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In computer science, anonymous recursion is recursion which does not explicitly call a function by name. This can be done either explicitly, by using a higher-order function – passing in a function as an argument and calling it – or implicitly, via reflection features which allow one to access certain functions depending on the current ...
where a represents the number of recursive calls at each level of recursion, b represents by what factor smaller the input is for the next level of recursion (i.e. the number of pieces you divide the problem into), and f(n) represents the work that the function does independently of any recursion (e.g. partitioning, recombining) at each level ...
Most modern implementations of a function call use a call stack, a special case of the stack data structure, to implement function calls and returns. Each procedure call creates a new entry, called a stack frame , at the top of the stack; when the procedure returns, its stack frame is deleted from the stack, and its space may be used for other ...
In computer science, corecursion is a type of operation that is dual 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.
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 ...
Common patterns of recursion can be abstracted away using higher-order functions, with catamorphisms and anamorphisms (or "folds" and "unfolds") being the most obvious examples. Such recursion schemes play a role analogous to built-in control structures such as loops in imperative languages.
Thus, it is provably total. One can use a diagonalization argument to show that f is not recursive primitive in itself: had it been such, so would be h(n) = f(n,n)+1. But if this equals some primitive recursive function, there is an m such that h(n) = f(m,n) for all n, and then h(m) = f(m,m), leading to contradiction.
This can also be done without mutual recursion, for example by still having separate functions for each production rule, but having them called by a single controller function, or by putting all the grammar in a single function. Mutual recursion can also implement a finite-state machine, with one function for each state, and single recursion in ...