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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 ...
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 ...
Most recursive definitions have two foundations: a base case (basis) and an inductive clause. The difference between a circular definition and a recursive definition is that a recursive definition must always have base cases, cases that satisfy the definition without being defined in terms of the definition itself, and that all other instances in the inductive clauses must be "smaller" in some ...
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 total recursive function is a partial recursive function that is defined for every input. Every primitive recursive function is total recursive, but not all total recursive functions are primitive recursive. The Ackermann function A(m,n) is a well-known example of a total recursive function (in fact, provable total), that is not primitive ...
A constant-recursive sequence is any sequence of integers, rational numbers, algebraic numbers, real numbers, or complex numbers,,,, … (written as () = as a shorthand) satisfying a formula of the form
Structural recursion is usually proved correct by structural induction; in particularly easy cases, the inductive step is often left out. The length and ++ functions in the example below are structurally recursive. For example, if the structures are lists, one usually introduces the partial order "<", in which L < M whenever list L is the tail ...
A recursive operator is an enumeration operator that, when given the graph of a partial recursive function, always returns the graph of a partial recursive function. A fixed point of an enumeration operator Φ is a set F such that Φ(F) = F. The first enumeration theorem shows that fixed points can be effectively obtained if the enumeration ...