An Initial Model for the Monadic Calculus

When it comes to compiler design, there has been a dispute as to whether it is useful to transform the source code using the continuation passing style (CPS) transform first. Some have observed that it is easier to optimize CPS transformed code, while others maintain that direct style compilation is better. Sabry and Felleisen resolve the dispute by showing how get the benefits of CPS without the CPS transform. They note that it is, indeed, sometimes easier to reason about CPS transformed terms than to reason directly about Λ-terms. They set about determining what axioms would make it just as easy to reason about Λ-terms. Their result is summarized in the following theorem, where λC denotes the computational λ-calculus, and cps(−) denotes the CPS transform. Theorem(SF): λC `M = N iff λβ,eta ` cps(M) = cps(N) Filinski and others have pointed out that the CPS transform can be seen as an instance of the monadic transform. (See table 5 for the monadic transform.) Instantiate f∗ and η as follows to get the CPS transform from the monadic transform: η = λa.λk.(ka), f∗ = λt.λk.(t(λa.((fa)k))). In this paper we generalize Sabry and Felleisen’s result and prove the following theorem. Theorem : λC `M = N iff λM `M = N To prove this theorem, we follow Sabry and Felleisen in our approach. After proving that λC `M = N =⇒ λM `M = N , we define a grammar for monadic terms, and an inverse transform. We show that the inverse transform does, in fact, invert the monadic transform. The result then quickly follows.