Effectful applicative similarity for call-by-name lambda calculi

International audienceWe introduce a notion of applicative similarity in which not terms but monadic values arising from the evaluation of effectful terms, can be compared. We prove this notion to be fully abstract whenever terms are evaluated in call-by-name order. This is the first full-abstraction result for such a generic, coinductive methodology for program equivalence

Effectful applicative similarity for call-by-name lambda calculi

International audienceWe introduce a notion of applicative similarity in which not terms but monadic values arising from the evaluation of effectful terms, can be compared. We prove this notion to be fully abstract whenever terms are evaluated in call-by-name order. This is the first full-abstraction result for such a generic, coinductive methodology for program equivalence

Effectful Normal Form Bisimulation

International audienceNormal form bisimulation, also known as open bisimulation, is a coinductive technique for higher-order program equivalence in which programs are compared by looking at their essentially infinitary tree-like normal forms, i.e. at their Böhm or Lévy-Longo trees. The technique has been shown to be useful not only when proving metatheorems about λ-calculi and their semantics, but also when looking at concrete examples of terms. In this paper, we show that there is a way to generalise normal form bisimulation to calculi with algebraic effects, à la Plotkin and Power. We show that some mild conditions on monads and relators, which have already been shown to guarantee effectful applicative bisimi-larity to be a congruence relation, are enough to prove that the obtained notion of bisimilarity, which we call effectful normal form bisimilarity, is a congruence relation, and thus sound for contextual equivalence. Additionally , contrary to applicative bisimilarity, normal form bisimilarity allows for enhancements of the bisimulation proof method, hence proving a powerful reasoning principle for effectful programming languages

Resource transition systems and full abstraction for linear higher-order effectful programs

We investigate program equivalence for linear higher-order (sequential) languages endowed with primitives for computational effects. More specifically, we study operationally-based notions of program equivalence for a linear \u3b3-calculus with explicit copying and algebraic effects \ue0 la Plotkin and Power. Such a calculus makes explicit the interaction between copying and linearity, which are intensional aspects of computation, with effects, which are, instead, extensional. We review some of the notions of equivalences for linear calculi proposed in the literature and show their limitations when applied to effectful calculi where copying is a first-class citizen. We then introduce resource transition systems, namely transition systems whose states are built over tuples of programs representing the available resources, as an operational semantics accounting for both intensional and extensional interactive behaviours of programs. Our main result is a sound and complete characterization of contextual equivalence as trace equivalence defined on top of resource transition systems

Resource Transition Systems and Full Abstraction for Linear Higher-Order Effectful Programs

International audienceWe investigate program equivalence for linear higher-order (sequential) languages endowed with primitives for computational effects. More specifically, we study operationally-based notions of program equivalence for a linear λ-calculus with explicit copying and algebraic effects à la Plotkin and Power. Such a calculus makes explicit the interaction between copying and linearity, which are intensional aspects of computation, with effects, which are, instead, extensional. We review some of the notions of equivalences for linear calculi proposed in the literature and show their limitations when applied to effectful calculi where copying is a first-class citizen. We then introduce resource transition systems, namely transition systems whose states are built over tuples of programs representing the available resources, as an operational semantics accounting for both intensional and extensional interactive behaviours of programs. Our main result is a sound and complete characterization of contextual equivalence as trace equivalence defined on top of resource transition systems

Effectful Applicative Bisimilarity: Monads, Relators, and Howe's Method

International audienceWe study Abramsky's applicative bisimilarity abstractly , in the context of call-by-value λ-calculi with algebraic effects. We first of all endow a computational λ-calculus with a monadic operational semantics. We then show how the theory of relators provides precisely what is needed to generalise applicative bisimilarity to such a calculus, and to single out those monads and relators for which applicative bisimilarity is a congruence, thus a sound methodology for program equivalence. This is done by studying Howe's method in the abstract

Quantitative Behavioural Reasoning for Higher-order Effectful Programs: Applicative Distances (Extended Version)

This paper studies the quantitative refinements of Abramsky's applicative similarity and bisimilarity in the context of a generalisation of Fuzz, a call-by-value $\lambda$-calculus with a linear type system that can express programs sensitivity, enriched with algebraic operations \emph{\`a la} Plotkin and Power. To do so a general, abstract framework for studying behavioural relations taking values over quantales is defined according to Lawvere's analysis of generalised metric spaces. Barr's notion of relator (or lax extension) is then extended to quantale-valued relations adapting and extending results from the field of monoidal topology. Abstract notions of quantale-valued effectful applicative similarity and bisimilarity are then defined and proved to be a compatible generalised metric (in the sense of Lawvere) and pseudometric, respectively, under mild conditions