Step-Indexed Logical Relations for Probability (long version)

It is well-known that constructing models of higher-order probabilistic programming languages is challenging. We show how to construct step-indexed logical relations for a probabilistic extension of a higher-order programming language with impredicative polymorphism and recursive types. We show that the resulting logical relation is sound and complete with respect to the contextual preorder and, moreover, that it is convenient for reasoning about concrete program equivalences. Finally, we extend the language with dynamically allocated first-order references and show how to extend the logical relation to this language. We show that the resulting relation remains useful for reasoning about examples involving both state and probabilistic choice.Comment: Extended version with appendix of a FoSSaCS'15 pape

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

Quantitative Behavioural Reasoning for Higher-order Effectful Programs: Applicative Distances

International audienceThis paper studies quantitative refinements of Abramsky's applica-tive similarity and bisimilarity in the context of a generalisation of Fuzz, a call-by-value λ-calculus with a linear type system that can express program sensitivity, enriched with algebraic operations à la Plotkin and Power. To do so a general, abstract framework for studying behavioural relations taking values over quantales is introduced 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

Towards Probabilistic Reasoning in Type Theory - The Intersection Type Case

The development of different probabilistic models of uncertainty has been inspired by the rapid progress in various fields, e.g. in AI, probabilistic programming, etc. Lambda calculus is a universal model of computation suitable to express programming languages concepts. Hence, different methods for probabilistic reasoning in lambda calculus have been investigated. In this paper, we develop a formal model for probabilistic reasoning about lambda terms with intersection types, which is a combination of lambda calculus and probabilistic logic. The language of lambda calculus with intersection types is endowed with a probabilistic operator. We propose a semantics based on the possible world approach. An infinitary axiomatization is given for this system and it is proved to be sound with respect to the proposed semantics

An Assertion-Based Program Logic for Probabilistic Programs

International audienceResearch on deductive verification of probabilistic programs has considered expectation-based logics, where pre-and post-conditions are real-valued functions on states, and assertion-based logics, where pre-and post-conditions are boolean predicates on state distributions. Both approaches have developed over nearly four decades, but they have different standings today. Expectation-based systems have managed to formalize many sophisticated case studies, while assertion-based systems today have more limited expressivity and have targeted simpler examples. We present Ellora, a sound and relatively complete assertion-based program logic, and demonstrate its expressivity by verifying several classical examples of randomized algorithms using an implementation in the EasyCrypt proof assistant. Ellora features new proof rules for loops and adversarial code, and supports richer assertions than existing program logics. We also show that Ellora allows convenient reasoning about complex probabilistic concepts by developing a new program logic for probabilistic independence and distribution law, and then smoothly embedding it into Ellora. Our work demonstrates that the assertion-based approach is not fundamentally limited and suggests that some notions are potentially easier to reason about in assertion-based systems