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Concept# Fold (higher-order function)

Summary

In functional programming, fold (also termed reduce, accumulate, aggregate, compress, or inject) refers to a family of higher-order functions that analyze a recursive data structure and through use of a given combining operation, recombine the results of recursively processing its constituent parts, building up a return value. Typically, a fold is presented with a combining function, a top node of a data structure, and possibly some default values to be used under certain conditions. The fold then proceeds to combine elements of the data structure's hierarchy, using the function in a systematic way.
Folds are in a sense dual to unfolds, which take a seed value and apply a function corecursively to decide how to progressively construct a corecursive data structure, whereas a fold recursively breaks that structure down, replacing it with the results of applying a combining function at each node on its terminal values and the recursive results (catamorphism, versus anamorphism of unfolds).
Folds can be regarded as consistently replacing the structural components of a data structure with functions and values. Lists, for example, are built up in many functional languages from two primitives: any list is either an empty list, commonly called nil ([]), or is constructed by prefixing an element in front of another list, creating what is called a cons node ( Cons(X1,Cons(X2,Cons(...(Cons(Xn,nil))))) ), resulting from application of a cons function (written down as a colon (:) in Haskell). One can view a fold on lists as replacing the nil at the end of the list with a specific value, and replacing each cons with a specific function. These replacements can be viewed as a diagram:
There's another way to perform the structural transformation in a consistent manner, with the order of the two links of each node flipped when fed into the combining function:
These pictures illustrate right and left fold of a list visually. They also highlight the fact that foldr (:) [] is the identity function on lists (a shallow copy in Lisp parlance), as replacing cons with cons and nil with nil will not change the result.

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