Recursion from Iteration

In a simply-typed, call-by-value (CBV) language with rst-class continuations , the usual CBV xpoint operator can be deened in terms of a simple, innnitely-looping iteration primitive. We rst consider a natural but awed deenition, based on exceptions and \iterative deepening" of nite unfoldings, and point out some of its shortcomings. Then we present the proper construction using full rst-class continuations , with both an informal derivation and a proof that the behavior of the deened operator faithfully mimics a \built-in" recursion primitive. In fact, given an additional uniformity assumption, the construction is a two-sided inverse of the usual deenition of iteration from recursion. Continuing, we show that the CBV looping primitive is in fact the direct-style equivalent of a continuation-passing-style xpoint, and that this correspondence extends all the way to traditional deenitions of these operators in terms of reeexive types.