It has been suggested that this article or section be merged into computable function. (Discuss) In mathematical logic and computer science, the recursive functions are a class of functions from natural numbers to natural numbers which are "computable" in some intuitive sense. In fact, in computability theory it is shown that the recursive functions are precisely the functions that can be computed by Turing machines. Recursive functions are related to primitive recursive functions, and their inductive definition (below) builds upon that of the primitive recursive functions. However, not every recursive function is a primitive recursive function — the most famous example of one which is not is the Ackermann function. Wikipedia does not have an article with this exact name. ...
In computability theory computable functions or Turing computable functions are the basic objects of study. ...
Mathematical logic is a discipline within mathematics, studying formal systems in relation to the way they encode intuitive concepts of proof and computation as part of the foundations of mathematics. ...
Computer science is the study of the theoretical foundations of information and computation and their implementation and application in computer systems. ...
Partial plot of a function f. ...
In mathematics, a natural number is either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). The former definition is generally used in number theory, while the latter is preferred in set theory and computer science. ...
In mathematics, a natural number is either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). The former definition is generally used in number theory, while the latter is preferred in set theory and computer science. ...
In computer science, computability theory is the branch of the theory of computation that studies which problems are computationally solvable using different models of computation. ...
An artistic representation of a Turing Machine . ...
In computability theory, primitive recursive functions are a class of functions which form an important building block on the way to a full formalization of computability. ...
In the theory of computation, the Ackermann function or Ackermann-Peter function is a simple example of a recursive function that is not primitively recursive. ...
Other equivalent function classes are the λ-recursive functions and the functions that can be computed by Markov algorithms. The lambda calculus is a formal system designed to investigate function definition, function application, and recursion. ...
A Markov algorithm is a string rewriting system that uses grammar-like rules to operate on strings of symbols. ...
The set of all recursive functions is known as R. The class of decision problems solvable by a Turing machine. ...
Definition Take as axioms the axioms of the primitive recursive functions, but extend the definitions so as to allow for partial functions. Add one further operator, the unbounded search operator, defined as follows: In computability theory, primitive recursive functions are a class of functions which form an important building block on the way to a full formalization of computability. ...
This article needs to be cleaned up to conform to a higher standard of quality. ...
- If f(x,z1,z2,...,zn) is a partial function on the natural numbers with n+1 arguments x, z1,...,zn, then the function μx f is the partial function with arguments z1,...,zn that returns the least x such that f(0,z1,z2,...,zn), f(1,z1,z2,...,zn), ..., f(x,z1,z2,...,zn) are all defined and f(x,z1,z2,...,zn) = 0, if such an x exists; if no such x exists, then μx f is not defined for the particular arguments z1,...,zn.
It is easy to see that this least search axiom (along with the simple primitive recursion axioms) implies the bounded search axiom of primitive recursion. Natural number can mean either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). Natural numbers have two main purposes: they can be used for counting (there are 3 apples on the table), or they can be used for ordering (this is...
The set of partial recursive functions is defined as the smallest set of partial functions of any arity from natural numbers to natural numbers which contains the zero, successor, and projection functions, and which is closed under composition, primitive recursion, and unbounded search. In mathematics and computer programming the arity of a function or an operator is the number of arguments or operands it takes (arity is sometimes referred to as valency, although that actually refers to another meaning of valency in mathematics). ...
The set of total recursive functions is the subset of partial recursive functions which are total. In mathematics and computer science, a partial function from the domain X to the codomain Y is a binary relation over X and Y which associates with every element in the set X at most one element in the set Y. If a partial function associates with every element in...
In the equivalence of models of computability the parallel is drawn between Turing machines which do not terminate for certain inputs and an undefined result for that input in the corresponding partial recursive function. The unbounded search operator is not definable by the rules of primitive recursion as those do not provide a mechanism for "infinite loops" (undefined values). In computability theory the Church-Turing thesis, Churchs thesis, Churchs conjecture or Turings thesis, named after Alonzo Church and Alan Turing, is a hypothesis about the nature of mechanical calculation devices, such as electronic computers. ...
An artistic representation of a Turing Machine . ...
It is interesting to note that if the application of the unbounded search operator in the definition above is limited strictly to regular functions (functions which are guaranteed to be total when the unbounded search operator is applied to them), the resulting set (historically called the general recursive functions) is the same as the set of recursive functions -- in other words, the requirement for partial functions can be partially obviated.
Examples In mathematics, the Fibonacci numbers form a sequence defined recursively by: In other words, one starts with 0 and 1, and then produces the next Fibonacci number by adding the two previous Fibonacci numbers. ...
In discrete mathematics, the McCarthy 91 function is a recursive function which returns 91 for all positive integer arguments n ≤ 101 and returns for n > 101. ...
See also A Sierpinski triangle —a confined recursion of triangles to form a geometric lattice. ...
A common method of simplification is to divide a problem into subproblems of the same type. ...
External links - Stanford Encyclopedia of Philosophy entry
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