Three Steps for the CPS Transformation

Transforming a λ-term into continuation-passing style (CPS) might seem mystical at first, but in fact it can be characterized by three separate aspects: • The values of all intermediate applications are given a name. • The evaluation of these applications is sequentialized based on a traversal of their syntax tree. This traversal mimics the reduction strategy. • The resulting term is equipped with a continuation — a λ-abstraction whose application to intermediate values yields the final result of the whole evaluation. The first point is fulfilled using the uniform naming mechanism of λ-abstraction (Church encoding), which explains why continuations are represented as functions. The second point justifies why CPS terms are evaluation-order independent — their evaluation order is determined by the syntax tree traversal of the CPS transformation. The third point captures the essence of the CPS transformation. We have staged Fischer and Plotkin’s original CPS transformer according to the three points above. For call-by-value λ-terms, the tree traversal is post-order. The method is applicable to other evaluation orders, e.g., call-by-value with evaluation from right to left, or call-by-name. This staging originates in the need for a clearer formulation of the CPS transformation, suitable for deriving its inverse.