Effectful applicative bisimilarity: Monads, relators, and Howe's method

Abstract

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.

Quantitative Behavioural Reasoning for Higher-order Effectful Programs

“…To prove compatibility of applicative Γ-similarity we design a generalisition of the so-called Howe's method [26] combining and extending ideas from [12] and [14]. We start by defining the notion of It is easy to see that compatible refinement− is monotone, and thus so is the map Φ α defined by Φ α (β) α ·β.…”

“…We also use monads to give operational semantics to V-Fuzz [14]. Intuitively, a program e evaluates to a monadic value v ∈ T V, where V denotes the set of values.…”

“…It is immediate to see that when instantiated with V = 2, the above definition gives the usual notion of relator, with some minor differences. In [14] and [33] a kernel preservation condition is required in place of (V-rel 3). Such condition is also known as stability in [27].…”

“…Let us examine premise (14). First of all, since u is a closed value of type σ ⊸ τ it must be of the form λx.…”

“…The present work originated from the idea of generalising the theory developed in [14] to relations taking values over arbitrary quantitative domains (such as the real extended half-line [0, ∞] or the unit interval [0, 1]). Such generalisation requires three major improvements in the current theory of effectful applicative (bi)similarity:…”