On the Expressive Power of User - Defined E ects

We compare the expressive power of three programming abstractions for user-de ned computational e ects: Bauer and Pretnar’s e ect handlers, Filinski’s monadic re ection, and delimited control without answer-type-modi cation. This comparison allows a precise discussion about the relative expressiveness of each programming abstraction. It also demonstrates the sensitivity of the relative expressiveness of user-de ned e ects to seemingly orthogonal language features. We present three calculi, one per abstraction, extending Levy’s call-by-push-value. For each calculus, we present syntax, operational semantics, a natural type-and-e ect system, and, for e ect handlers and monadic re ection, a set-theoretic denotational semantics. We establish their basic meta-theoretic properties: safety, termination, and, where applicable, soundness and adequacy. Using Felleisen’s notion of a macro translation, we show that these abstractions can macro-express each other, and show which translations preserve typeability. We use the adequate nitary set-theoretic denotational semantics for the monadic calculus to show that e ect handlers cannot be macro-expressed while preserving typeability either by monadic re ection or by delimited control. We supplement our development with a mechanised Abella formalisation.