Measurable cones, with linear and measurable functions as morphisms, are a model of intuitionistic linear logic and of call-by-name probabilistic PCF which accommodates "continuous data types" such as the real line. So far however, they lacked a major feature to make them a model of more general probabilistic programming languages (notably call-by-value and call-by-push-value languages): a theory of integration for functions whose codomain is a cone, which is the key ingredient for interpreting the sampling programming primitives. The goal of this paper is to develop such a theory: our definition of integrals is an adaptation to cones of Pettis integrals in topological vector spaces. We prove that such integrable cones, with integral-preserving linear maps as morphisms, form a model of Linear Logic for which we develop two exponential comonads: the first based on a notion of stable and measurable functions introduced in earlier work and the second based on a new notion of integrable analytic function on cones.

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We prove the following completeness result about classical realizability: given any Boolean algebra with at least two elements, there exists a Krivine-style classical realizability model whose characteristic Boolean algebra is elementarily equivalent to it. This is done by controlling precisely which combinations of so-called “angelic” (or “may”) and “demonic” (or “must”) nondeterminism exist in the underlying model of computation.

An approximate program transformation is a transformation that can change the semantics of a program within a specified empirical error bound. Such transformations have wide applications: they can decrease computation time, power consumption, and memory usage, and can, in some cases, allow implementations of incomputable operations. Correctness proofs of approximate program transformations are by definition quantitative. Unfortunately, unlike with standard program transformations, there is as of yet no modular way to prove correctness of an approximate transformation itself. Error bounds must be proved for each transformed program individually, and must be re-proved each time a program is modified or a different set of approximations are applied. In this paper, we give a semantics that enables quantitative reasoning about a large class of approximate program transformations in a local, composable way. Our semantics is based on a notion of distance between programs that defines what it means for an approximate transformation to be correct up to an error bound. The key insight is that distances between programs cannot in general be formulated in terms of metric spaces and real numbers. Instead, our semantics admits natural notions of distance for each type construct; for example, numbers are used as distances for numerical data, functions are used as distances for functional data, an polymorphic lambda-terms are used as distances for polymorphic data. We then show how our semantics applies to two example approximations: replacing reals with floating-point numbers, and loop perforation.

Abstract We develop a technique for representing and preserving cardinals in realizability models, and we apply this technique to define a realizability model of Zorn’s lemma restricted to an ordinal.

We show how the language of Krivine's classical realizability may be used to specify various forms of nondeterminism and relate them with properties of realizability models. More specifically, we introduce an abstract notion of multi-evaluation relation which allows us to finely describe various nondeterministic behaviours. This defines a hierarchy of computational models, ordered by their degree of nondeterminism, similar to Sazonov's degrees of parallelism. What we show is a duality between the structure of the characteristic Boolean algebra of a realizability model and the degree of nondeterminism in its underlying computational model.