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

Behavioural preorders and equivalences for higher-order languages have been extensively studied in the last decades, often leading to rich and satisfactory theories. However, in presence of effectful computations ordinary behavioural relations can be too discriminating, as highlighted by probabilistic higher-order languages. An elegant way to avoid such problem is to move from qualitative, boolean-valued relations to quantitative relations describing weakened notions of metrics. This paper studies the quantitative refinements of Absramsky’s applicative 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—algebraic structures representing sets of abstract quantities—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. An abstract notion of quantalevalued effectful applicative similarity is then defined and proved to be a compatible generalised metric (in the sense of Lawvere). Notions of quantale-valued applicative bisimilarity and two-way similarity are also defined and proved to be compatible symmetric generalised metrics (viz. pseudometrics) under mild conditions. The resulting theory refines the existing theory of effectful applicative similarity and can be instantiated to come up with applicative distances for concrete calculi. A remarkable example is given by the probabilistic instantiation of Fuzz where a compatible probabilistic applicative distance based on the Wasserstein-Kantorovich distance is obtained. To the best of the author’s knowledge this is the first example of a (non-trivial) compatible applicative distance for a universal (i.e. Turing complete) probabilistic λ-calculus.