Question

Explain what the following Scheme code is doing:

(define (make-stream n f)
  (define (next m)
    (cons m (lambda () (next (f m)))))
  (next n))

(define head car)

(define (tail stream)
  ((cdr stream)))

(define (nth stream n)
  (if (= n 0) (head stream)
    (nth (tail stream) (- n 1))))

(define even (make-stream 0 (lambda (n) (+ n 2))))

Try it out in Scheme48 and check the values of the following expressions:

even
(head even)
(head (tail even))
(head (tail (tail even)))
(head (tail (tail (tail even))))
(nth even 5)
(nth even 1000)

Explain what the lambda in make-stream is good for, where this function is called, and how tail and nth work. To see what’s going on, trace manually through the execution of

(head (tail (tail even)))

Answer 1

ANSWER:

When a function is called, the computer must "remember" the place it was called from, the return address, so that it can return to that location with the result once the call is complete. Typically, this information is saved on the call stack, a simple list of return locations in order of the times that the call locations they describe were reached. For tail calls, there is no need to remember the place we are calling from – instead, we can perform tail call elimination by leaving the stack alone (except possibly for function arguments and local variables[3]), and the newly called function will return its result directly to the original caller. Note that the tail call doesn't have to appear lexically after all other statements in the source code; it is only important that the calling function return immediately after the tail call, returning the tail call's result if any, since the calling function will never get a chance to do anything after the call if the optimization is performed.

For non-recursive function calls, this is usually an optimization that saves little time and space, since there are not that many different functions available to call. When dealing with recursive or mutually recursive functions where recursion happens through tail calls, however, the stack space and the number of returns saved can grow to be very significant, since a function can call itself, directly or indirectly, creating a new call stack frame each iteration. In fact, it often asymptotically reduces stack space requirements from linear, or O(n), to constant, or O(1). Tail call elimination is thus required by the standard definitions of some programming languages, such as Scheme,[4][5] and languages in the ML family among others. In the case of Scheme, the language definition formalizes the intuitive notion of tail position exactly, by specifying which syntactic forms allow having results in tail context.[6] Implementations allowing an unlimited number of tail calls to be active at the same moment, thanks to tail call elimination, can also be called 'properly tail-recursive'.[4]

Besides space and execution efficiency, tail call elimination is important in the functional programming idiom known as continuation-passing style (CPS), which would otherwise quickly run out of stack space.