The Lazy Fibonacci List

In a project I am working on, I want to implement a large list using lazy evaluation in Scheme. The list is large enough to be too unwieldy to store entirely in memory, but I still want to represent it in my program as if it was. The solution is lazy evaluation.

One use of lazy evaluation is allowing a program to have infinitely sized data structures without going into the impossible task of actually creating them. Instead, the structure is created on the fly as needed. As a prototype for getting it right, I made an infinitely long list in Scheme that contains the entire Fibonacci series.

This function, given two numbers from the series, returns the lazy list. It uses delay to delay evaluation of the list.

(define (fib f)
  (cons (cadr f)
        (delay (fib (list (cadr f)
                          (apply + f))))))

Notice the recursion here as no base case, so without lazy evaluation it would continue along forever without halting. Now run it,

> (fib '(0 1))
(1 . #<promise>)

The rest of the list is stored as a promise, which will later be teased out using force. This forces evaluation of the promise. Here is a function to traverse the list to the nth element and return it. Notice, this does have a base case.

(define (nth-fib f n)
  (if (= n 1) (car f)
      (nth-fib (force (cdr f)) (- n 1))))

Here it is in action. It is retrieving the 30th element.

> (define f (fib '(0 1)))
> f
(1 . #<promise>)
> (nth-fib f 30)

If you examine f, it contains the first 30 numbers until running into an unevaluated promise. This behavior is very similar to memoization, as calculated values are stored instead of being recalculated later.

These two functions are also behaving as coroutines. When nth-fib reaches a promise, it yields to fib, which continues its non-halting definition. After producing a new value in f, it yields back to nth-fib.

The way I called these functions above, however, can lead to problems. We are storing all the calculated values in f, which can take up a lot of memory. For example, this probably won't work,

> (nth-fib f 1000000)

We will run out of memory before it halts. Instead, we can do this,

> (nth-fib (fib '(0 1)) 1000000)

Because nth-fib uses tail recursion as it traverses the list, unneeded calculated values are tossed (which the garbage collector will handle) and no additional function stack is used. All Scheme implementations optimize tail recursion in this way. This will continue along until it hits the millionth Fibonacci number, all while using a constant amount of memory.

It turns out that Scheme calls this type of data structure a stream, and some implementations have functions and macros defined so that they are ready to use.

So there you go: memoization, lazy evaluation, and coroutines all packed into one example.

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Chris Wellons (PGP)
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