource demands, variable access, notions of linearity, and data input requirements. The effectful or coeffectful behaviour of a program can be captured and described via type-based analyses, with fine grained information provided by monoidal effect annotations and semiring coeffects. Various recent work has proposed models for such typed calculi in terms of $\textit{graded (strong) monads}$ for effects and $\textit{graded (monoidal) comonads}$ for coeffects. Effects and coeffects have been studied separately so far, but in practice many computations are both effectful and coeffectful, e.g., possibly throwing exceptions but with resource requirements. To remedy this, we introduce a new general calculus with a combined $\textit{effect-coeffect system}$. This can describe both the $\textit{changes}$ and $\textit{requirements}$ that a program has on its context, as well as interactions between these effectful and coeffectful features of computation. The effect-coeffect system has a denotational model in terms of effect-graded monads and coeffect-graded comonads where interaction is expressed via the novel concept of $\textit{graded distributive laws}$. This graded semantics unifies the syntactic type theory with the denotational model. We show that our calculus can be instantiated to describe in a natural way various different kinds of interaction between a program and its evaluation context.Orchard was supported by EPSRC grant EP/M026124/1 and EP/K011715/1 (whilst previously at Imperial College London), Katsumata by JSPS KAKENHI grant JP15K00014, Uustalu by Estonian Min. of Educ. and Res. grant IUT33-13 and Estonian Sci. Found. grant 9475. Gaboardi’s work was done in part while at the University of Dundee, UK supported by EPSRC grant EP/M022358/1

Quantitative program reasoning with graded modal types

In programming, data is often considered to be infinitely copiable, arbitrarily discardable, and universally unconstrained. However this view is naive: some data encapsulates resources that are subject to protocols (e.g., file and device handles, channels); some data should not be arbitrarily copied or communicated (e.g., private data). Linear types provide a partial remedy by delineating data in two camps: "resources" to be used but never copied or discarded, and unconstrained values. However, this binary distinction is too coarse-grained. Instead, we propose the general notion of graded modal types, which in combination with linear and indexed types, provides an expressive type theory for enforcing fine-grained resource-like properties of data. We present a type system drawing together these aspects (linear, graded, and indexed) embodied in a fully-fledged functional language implementation, called Granule. We detail the type system, including its metatheoretic properties, and explore examples in the concrete language. This work advances the wider goal of expanding the reach of type systems to capture and verify a broader set of program properties

New Results on Morris's Observational Theory: the benefits of separating the inseparable

International audienceWorking in the untyped lambda calculus, we study Morris's λ-theory H +. Introduced in 1968, this is the original extensional theory of contextual equivalence. On the syntactic side, we show that this λ-theory validates the ω-rule, thus settling a long-standing open problem. On the semantic side, we provide sufficient and necessary conditions for relational graph models to be fully abstract for H +. We show that a relational graph model captures Morris's observational preorder exactly when it is extensional and λ-König. Intuitively, a model is λ-König when every λ-definable tree has an infinite path which is witnessed by some element of the model. Both results follows from a weak separability property enjoyed by terms differing only because of some infinite η-expansion, which is proved through a refined version of the Böhm-out technique