Adequacy for Algebraic Effects

Abstract

Abstract. 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 withand without-recursion, an extension of λ c with arithmetic. We prove general adequacy theorems, and illustrate these with two examples: nondeterminism and probabilistic nondeterminism.

Combining Computational Effects: Commutativity and Sum

“…Modulo a few side conditions, the answer is yes; we make fundamental use of the correspondence between strong monads and a generalised notion of Lawvere theory in order to provide it (23]. That correspondence is computationally natural and is already implicit in our previous work on computational effects (20,21,22]. We are unaware of any direct justification for the existence of T ® T'.…”

“…Here, we are following an algebraic programme that shifts focus away from monads to the study of natural operations that yield the required effects (see (20,21] for other recent work along these lines), with the monads then corresponding to natural theories for these operations (22]. For instance, rather than emphasise the side-effects monad (S x -) 8 , we emphasise the operations lookup and update associated with side-effects, and the equations that relate them (22].…”

Combining effects and coeffects via grading

“…Previous work by Plotkin and Power [42] showed how to design an operational semantics that collects effect information. Similarly, previous work on coeffects included an instrumented operational semantics collecting information on observable coeffect actions [8].…”

Handlers of Algebraic Effects

“…The most important is how to simultaneously handle two computations to describe parallel operators, e.g., that of CCS or the Unix pipe combinator; that would bring parallelism within the ambit of the algebraic theory of effects. More routinely, perhaps, the logic should be worked out more fully, the work done on combinations of effects in [10] should be extended to combinations of handlers, and there should be a general operational semantics [18] including that of Benton and Kennedy in [2].…”

“…Examples include exceptions, nondeterminism, interactive input/output, concurrency, state, time, continuations, and combinations thereof. Plotkin and Power later proposed to focus on algebraic effects, that is, effects that allow a representation by operations and equations [18,20,21]; the operations give rise to the effects at hand. All of the effects mentioned above are algebraic, with the notable exception of continuations [6], which have to be treated differently: see [9] for initial ideas.…”