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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 ...
Recursion (computer science) Tree created using the Logo programming language and relying heavily on recursion. Each branch can be seen as a smaller version of a tree. Recursive drawing of a Sierpiński Triangle through turtle graphics. In computer science, recursion is a method of solving a computational problem where the solution depends on ...
Computability theory. 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.
Proof theory is a major branch [1] of mathematical logic and theoretical computer science within which proofs are treated as formal mathematical objects, facilitating their analysis by mathematical techniques. Proofs are typically presented as inductively-defined data structures such as lists, boxed lists, or trees, which are constructed ...
Church–Turing thesis. In computability theory, the Church–Turing thesis (also known as computability thesis, [1] the Turing–Church thesis, [2] the Church–Turing conjecture, Church's thesis, Church's conjecture, and Turing's thesis) is a thesis about the nature of computable functions. It states that a function on the natural numbers can ...
Kleene's recursion theorem. In computability theory, Kleene's recursion theorems are a pair of fundamental results about the application of computable functions to their own descriptions. The theorems were first proved by Stephen Kleene in 1938 [1] and appear in his 1952 book Introduction to Metamathematics. [2]
Structural recursion is a recursion method bearing the same relationship to structural induction as ordinary recursion bears to ordinary mathematical induction. Structural induction is used to prove that some proposition P ( x ) holds for all x of some sort of recursively defined structure, such as formulas , lists , or trees .
The Ackermann function grows faster than any primitive recursive function and therefore is not itself primitive recursive. The sketch of the proof is this: a primitive recursive function defined using up to k recursions must grow slower than + (), the (k+1)-th function in the fast-growing hierarchy, but the Ackermann function grows at least as ...