# aliquote

## < a quantity that can be divided into another a whole number of time />

In my previous post I discussed the repeated application of a function to build a list of results. This was done using a named let form first, then through currying. See what’s happening?

(import (chicken random))

(define (curry n f)
(if (= n 0)
(lambda (x) x)
(lambda (x) (f ((curry (- n 1) f) x)))))

(define (square x) (* x x))
((curry 5 (lambda (x) (cons (square 2) x))) '())
;; => (4 4 4 4 4)

(define (random max) (pseudo-random-integer max))
((curry 5 (lambda (x) (cons (random 10) x))) '())
;; => (9 6 5 3 7)


Since intermediate results are not exploited in successive calls to f in the above example, this amounts to make n independent calls to f, unlike the following use of curry, where intermediate results are reused:

((curry 2 (lambda (x) (square x))) 2)
;; => 16
((curry 3 (lambda (x) (square x))) 2)
;; => 256


In the first case, we compute $2\times 2 = 4$, then $4\times 4 = 16$, while in the second case we add an extra step with $16\times 16 = 256$.

The most natural application of currying is probably to compute an iterated or cumulative sum in statistical computing. As an alternative one may consider a “reducer”, like reduce or foldl. Currying is also important in functional programming when it comes to manage function multiple arguments, since it allows to express $f(x, y)$ as $g(x)(y)$, for instance. In Python, the partial function from the functools package does just that. In Scheme, we can introduce as many lambdas as we need in a function, or use specialized macros.1 As an example, consider a function that expect two arguments, (define (greater? x n) (> x n)) – I use such constructs in combination with filter to select observations in a sequence of numerical values; it can be converted to a one-argument function using an extra lambda, (define (greater? n) (lambda (x) (> x n))). Then instead of (greater? 3 1), we would simply write ((greater? 1) 3). The function now returns a new function that knows the value of n.

The SICP has another application of repeated call to a function in exercise 1.43:

If f is a numerical function and n is a positive integer, then we can form the nth repeated application of f, which is defined to be the function whose value at x is f(f(…(f(x))…)). For example, if f is the function x + 1, then the nth repeated application of f is the function x + n. If f is the operation of squaring a number, then the nth repeated application of f is the function that raises its argument to the 2^nth power. Write a procedure that takes as inputs a procedure that computes f and a positive integer n and returns the procedure that computes the nth repeated application of f.

Unlike the previous case, this times it’s all about composing function using an auxiliary function, compose:

(define (compose f g)
(lambda (x)
(f (g x))))


Now, a recursive call to a function that calls itself $n$ times, e.g. $f(x) = x^2$ which for $x=5$ and $n=4$ is $(5 \times (5 \times (5 \times 5)))$, can be written as:

(define (repeated fn n)
(if (= n 1)
fn
(compose fn (repeated fn (- n 1)))))


In the above, fn isn’t called n times, instead this function returns the function fn itself $n-1$ times and the lambda from compose takes care of calling those new functions, plus an extra call via the guard which makes for a total of $n$ calls. Getting ride of the compose helper function, we simply have the following lambda in action:

(define (repeated fn n)
(if (= n 1)
fn
(lambda (x) (fn ((repeated fn (- n 1)) x)))))


See the Scheme wiki for further discussion on this exercise.

As a sidenote, the following function is defined (and explained) in Simply Scheme:

(define repeated
(let ((= =) (- -))
(lambda (fn number)
(if (= number 0)
(lambda (x) x)
(lambda (x)
((repeated fn (- number 1)) (fn x)))))))


♪ The Fall • Totally Wired

1. There’s even an SRFI under review. ↩︎