A functional correspondence between monadic evaluators and abstract machines for languages with computational effects

Abstract

We show how to derive new abstract machines from monadic evaluators for the computational λ-calculus. Starting from (1) a generic evaluator parameterized by a monad and (2) a monad specifying a computational effect, we inline the components of the monad in the generic evaluator to obtain an evaluator written in a style that is specific to this computational effect. We then derive the corresponding abstract machine by closure-converting, CPS-transforming, and defunctionalizing this specific evaluator. We illustrate the construction with the identity monad, obtaining yet again the CEK machine, and with a lifted state monad, obtaining a variant of the CEK machine with error and state.In addition, we characterize the tail-recursive stack inspection presented by Clements and Felleisen at ESOP 2003 as a lifted state monad. This enables us to combine the stackinspection monad with other monads and to construct abstract machines for languages with properly tail-recursive stack inspection and other computational effects. The construction scales to other monads-including one more properly dedicated to stack inspection than the lifted state monad-and other monadic evaluators.

Combining effects: Sum and tensor

“…In regard to the latter it is worth noting that the combination supports a nonstandard exception-handling mechanism, that, when an exception is raised, restores the state to what it was when the handler was defined, and then executes the handler. This combination of state and exceptions has been termed 'transitional' in [11], and 'snapback' in [4]. In practice, monads for various kinds of exceptions, and even states, would be combined; some sample calculations are given in Section 7.…”

A functional correspondence between call-by-need evaluators and lazy abstract machines

“…To make it follow call by need [14,17,27], we thread a heap of updatable cells (Section 2.2). Threading this heap is akin to inlining a state monad [2]. Using updatable cells to implement call by need is traditional [3, p. 333], [28, p. 81].…”

“…Sestoft concentrates on deriving one particular abstract machine out of one natural semantics, whereas the present article is part of a general investigation of a correspondence between evaluators and abstract machines [1,2,7].…”

“…The functional correspondence can also provide guidelines for constructing complex abstract machines. For example, along the lines described in the present article, we can derive a lazy abstract machine handling arbitrarily complex computational effects-e.g., stack inspection-given a call-by-need evaluator equipped with the corresponding monad [2].…”

“…For example, one can derive "new" abstract machines by inlining monads in an evaluator expressed in Moggi's computational meta-language [2]. One can also reflect refinements and variations from an abstract machine to the corresponding evaluator.…”