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If the target of a tail is the same subroutine, the subroutine is said to be tail recursive, which is a special case of direct recursion. Tail recursion (or tail-end recursion) is particularly useful, and is often easy to optimize in implementations. Tail calls can be implemented without adding a new stack frame to the call stack.
A common algorithm design tactic is to divide a problem into sub-problems of the same type as the original, solve those sub-problems, and combine the results. This is often referred to as the divide-and-conquer method; when combined with a lookup table that stores the results of previously solved sub-problems (to avoid solving them repeatedly and incurring extra computation time), it can be ...
The 91 function was chosen for being nested-recursive (contrasted with single recursion, such as defining () by means of ()). The example was popularized by Manna's book, Mathematical Theory of Computation (1974). As the field of Formal Methods advanced, this example appeared repeatedly in the research literature.
Mathematically, a set of mutually recursive functions are primitive recursive, which can be proven by course-of-values recursion, building a single function F that lists the values of the individual recursive function in order: = (), (), (), (), …, and rewriting the mutual recursion as a primitive recursion.
A classic example of recursion is the definition of the factorial function, given here in Python code: def factorial ( n ): if n > 0 : return n * factorial ( n - 1 ) else : return 1 The function calls itself recursively on a smaller version of the input (n - 1) and multiplies the result of the recursive call by n , until reaching the base case ...
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 ...
int foo; //foo might be defined somewhere in this file extern int bar; //bar must be defined in some other file In Pascal and other Wirth programming languages, it is a general rule that all entities must be declared before use, and thus forward declaration is necessary for mutual recursion, for instance.
This mutually recursive definition can be converted to a singly recursive definition by inlining the definition of a forest: t: v [t[1], ..., t[k]] A tree t consists of a pair of a value v and a list of trees (its children). This definition is more compact, but somewhat messier: a tree consists of a pair of one type and a list another, which ...