The Vectorial λ\lambda-Calculus

We describe a type system for the linear-algebraic $\lambda$-calculus. The type system accounts for the linear-algebraic aspects of this extension of $\lambda$-calculus: it is able to statically describe the linear combinations of terms that will be obtained when reducing the programs. This gives rise to an original type theory where types, in the same way as terms, can be superposed into linear combinations. We prove that the resulting typed $\lambda$-calculus is strongly normalising and features weak subject reduction. Finally, we show how to naturally encode matrices and vectors in this typed calculus.Comment: Long and corrected version of arXiv:1012.4032 (EPTCS 88:1-15), to appear in Information and Computatio

Intersection types and (positive) almost-sure termination

Randomized higher-order computation can be seen as being captured by a λ-calculus endowed with a single algebraic operation, namely a construct for binary probabilistic choice. What matters about such computations is the probability of obtaining any given result, rather than the possibility or the necessity of obtaining it, like in (non)deterministic computation. Termination, arguably the simplest kind of reachability problem, can be spelled out in at least two ways, depending on whether it talks about the probability of convergence or about the expected evaluation time, the second one providing a stronger guarantee. In this paper, we show that intersection types are capable of precisely characterizing both notions of termination inside a single system of types: the probability of convergence of any λ-term can be underapproximated by its type, while the underlying derivation's weight gives a lower bound to the term's expected number of steps to normal form. Noticeably, both approximations are tight-not only soundness but also completeness holds. The crucial ingredient is non-idempotency, without which it would be impossible to reason on the expected number of reduction steps which are necessary to completely evaluate any term. Besides, the kind of approximation we obtain is proved to be optimal recursion theoretically: no recursively enumerable formal system can do better than that

Intersection Types and (Positive) Almost-Sure Termination

Randomized higher-order computation can be seen as being captured by a lambda calculus endowed with a single algebraic operation, namely a construct for binary probabilistic choice. What matters about such computations is the probability of obtaining any given result, rather than the possibility or the necessity of obtaining it, like in (non)deterministic computation. Termination, arguably the simplest kind of reachability problem, can be spelled out in at least two ways, depending on whether it talks about the probability of convergence or about the expected evaluation time, the second one providing a stronger guarantee. In this paper, we show that intersection types are capable of precisely characterizing both notions of termination inside a single system of types: the probability of convergence of any lambda-term can be underapproximated by its type, while the underlying derivation's weight gives a lower bound to the term's expected number of steps to normal form. Noticeably, both approximations are tight -- not only soundness but also completeness holds. The crucial ingredient is non-idempotency, without which it would be impossible to reason on the expected number of reduction steps which are necessary to completely evaluate any term. Besides, the kind of approximation we obtain is proved to be optimal recursion theoretically: no recursively enumerable formal system can do better than that

The probability of non-confluent systems

We show how to provide a structure of probability space to the set of execution traces on a non-confluent abstract rewrite system, by defining a variant of a Lebesgue measure on the space of traces. Then, we show how to use this probability space to transform a non-deterministic calculus into a probabilistic one. We use as example Lambda+, a recently introduced calculus defined through type isomorphisms.Comment: In Proceedings DCM 2013, arXiv:1403.768

Taylor expansion for Call-By-Push-Value

The connection between the Call-By-Push-Value lambda-calculus introduced by Levy and Linear Logic introduced by Girard has been widely explored through a denotational view reflecting the precise ruling of resources in this language. We take a further step in this direction and apply Taylor expansion introduced by Ehrhard and Regnier. We define a resource lambda-calculus in whose terms can be used to approximate terms of Call-By-Push-Value. We show that this approximation is coherent with reduction and with the translations of Call-By-Name and Call-By-Value strategies into Call-By-Push-Value

On Probabilistic Applicative Bisimulation and Call-by-Value λ\lambda-Calculi (Long Version)

Probabilistic applicative bisimulation is a recently introduced coinductive methodology for program equivalence in a probabilistic, higher-order, setting. In this paper, the technique is applied to a typed, call-by-value, lambda-calculus. Surprisingly, the obtained relation coincides with context equivalence, contrary to what happens when call-by-name evaluation is considered. Even more surprisingly, full-abstraction only holds in a symmetric setting.Comment: 30 page

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