Higher order procedures
Function as data
data : functions :: nouns : verbs Information and actions naturally exist as distinct categories in human knowledge.
Applicative-order vs normal-order evaluation
Scheme is an applicative-order language. Arguments are evaluated first, then the procedure is applied:
Given these functions:
(define (square x) (* x x))
(define (sum-of-squares x y)
(+ (square x) (square y)))
(define (f a)
(sum-of-squares (+ a 1) (* a 2)))applicative-order evaluation would evaluate all argument expressons first (starting with the deepest-nested), then apply the procedure:
(f 5)
(sum-of-squares (+ 5 1) (* 5 2))
(sum-of-squares 6 10)
(+ (square 6) (square 10))
(+ (* 6 6) (* 10 10))
(+ 36 100)
136Normal-order evaluation would start by expanding all expressions, then reduce them:
(f 5)
(sum-of-squares (+ 5 1) (* 5 2))
(+ (square (+ 5 1)) (square (* 5 2)))
(+ (* (+ 5 1) (+ 5 1)) (* (* 5 2) (* 5 2)))
(+ (* 6 6) (* 10 10))
(+ 36 100)
136Applicative order and normal order will yield the same results if the procedure is a function. If the procedure relies on external variables, the results may be different.
To check if a language is applicative-order or normal-order, see if an if-else statement evaluates both the if action and the else action immediately.