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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.
Official source
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Ontological neighbourhood
Related people (7)
Related concepts (16)
Corecursion
In computer science, corecursion is a type of operation that is to recursion. Whereas recursion works analytically, starting on data further from a base case and breaking it down into smaller data and repeating until one reaches a base case, corecursion works synthetically, starting from a base case and building it up, iteratively producing data further removed from a base case. Put simply, corecursive algorithms use the data that they themselves produce, bit by bit, as they become available, and needed, to produce further bits of data.
Clojure
Clojure (ˈkloʊʒər, like closure) is a dynamic and functional dialect of the Lisp programming language on the Java platform. Like most other Lisps, Clojure's syntax is built on S-expressions that are first parsed into data structures by a reader before being compiled. Clojure's reader supports literal syntax for maps, sets and vectors along with lists, and these are compiled to the mentioned structures directly. Clojure treats code as data and has a Lisp macro system.
Map (higher-order function)
In many programming languages, map is the name of a higher-order function that applies a given function to each element of a collection, e.g. a list or set, returning the results in a collection of the same type. It is often called apply-to-all when considered in functional form. The concept of a map is not limited to lists: it works for sequential containers, tree-like containers, or even abstract containers such as futures and promises. Suppose we have a list of integers [1, 2, 3, 4, 5] and would like to calculate the square of each integer.