On the Versatility of Open Logical Relations

Abstract

Logical relations are one among the most powerful techniques in the theory of programming languages, and have been used extensively for proving properties of a variety of higher-order calculi. However, there are properties that cannot be immediately proved by means of logical relations, for instance program continuity and differentiability in higher-order languages extended with real-valued functions. Informally, the problem stems from the fact that these properties are naturally expressed on terms of non-ground type (or, equivalently, on open terms of base type), and there is no apparent good definition for a base case (i.e. for closed terms of ground types). To overcome this issue, we study a generalization of the concept of a logical relation, called open logical relation, and prove that it can be fruitfully applied in several contexts in which the property of interest is about expressions of first-order type. Our setting is a simply-typed λ-calculus enriched with real numbers and real-valued first-order functions from a given set, such as the one of continuous or differentiable functions. We first prove a containment theorem stating that for any collection of real-valued firstorder functions including projection functions and closed under function composition, any well-typed term of first-order type denotes a function belonging to that collection. Then, we show by way of open logical relations the correctness of the core of a recently published algorithm for forward automatic differentiation. Finally, we define a refinement-based type system for local continuity in an extension of our calculus with conditionals, and prove the soundness of the type system using open logical relations.

Densities of Almost Surely Terminating Probabilistic Programs are Differentiable Almost Everywhere

“…Finally we note that open logical relations [1] are a convenient proof technique for establishing properties of programs which hold at first order, such as almost everywhere differentiability. This approach remains to be investigated in this context, as the connection with probabilistic densities is not immediate.…”

A language for evaluating derivatives of functionals using automatic differentiation

“…In these works, correctness of automatic differentiation is proved in the context of first order languages [1], or higher order functional languages [3,11,21,40,43]. Calculi that use logical relations to prove the correctness of derivative evaluation, a key aspect of our work, are provided in [7,15]. In several of these works, the semantics of differentiation is given using a categorical setting, [3,29,40], in contrast to using the more concrete space of real numbers or its extension in domain theory.…”

“…Consider the one dimensional initial value problem, ẏ′ (x) = v(y(x)), y(0) = 0 (7) where v : O → R is a continuous one dimensional vector field in an open neighbourhood O ⊂ R of the origin.…”

“…Logical relations are a standard proof technique used in the semantics of functional languages; they are used for proving that the semantic interpretation of terms satisfies some desired properties. A general introduction to logical relations can be found in [44], while in [7,15], logical relations are used in a way similar to our work.…”

“…By Kleene's theorem, the least fixed point y of P u satisfies y = P u (y). In addition, if u is an extension function v and f is a solution of the IVP in Equation (7) then y contains f i.e., y ⊑ f , whereas if y is maximal then it is the maximal extension of the unique solution of the initial value theorem; cf. [19,Proposition 3.7].…”