|
Upper Semicomputable
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 definable set, 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 (mathematics), 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 computational complexity theory ... [...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 the whole itself) to . The subset , that is, the '' domain'' of viewed as a function, is called the domain of definition or natural domain of . If equals , that is, if is defined on every element in , then is said to be a total function. In other words, a partial function is a binary relation over two sets that associates to every element of the first set ''at most'' one element of the second set; it is thus a univalent relation. This generalizes the concept of a (total) function by not requiring ''every'' element of the first set to be associated to an element of the second set. A partial function is often used when its exact domain of definition is not known, or is difficult to specify. However, even when the exact domain of definition is known, partial functions are often used for simplicity or brevity. This is the case in calculus, where, for example, the quotien ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computable Function
Computable functions are the basic objects of study in computability theory. Informally, a function is ''computable'' if there is an algorithm that computes the value of the function for every value of its argument. Because of the lack of a precise definition of the concept of algorithm, every formal definition of computability must refer to a specific model of computation. Many such models of computation have been proposed, the major ones being Turing machines, register machines, lambda calculus and general recursive functions. Although these four are of a very different nature, they provide exactly the same class of computable functions, and, for every model of computation that has ever been proposed, the computable functions for such a model are computable for the above four models of computation. The Church–Turing thesis is the unprovable assertion that every notion of computability that can be imagined can compute only functions that are computable in the above sense. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |