Derivative example, GCD example, Ackerman's function example
One common mathematical problem is to find the slope of some function f at some point x. Given f, its derivative f' is a function such that f'(x) is the slope of f at x. Determining the derivative of a function exactly (symbolically) is somewhat difficult, and part of calculus. Calculating an approximation of the derivation is much simpler, and what we'll do.
Consider the following picture:
; deriv-at-point : (num -> num) num num -> num (define (deriv-at-point f x eps) ...)
; deriv : (num -> num) -> (num -> num) (define (deriv f) ...)choosing an appropriately small constant value of eps.
GCD (Greatest Common Divisor) is a common mathematical problem. Given two positive natural numbers, what is the largest (positive) natural number which is a divisor for both inputs? The benefit for us is that it provides a good example of the tradeoffs between structural recursion and generative recursion.
A structurally recursive version is rather straightforward -- count down until you find a numbers that divides both inputs. Q: What should we count down from? Q: Why does this find the greatest common divisor? Here's the code:
;; gcd-structural : N[>=1] N[>=1] -> N[>=1] ;; find the greatest common divisor of the two inputs ;; Examples: ;; (gcd 6 25) = 1 ;; (gcd 18 24) = 6 ;; based on structural recursion (define (gcd-structural n m) (local (; divides? : N N -> boolean ; returns whether i divides n with zero remainder (define (divides? n i) (zero? (remainder n i))) ; first-divisor-<= : N[>=1] -> N[>=1] ; find the greatest common divisor of m and n ; which is less than or equal to i (define (first-divisor-<= i) (cond [(= i 1) 1] [else (cond [(and (divides? n i) (divides? m i)) i] [else (first-divisor-<= (sub1 i))])]))) (first-divisor-<= (min m n))))
A generative recursive version devised by Euclid requires a "Eureka!". To do: Look at the following and figure out what it does. Q: Why does this work?
;; gcd-generative : N[>=1] N[>=1] -> N[>=1] ;; find the greatest common divisor of the two inputs ;; Examples: ;; (gcd 6 25) = 1 ;; (gcd 18 24) = 6 ;; based on generative recursion (define (gcd-generative n m) (local (; euclid-gcd : N N -> N ; find the greatest common divisor of the two inputs ; assume that larger >= smaller (define (euclid-gcd larger smaller) (cond [(= smaller 0) larger] [else (euclid-gcd smaller (remainder larger smaller))]))) (euclid-gcd (max m n) (min m n))))
The advantage of the structural version is obvious -- it is easier to understand. To do: To see the advantage of the generative version, guesstimate how many recursive calls each version takes to find the answer. Try out your idea on some more examples, including some larger ones, such as
(gcd-structural 101135853 45014640) (gcd-generative 101135853 45014640)
Here's the definition of a strange numerical function on natural numbers, called Ackerman's Function:
A(m,n) = 0, if n=0 = 2n, if n>0 and m=0 = 2, if n=1 and m>0 = A(m-1,A(m,n-1)), if n>1 and m>0Note that this definition is not structurally recursive. (In fact, it cannot be defined in a structurally recursive manner.)
This function isn't very useful, except as a function that grows faster than probably any one you've seen before. (In technical jargon, the function is not primitive recursive.)