Abstract-We study Abramsky's applicative bisimilarity abstractly, in the context of call-by-value λ-calculi with algebraic effects. We first of all endow a computational λ-calculus with a monadic operational semantics. We then show how the theory of relators provides precisely what is needed to generalise applicative bisimilarity to such a calculus, and to single out those monads and relators for which applicative bisimilarity is a congruence, thus a sound methodology for program equivalence. This is done by studying Howe's method in the abstract.
This paper studies quantitative refinements of Abramsky'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 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.
Semantics is traditionally concerned with program equivalence, in which all pairs of programs which are
not
equivalent are treated the same, and simply dubbed as incomparable. In recent years, various forms of program
metrics
have been introduced such that the distance between non-equivalent programs is measured as an element of an appropriate quantale. By letting the underlying quantale
vary
as the type of the compared programs become more complex, the recently introduced framework of differential logical relations allows for a new contextual form of reasoning. In this paper, we show that all this can be generalised to
effectful
higher-order programs, in which not only the
values
, but also the
effects
computations produce can be appropriately distanced in a principled way. We show that the resulting framework is flexible, allowing various forms of effects to be handled, and that it provides compact and informative judgments about program differences.
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