Behavioural Equivalence via Modalities for Algebraic Effects

Abstract

The paper investigates behavioural equivalence between programs in a call-by-value functional language extended with a signature of (algebraic) effect-triggering operations. Two programs are considered as being behaviourally equivalent if they enjoy the same behavioural properties. To formulate this, we define a logic whose formulas specify behavioural properties. A crucial ingredient is a collection of modalities expressing effect-specific aspects of behaviour. We give a general theory of such modalities. If two conditions, openness and decomposability, are satisfied by the modalities then the logically specified behavioural equivalence coincides with a modality-defined notion of applicative bisimilarity, which can be proven to be a congruence by a generalisation of Howe's method. We show that the openness and decomposability conditions hold for several examples of algebraic effects: nondeterminism, probabilistic choice, global store and input/output. *

A Sound and Complete Logic for Algebraic Effects

“…It is easy to see that for all running examples of effects the set P is consistent. The proof that each P ∈ P is Scott-open is similar to that for modalities from [38]. It remains to show that for all our examples P is decomposable.…”

“…The elements of P are subsets of Trees Σ . Observations play a similar role to the modalities from [38]. For our running examples of effects, P is defined as follows:…”

“…The full proof can be found in [21]. First, we define applicative bisimilarity for ECPS, similarly to the way Simpson and Voorneveld [38] define it for PCF with algebraic effects. Then, we prove in turn that F-logical equivalence coincides with applicative bisimilarity, and that applicative bisimilarity coincides with contextual equivalence.…”

“…Dal Lago, Gavazzo and Levy [7] provide an abstract treatment of applicative bisimilarity in the presence of algebraic effects. Working in a typed, call-by-value setting, Simpson and Voorneveld [38] propose a modal logic for effectful programs whose induced program equivalence coincides with applicative bisimilarity, but not with contextual equivalence (see counterexample in Sect. 5).…”

Quantitative Behavioural Reasoning for Higher-order Effectful Programs

“…Proving compatibility of the latter is straightforward. However, proving compatibility of applicative Γ-bisimilarity is not trivial and requires a variation of the so-called transitive closure trick [26,31,39] based on ideas in [46].…”