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In mathematics and computer science, mutual recursion is a form of recursion where two mathematical or computational objects, such as functions or datatypes, are defined in terms of each other. [1] Mutual recursion is very common in functional programming and in some problem domains, such as recursive descent parsers, where the datatypes are ...
As the field of Formal Methods advanced, this example appeared repeatedly in the research literature. In particular, it is viewed as a "challenge problem" for automated program verification. It is easier to reason about tail-recursive control flow, this is an equivalent ( extensionally equal ) definition:
Recursion that contains only a single self-reference is known as single recursion, while recursion that contains multiple self-references is known as multiple recursion. Standard examples of single recursion include list traversal, such as in a linear search, or computing the factorial function, while standard examples of multiple recursion ...
tarai is short for たらい回し (tarai mawashi, "to pass around") in Japanese.. John McCarthy named this function tak() after Takeuchi. [5]However, in certain later references, the y somehow got turned into the z.
A classic example of recursion is computing the factorial, which is defined recursively by 0! := 1 and n! := n × (n - 1)!. To recursively compute its result on a given input, a recursive function calls (a copy of) itself with a different ("smaller" in some way) input and uses the result of this call to construct its result.
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 ...
The importance of primitive recursive functions lies in the fact that most computable functions that are studied in number theory (and more generally in mathematics) are primitive recursive. For example, addition and division, the factorial and exponential function, and the function which returns the nth prime are all primitive recursive. [1]
Data types can also be defined by mutual recursion. The most important basic example of this is a tree, which can be defined mutually recursively in terms of a forest (a list of trees). Symbolically: f: [t[1], ..., t[k]] t: v f A forest f consists of a list of trees, while a tree t consists of a pair of a value v and a forest f (its children ...