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Cylindrification
In computability theory a cylindrification is a construction that associates a cylindric numbering to each numbering. The concept was first introduced by Yuri L. Ershov in 1973. Definition Given a numbering \nu, the cylindrification c(\nu) is defined as :\mathrm(c(\nu)) := \{\langle n, k \rangle , n \in \mathrm{Domain}(\nu)\} :c(\nu)\langle n, k \rangle := \nu(n) where \langle n, k \rangle is the Cantor pairing function In mathematics, a pairing function is a process to uniquely encode two natural numbers into a single natural number. Any pairing function can be used in set theory to prove that integers and rational numbers have the same cardinality as natural n .... Note that the cylindrification operation increases the input arity by 1. Properties * Given two numberings \nu and \mu then \nu \le \mu \Leftrightarrow c(\nu) \le_1 c(\mu) * \nu \le_1 c(\nu) References * Yu. L. Ershov, "Theorie der Numerierungen I." Zeitschrift für mathematische Logik und Grundlagen der ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Numbering (computability Theory)
In computability theory a numbering is the assignment of natural numbers to a set of objects such as functions, rational numbers, graphs, or words in some formal language. A numbering can be used to transfer the idea of computability and related concepts, which are originally defined on the natural numbers using computable functions, to these different types of objects. Common examples of numberings include Gödel numberings in first-order logic, the description numbers that arise from universal Turing machines and admissible numberings of the set of partial computable functions. Definition and examples A numbering of a set S is a surjective partial function from \mathbb to ''S'' (Ershov 1999:477). The value of a numbering ''ν'' at a number ''i'' (if defined) is often written ''ν''''i'' instead of the usual \nu(i) \!. Examples of numberings include: * The set of all finite subsets of \mathbb has a numbering \gamma , defined so that \gamma(0) = \emptyset and so that, for eac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computability 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] |
Cylindric Numbering
In computability theory a cylindric numbering is a special kind of numbering first introduced by Yuri L. Ershov in 1973. If a numbering \nu is reducible to \mu then there exists a computable function f with \nu = \mu \circ f. Usually f is not injective, but if \mu is a cylindric numbering we can always find an injective f. Definition A numbering \nu is called cylindric if :\nu \equiv_1 c(\nu). That is if it is one-equivalent to its cylindrification A set S is called cylindric if its indicator function :1_S: \mathbb \to \{0,1\} is a cylindric numbering. Examples * Every Gödel numbering is cylindric Properties * Cylindric numberings are idempotent Idempotence (, ) is the property of certain operation (mathematics), operations in mathematics and computer science whereby they can be applied multiple times without changing the result beyond the initial application. The concept of idempotence ...: \nu \circ \nu = \nu References * Yu. L. Ershov, "Theorie der ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Yuri L
Yuri may refer to: People and fictional characters Given name *Yuri (Slavic name), the Slavic masculine form of the given name George, including a list of people with the given name Yuri, Yury, etc. *Yuri (Japanese name), also Yūri, feminine Japanese given names, including a list of people and fictional characters *Yu-ri (Korean name), Korean unisex given name, including a list of people and fictional characters Singers *Yuri (Japanese singer), vocalist of the band Move *Yuri (Korean singer), member of Girl Friends *Yuri (Mexican singer) *Kwon Yu-ri, member of Girls' Generation Footballers *Yuri (footballer, born 1982), full name Yuri de Souza Fonseca, Brazilian football forward *Yuri (footballer, born 1984), full name Yuri Adriano Santos, Brazilian footballer *Yuri (footballer, born 1986), full name Yuri Vera Cruz Erbas, Brazilian footballer *Yuri (footballer, born 1989), full name Yuri Naves Roberto, Brazilian football defensive midfielder * Yuri (footballer, born 1990), full ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cantor Pairing Function
In mathematics, a pairing function is a process to uniquely encode two natural numbers into a single natural number. Any pairing function can be used in set theory to prove that integers and rational numbers have the same cardinality as natural numbers. Definition A pairing function is a bijection :\pi:\mathbb \times \mathbb \to \mathbb. More generally, a pairing function on a set ''A'' is a function that maps each pair of elements from ''A'' into an element of ''A'', such that any two pairs of elements of ''A'' are associated with different elements of ''A,'' or a bijection from A^2 to ''A''. Hopcroft and Ullman pairing function Hopcroft and Ullman (1979) define the following pairing function: \langle i, j\rangle := \frac(i+j-2)(i+j-1) + i, where i, j\in\. This is the same as the Cantor pairing function below, shifted to exclude 0 (i.e., i=k_2+1, j=k_1+1, and \langle i, j\rangle - 1 = \pi(k_2,k_1)). Cantor pairing function The Cantor pairing function is a primitive recursi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
