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Semicomputable Function
In computability theory, a semicomputable function is a partial function f : \mathbb \rightarrow \mathbb that can be approximated either from above or from below by a computable function. More precisely a partial function f : \mathbb \rightarrow \mathbb is upper semicomputable, meaning it can be approximated from above, if there exists a computable function \phi(x,k) : \mathbb \times \mathbb \rightarrow \mathbb, where x is the desired parameter for f(x) and k is the level of approximation, such that: * \lim_ \phi(x,k) = f(x) * \forall k \in \mathbb : \phi(x,k+1) \leq \phi(x,k) Completely analogous a partial function f : \mathbb \rightarrow \mathbb is lower semicomputable if and only if -f(x) is upper semicomputable or equivalently if there exists a computable function \phi(x,k) such that: * \lim_ \phi(x,k) = f(x) * \forall k \in \mathbb : \phi(x,k+1) \geq \phi(x,k) If a partial function is both upper and lower semicomputable it is called computable. See a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Recursion 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. In these areas, computability theory overlaps with proof theory and effective descriptive set theory. Basic questions addressed by computability theory include: * What does it mean for a function on the natural numbers to be computable? * How can noncomputable functions be classified into a hierarchy based on their level of noncomputability? Although there is considerable overlap in terms of knowledge and methods, mathematical computability theorists study the theory of relative computability, reducibility notions, and degree structures; those in the computer science field focus on the theory of subrecursive hierarchies, formal methods, and formal languages. I ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partial Function
In mathematics, a partial function from a set to a set is a function from a subset of (possibly itself) to . The subset , that is, the domain of viewed as a function, is called the domain of definition of . If equals , that is, if is defined on every element in , then is said to be total. More technically, a partial function is a binary relation over two sets that associates every element of the first set to ''at most'' one element of the second set; it is thus a functional binary relation. It generalizes the concept of a (total) function by not requiring every element of the first set to be associated to ''exactly'' one element of the second set. A partial function is often used when its exact domain of definition is not known or difficult to specify. This is the case in calculus, where, for example, the quotient of two functions is a partial function whose domain of definition cannot contain the zeros of the denominator. For this reason, in calculus, and more gene ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computable Function
Computable functions are the basic objects of study in computability theory. Computable functions are the formalized analogue of the intuitive notion of algorithms, in the sense that a function is computable if there exists an algorithm that can do the job of the function, i.e. given an input of the function domain it can return the corresponding output. Computable functions are used to discuss computability without referring to any concrete model of computation such as Turing machines or register machines. Any definition, however, must make reference to some specific model of computation but all valid definitions yield the same class of functions. Particular models of computability that give rise to the set of computable functions are the Turing-computable functions and the general recursive functions. Before the precise definition of computable function, mathematicians often used the informal term ''effectively calculable''. This term has since come to be identified with the com ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |