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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] |
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] |
Vladimir Andreyevich Uspensky
Vladimir Andreyevich Uspensky (Russian: Влади́мир Андре́евич Успе́нский; 27 November 1930 – 27 June 2018) was a Russian mathematician, linguist, writer, doctor of physics and mathematics (1964). He was the author of numerous papers on mathematical logic and linguistics. In addition, he also penned a number of memoir essays. Uspensky initiated a reform of linguistic education in Russia. Biography Uspensky graduated in 1952 from the MSU Faculty of Mechanics and Mathematics (Lomonosov Moscow State University). He was a student of Andrey Kolmogorov. He was the head of the Chair of Mathematical Logic and Theory of Algorithms in the MSU Faculty of Mechanics and Mathematics (1995) and one of the founders of the Structural Linguistics branch (now the Theoretical and Applied Linguistics branch) in the MSU Faculty of Philology, where he also taught. He was the author of many books and of over 100 research papers. He prepared 25 candidate A candidate, or ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Yuri Ershov
Yury Leonidovich Yershov (, born 1 May 194 is a Soviet and Russian mathematician. Yury Yershov was born in 1940 in Novosibirsk. In 1958 he entered the Tomsk State University and in 1963 graduated from the Mathematical Department of the Novosibirsk State University. In 1964 he successfully defended his PhD thesis "Decidable and Undecidable Theories" (advisor Anatoly Maltsev). In 1966 he successfully defended his DrSc thesis "Elementary Theory of Fields" (Элементарные теория полей). Apart from being a mathematician, Yershov was a member of the Communist Party and had different distinguished administrative duties in Novosibirsk State University. Yershov has been accused of antisemitic practices, and his visit to the U.S. in 1980 drew public protests by a number of U.S. mathematicians. Yershov himself denied the validity of these accusations. Yury Yershov is a member of the Russian Academy of Sciences, professor emeritus of Novosibirsk State University an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Description Number
Description numbers are numbers that arise in the theory of Turing machines. They are very similar to Gödel numbers, and are also occasionally called "Gödel numbers" in the literature. Given some universal Turing machine, every Turing machine can, given its encoding on that machine, be assigned a number. This is the machine's description number. These numbers play a key role in Alan Turing's proof of the undecidability of the halting problem, and are very useful in reasoning about Turing machines as well. An example of a description number Say we had a Turing machine ''M'' with states q1, ... qR, with a tape alphabet with symbols s1, ... sm, with the blank denoted by s0, and transitions giving the current state, current symbol, and actions performed (which might be to overwrite the current tape symbol and move the tape head left or right, or maybe not move it at all), and the next state. Under the original universal machine described by Alan Turing, this machine would be enco ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Complete Numbering
In computability theory complete numberings are generalizations of Gödel numbering first introduced by A.I. Mal'tsev in 1963. They are studied because several important results like the Kleene's recursion theorem and Rice's theorem, which were originally proven for the Gödel-numbered set of computable functions, still hold for arbitrary sets with complete numberings. Definition A numbering \nu of a set A is called complete (with respect to an element a \in A) if for every partial computable function f there exists a total computable function h so that (Ershov 1999:482): : \nu \circ h(i) = \begin \nu \circ f(i) & \mbox ~ i \in \operatorname(f), \\ a & \mbox. \end Ershov refers to the element ''a'' as a "special" element for the numbering. A numbering \nu is called precomplete if the weaker property holds: : \nu \circ f(i) = \nu \circ h(i) \qquad i \in \operatorname(f). Examples * Any numbering of a singleton set is complete * The identity function on the natural n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Preorder
In mathematics, especially in order theory, a preorder or quasiorder is a binary relation that is reflexive and transitive. Preorders are more general than equivalence relations and (non-strict) partial orders, both of which are special cases of a preorder: an antisymmetric (or skeletal) preorder is a partial order, and a symmetric preorder is an equivalence relation. The name comes from the idea that preorders (that are not partial orders) are 'almost' (partial) orders, but not quite; they are neither necessarily antisymmetric nor asymmetric. Because a preorder is a binary relation, the symbol \,\leq\, can be used as the notational device for the relation. However, because they are not necessarily antisymmetric, some of the ordinary intuition associated to the symbol \,\leq\, may not apply. On the other hand, a preorder can be used, in a straightforward fashion, to define a partial order and an equivalence relation. Doing so, however, is not always useful or worth ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Friedberg Numbering
In computability theory, a Friedberg numbering is a numbering There are many different numbering schemes for assigning nominal numbers to entities. These generally require an agreed set of rules, or a central coordinator. The schemes can be considered to be examples of a primary key of a database management ... (enumeration) of the set of all uniformly recursively enumerable sets that has no repetitions: each recursively enumerable set appears exactly once in the enumeration (Vereščagin and Shen 2003:30). The existence of such numberings was established by Richard M. Friedberg in 1958 (Cutland 1980:78). References * Nigel Cutland (1980), ''Computability: An Introduction to Recursive Function Theory'', Cambridge University Press. . * Richard M. Friedberg (1958), ''Three Theorems on Recursive Enumeration. I. Decomposition. II. Maximal Set. III. Enumeration Without Duplication'', ''Journal of Symbolic Logic'' 23:3, pp. 309–316. * Nikolaj K. Vereščagin and A. Shen (2 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Recursively Enumerable
In computability theory, a set ''S'' of natural numbers is called computably enumerable (c.e.), recursively enumerable (r.e.), semidecidable, partially decidable, listable, provable or Turing-recognizable if: *There is an algorithm such that the set of input numbers for which the algorithm halts is exactly ''S''. Or, equivalently, *There is an algorithm that enumerates the members of ''S''. That means that its output is simply a list of all the members of ''S'': ''s''1, ''s''2, ''s''3, ... . If ''S'' is infinite, this algorithm will run forever. The first condition suggests why the term ''semidecidable'' is sometimes used. More precisely, if a number is in the set, one can ''decide'' this by running the algorithm, but if the number is not in the set, the algorithm runs forever, and no information is returned. A set that is "completely decidable" is a computable set. The second condition suggests why ''computably enumerable'' is used. The abbreviations c.e. and r.e. are oft ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Domain (function)
In mathematics, the domain of a function is the set of inputs accepted by the function. It is sometimes denoted by \operatorname(f) or \operatornamef, where is the function. More precisely, given a function f\colon X\to Y, the domain of is . Note that in modern mathematical language, the domain is part of the definition of a function rather than a property of it. In the special case that and are both subsets of \R, the function can be graphed in the Cartesian coordinate system. In this case, the domain is represented on the -axis of the graph, as the projection of the graph of the function onto the -axis. For a function f\colon X\to Y, the set is called the codomain, and the set of values attained by the function (which is a subset of ) is called its range or image. Any function can be restricted to a subset of its domain. The restriction of f \colon X \to Y to A, where A\subseteq X, is written as \left. f \_A \colon A \to Y. Natural domain If a real function is give ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Recursively Enumerable Set
In computability theory, a set ''S'' of natural numbers is called computably enumerable (c.e.), recursively enumerable (r.e.), semidecidable, partially decidable, listable, provable or Turing-recognizable if: *There is an algorithm such that the set of input numbers for which the algorithm halts is exactly ''S''. Or, equivalently, *There is an enumeration algorithm, algorithm that enumerates the members of ''S''. That means that its output is simply a list of all the members of ''S'': ''s''1, ''s''2, ''s''3, ... . If ''S'' is infinite, this algorithm will run forever. The first condition suggests why the term ''semidecidable'' is sometimes used. More precisely, if a number is in the set, one can ''decide'' this by running the algorithm, but if the number is not in the set, the algorithm runs forever, and no information is returned. A set that is "completely decidable" is a computable set. The second condition suggests why ''computably enumerable'' is used. The abbreviations ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
