Continuations, functions and jumps

Practically all programming languages have some form of control structure or jumping. The more advanced forms of control structure tend to resemble function calls, so much so that they are usually not even described as jumps. Consider for example the library function exit in C. Its use is much like a function, in that it may be called with an argument; but the whole point of exit is of course that its behaviour is utterly non-functional, in that it jumps out of arbitrarily many surrounding blocks and pending function calls. Such a "non-returning function" or "jump with arguments" is an example of a continuation in the sense which we are interested in here.On the other hand, a simple but fundamental idea in compiling is that a function call is broken down into two jumps: one from the caller to the callee for the call itself, and another jump back from the callee to the caller upon returning. (The return statement in C is in fact listed among the "jump statements" [5].) This is most obvious for void-accepting and -returning functions, but it generalizes to other functions as well, if one is willing to understand "jump" in the broader sense of jump with arguments, i.e. continuation.In this view, then, continuations are everywhere. Continuations have been used in many different settings, in which they appear in different guises, ranging from mathematical functions to machine addresses. Rather than confine ourselves to a definition of what a continuation is, we will focus on continuation-passing style (CPS), as it brings out the commonalities. The CPS transform compresses a great deal of insight into three little equations in λ-calculus. Making sense of it intuitively, however, requires some background knowledge and a certain fluency. The purpose of this article, therefore, is to help the reader uncompress the CPS transform by way of a rational reconstruction from jumps.In the sequel, we will attempt to illustrate the correspondence between continuations and jumps (even in the guise of the abhorred goto). The intent is partly historical, to retrace some of the analysis of jumps that led to the discoveries of continuations. At the same time, the language of choice for many researchers during the (pre)history of continuations, ALGOL 60, is not so different from today's mainstream languages (i.e. C); we hope that occasional snippets of C may be more easily accessible to many readers than a functional language would be. So in each of the four sections (Sections 2-5 below) that make up the body of this paper, some C code will be used to give a naive but concrete example of the issue under consideration, before generalizing to a more abstract setting.