Adequacy for Algebraic Effects

Moggi proposed a monadic account of computational effects. He also presented the computational λ-calculus, λc, a core call-by-value functional programming language for effects; the effects are obtained by adding appropriate operations. The question arises as to whether one can give a corresponding treatment of operational semantics. We do this in the case of algebraic effects where the operations are given by a single-sorted algebraic signature, and their semantics is supported by the monad, in a certain sense. We consider call-by-value PCF with— and without—recursion, an extension of λc with arithmetic. We prove general adequacy theorems, and illustrate these with two examples: nondeterminism and probabilistic nondeterminism.