Proper Tail Recursion and Space Eeciency

The IEEE/ANSI standard for Scheme requires implementations to be properly tail recursive. This ensures that portable code can rely upon the space eeciency of continuation-passing style and other idioms. On its face, proper tail recursion concerns the eeciency of procedure calls that occur within a tail context. When examined closely, proper tail recur-sion also depends upon the fact that garbage collection can be asymptotically more space-eecient than Algol-like stack allocation. Proper tail recursion is not the same as ad hoc tail call optimization in stack-based languages. Proper tail recursion often precludes stack allocation of variables, but yields a well-deened asymptotic space complexity that can be relied upon by portable programs. This paper ooers a formal and implementation-independent deenition of proper tail recursion for Scheme. It also shows how an entire family of reference implementations can be used to characterize related safe-for-space properties, and proves the asymptotic inequalities that hold between them. 1 Introduction Tail recursion is a phrase that has been used to refer to various syntactic notions, to particular techniques for implementing syntactic tail recursion, and to the space eeciency of those techniques. Syntactically, a call is a tail call if it appears within a function body that can reduce to the call; this is formalized in Section 2. Since the complete call graph is seldom available, a tail call is often said to be tail recur-sive regardless of whether it occurs within a cycle in the call graph. Scheme, Standard ML, and several other mostly-functional languages rely heavily on the eeciency of tail recursion. Common idioms, notably continuation-passing style (CPS), would quickly run out of stack space if tail calls were to consume space. To ensure that portable code can rely upon these idioms, the IEEE standard for Scheme IEE91] says Implementations of Scheme are required to be properly tail-recursive Ste78]. This allows the