|
Weihrauch Reducibility
In computable analysis, Weihrauch reducibility is a notion of reducibility between multi-valued functions on represented spaces that roughly captures the uniform computational strength of computational problems. It was originally introduced by Klaus Weihrauch in an unpublished 1992 technical report. Definition A ''represented space'' is a pair (X,\delta) of a set X and a surjective partial function \delta:\subset \mathbb^\rightarrow X. Let (X,\delta_X),(Y,\delta_Y) be represented spaces and let f:\subset X \rightrightarrows Y be a partial multi-valued function. A ''realizer'' for f is a (possibly partial) function F:\subset \mathbb^\mathbb \to \mathbb^\mathbb such that, for every p \in \mathrm f \circ \delta_X, \delta_Y \circ F (p) = f\circ \delta_X(p). Intuitively, a realizer F for f behaves "just like f" but it works on names. If F is a realizer for f we write F \vdash f. Let X,Y,Z,W be represented spaces and let f:\subset X \rightrightarrows Y, g:\subset Z \rightrigh ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computable Analysis
In mathematics and computer science, computable analysis is the study of mathematical analysis from the perspective of computability theory. It is concerned with the parts of real analysis and functional analysis that can be carried out in a computable manner. The field is closely related to constructive analysis and numerical analysis. A notable result is that integration (in the sense of the Riemann integral) is computable. This might be considered surprising as an integral is (loosely speaking) an infinite sum. While this result could be explained by the fact that every computable function from \mathbb ,1/math> to \mathbb R is uniformly continuous, the notable thing is that the modulus of continuity can always be computed without being explicitly given. A similarly surprising fact is that differentiation of complex functions is also computable, while the same result is ''false'' for real functions. The above motivating results have no counterpart in Bishop's constructive an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multi-valued Function
In mathematics, a multivalued function, also called multifunction, many-valued function, set-valued function, is similar to a function, but may associate several values to each input. More precisely, a multivalued function from a domain to a codomain associates each in to one or more values in ; it is thus a serial binary relation. Some authors allow a multivalued function to have no value for some inputs (in this case a multivalued function is simply a binary relation). However, in some contexts such as in complex analysis (''X'' = ''Y'' = C), authors prefer to mimic function theory as they extend concepts of the ordinary (single-valued) functions. In this context, an ordinary function is often called a single-valued function to avoid confusion. The term ''multivalued function'' originated in complex analysis, from analytic continuation. It often occurs that one knows the value of a complex analytic function f(z) in some neighbourhood of a point z=a. This is the case ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computational Problems
In theoretical computer science, a computational problem is a problem that may be solved by an algorithm. For example, the problem of factoring :"Given a positive integer ''n'', find a nontrivial prime factor of ''n''." is a computational problem. A computational problem can be viewed as a set of ''instances'' or ''cases'' together with a, possibly empty, set of ''solutions'' for every instance/case. For example, in the factoring problem, the instances are the integers ''n'', and solutions are prime numbers ''p'' that are the nontrivial prime factors of ''n''. Computational problems are one of the main objects of study in theoretical computer science. The field of computational complexity theory attempts to determine the amount of resources (computational complexity) solving a given problem will require and explain why some problems are intractable or undecidable. Computational problems belong to complexity classes that define broadly the resources (e.g. time, space/memory, en ... [...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
Computability is the ability to solve a problem in an effective manner. It is a key topic of the field of computability theory within mathematical logic and the theory of computation within computer science. The computability of a problem is closely linked to the existence of an algorithm to solve the problem. The most widely studied models of computability are the Turing-computable and μ-recursive functions, and the lambda calculus, all of which have computationally equivalent power. Other forms of computability are studied as well: computability notions weaker than Turing machines are studied in automata theory, while computability notions stronger than Turing machines are studied in the field of hypercomputation. Problems A central idea in computability is that of a (computational) problem, which is a task whose computability can be explored. There are two key types of problems: * A decision problem fixes a set ''S'', which may be a set of strings, natural numbers, or oth ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Baire Space (set Theory)
In set theory, the Baire space is the set of all infinite sequences of natural numbers with a certain topology. This space is commonly used in descriptive set theory, to the extent that its elements are often called "reals". It is denoted NN, ωω, by the symbol \mathcal or also ωω, not to be confused with the countable ordinal obtained by ordinal exponentiation. The Baire space is defined to be the Cartesian product of countably infinitely many copies of the set of natural numbers, and is given the product topology (where each copy of the set of natural numbers is given the discrete topology). The Baire space is often represented using the tree of finite sequences of natural numbers. The Baire space can be contrasted with Cantor space, the set of infinite sequences of binary digits. Topology and trees The product topology used to define the Baire space can be described more concretely in terms of trees. The basic open sets of the product topology are cylinder sets, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wadge Reducibility
In descriptive set theory, within mathematics, Wadge degrees are levels of complexity for sets of reals. Sets are compared by continuous reductions. The Wadge hierarchy is the structure of Wadge degrees. These concepts are named after William W. Wadge. Wadge degrees Suppose A and B are subsets of Baire space ωω. Then A is Wadge reducible to B or A ≤W B if there is a continuous function f on ωω with A = f^ /math>. The Wadge order is the preorder or quasiorder on the subsets of Baire space. Equivalence classes of sets under this preorder are called Wadge degrees, the degree of a set A is denoted by A.html" ;"title="math>A">math>Asub>W. The set of Wadge degrees ordered by the Wadge order is called the Wadge hierarchy. Properties of Wadge degrees include their consistency with measures of complexity stated in terms of definability. For example, if A ≤W B and B is a countable intersection of open sets, then so is A. The same works for all levels of the Borel hierarchy a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
