fold

Folding (also known as reduce or accumulate) is a method of reducing a sequence of terms down to a single term. This is accomplished by providing a fold function with a binary operator, an initial (or identity) value, and a sequence.

There are two kinds of fold: a right one and a left one.

Here's the definition for the fold-right function:

 (define (fold-right f init seq) 
   (if (null? seq) 
       init 
       (f (car seq) 
          (fold-right f init (cdr seq))))) 

And here is the definition of fold-left function:

 (define (fold-left f init seq) 
   (if (null? seq) 
       init 
       (fold-left f 
                  (f init (car seq)) 
                  (cdr seq)))) 

 (fold-right + 0 '(1 2 3 4)) 
   ; expands to (+ 1 (+ 2 (+ 3 (+ 4 0)))) 
  
 (fold-left + 0 '(1 2 3 4)) 
   ; expands to (+ (+ (+ (+ 0 1) 2) 3) 4) 

A list reversal function can easily be defined using fold-left:

 (define (reverse l) 
   (fold-left (lambda (x y) (cons y x)) 
              '() 
              l)) 

Further reading:

• Wikipedia Fold entry
• A tutorial on the universality and expressiveness of fold