In this paper we are concerned with understanding the nature of program metrics for calculi with higher-order types, seen as natural generalizations of program equivalences. Some of the metrics we are interested in are well-known, such as those based on the interpretation of terms in metric spaces and those obtained by generalizing observational equivalence. We also introduce a new one, called the interactive metric, built by applying the well-known Int-Construction to the category of metric complete partial orders. Our aim is then to understand how these metrics relate to each other, i.e., whether and in which cases one such metric refines another, in analogy with corresponding well-studied problems about program equivalences. The results we obtain are twofold. We first show that the metrics of semantic origin, i.e., the denotational and interactive ones, lie in between the observational and equational metrics and that in some cases, these inclusions are strict. Then, we give a result about the relationship between the denotational and interactive metrics, revealing that the former is less discriminating than the latter. All our results are given for a linear lambda-calculus, and some of them can be generalized to calculi with graded comonads, in the style of Fuzz.

Abstract We give two geometry of interaction models for a typed λ-calculus with recursion endowed with operators for sampling from a continuous uniform distribution and soft conditioning, namely a paradigmatic calculus for higher-order Bayesian programming. The models are based on the category of measurable spaces and partial measurable functions, and the category of measurable spaces and s-finite kernels, respectively. The former is proved adequate with respect to both a distribution-based and a sampling-based operational semantics, while the latter is proved adequate with respect to a sampling-based operational semantics.

We give a geometry of interaction model for a typed λ -calculus endowed with operators for sampling from a continuous uniform distribution and soft conditioning, namely a paradigmatic calculus for higher-order Bayesian programming. The model is based on the category of measurable spaces and partial measurable functions, and is proved adequate with respect to both a distribution-based and a sampling-based operational semantics.

A general framework of Memoryful Geometry of Interaction (mGoI) is introduced recently by the authors. It provides a sound translation of lambda-terms (on the high-level) to their realizations by stream transducers (on the low-level), where the internal states of the latter (called memories) are exploited for accommodating algebraic effects of Plotkin and Power. The translation is compositional, hence ``denotational,'' where transducers are inductively composed using an adaptation of Barbosa's coalgebraic component calculus. In the current paper we extend the mGoI framework and provide a systematic treatment of recursion---an essential feature of programming languages that was however missing in our previous work. Specifically, we introduce two new fixed-point operators in the coalgebraic component calculus. The two follow the previous work on recursion in GoI and are called Girard style and Mackie style: the former obviously exhibits some nice domain-theoretic properties, while the latter allows simpler construction. Their equivalence is established on the categorical (or, traced monoidal) level of abstraction, and is therefore generic with respect to the choice of algebraic effects. Our main result is an adequacy theorem of our mGoI translation, against Plotkin and Power's operational semantics for algebraic effects.

We propose a mathematical framework for step indexed realizability semantics of a call-by-value polymorphic lambda calculus with recursion, existential types and recursive types. Our framework subsumes step indexed realizability semantics by untyped call-by-value lambda calculi as well as categorical abstract machines. Starting from an extension of Hofstra's basic combinatorial objects, we construct a step indexed categorical realizability semantics. Our main result is soundness and adequacy of our step indexed realizability semantics. As an application, we show that a small step operational semantics captures the big step operational semantics of the call-by-value polymorphic lambda calculus. We also give a safe implementation of the call-by-value polymorphic lambda calculus into a categorical abstract machine.

While much of the current study on quantum computation employs low-level formalisms such as quantum circuits, several high-level languages/calculi have been recently proposed aiming at structured quantum programming. The current work contributes to the semantical study of such languages, by providing interaction-based semantics of a functional quantum programming language, the latter is based on linear lambda calculus and is equipped with features like the! modality and recursion. The proposed denotational model is the first one that supports the full features of a quantum functional programming language, we also prove adequacy of our semantics. The construction of our model is by a series of existing techniques taken from the semantics of classical computation as well as from process theory. The most notable among them is Girard's Geometry of Interaction (GoI), categorically formulated by Abramsky, Haghverdi and Scott. The mathematical genericity of these techniques - largely dueto their categorical formulation - is exploited for our move from classical to quantum.