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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 ...
Computability theory, also known as recursion theory, is a branch of mathematical logic, computer science, and the theory of computation that originated in the 1930s with the study of computable functions and Turing degrees. The field has since expanded to include the study of generalized computability and definability.
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 ...
If the function is total, it is also called a total recursive function (sometimes shortened to recursive function). [1] In computability theory, it is shown that the μ-recursive functions are precisely the functions that can be computed by Turing machines [2] [4] (this is one of the theorems that supports the Church–Turing thesis).
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]
The primitive recursive functionals are the smallest collection of objects of finite type such that: The constant function f(n) = 0 is a primitive recursive functional; The successor function g(n) = n + 1 is a primitive recursive functional; For any type σ×τ, the functional K(x σ, y τ) = x is a primitive recursive functional
For any recursive operator Ψ there is a partial computable function φ such that Ψ(φ) = φ and φ is the smallest partial computable function with this property. The first recursion theorem is also called Fixed point theorem (of recursion theory). [10] There is also a definition which can be applied to recursive functionals as follows:
This choice is motivated by the fact that in generalized recursion theories, such as α-recursion theory, the definition corresponding to domains has been found to be more natural. Other texts use the definition in terms of enumerations, which is equivalent for computably enumerable sets.