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The μ-recursive functions (or general recursive functions) are partial functions that take finite tuples of natural numbers and return a single natural number. They are the smallest class of partial functions that includes the initial functions and is closed under composition, primitive recursion, and the minimization operator μ .
A recursive step — a set of rules that reduces all successive cases toward the base case. For example, the following is a recursive definition of a person's ancestor. One's ancestor is either: One's parent (base case), or; One's parent's ancestor (recursive step). The Fibonacci sequence is another classic example of recursion: Fib(0) = 0 as ...
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 ...
Recursive function may refer to: Recursive function (programming), a function which references itself; General recursive function, a computable partial function from natural numbers to natural numbers Primitive recursive function, a function which can be computed with loops of bounded length; Another name for computable function
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 n th prime are all primitive recursive. [ 1 ]
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.
Recursion A recursive algorithm invokes itself repeatedly until meeting a termination condition, and is a common functional programming method. Iterative algorithms use repetitions such as loops or data structures like stacks to solve problems. Problems may be suited for one implementation or the other.
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.