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Low (computability)
In computability theory, a Turing degree 'X''is low if the Turing jump 'X''′is 0′. A set is low if it has low degree. Since every set is computable from its jump, any low set is computable in 0′, but the jump of sets computable in 0′ can bound any degree r.e. in 0′ (Schoenfield Jump Inversion). ''X'' being low says that its jump ''X''′ has the least possible degree in terms of Turing reducibility for the jump of a set. There are various related properties to low degrees: * A degree is ''lown'' if its n'th jump is the n'th jump of 0.C. J. Ash, J. Knight, ''Computable Structures and the Hyperarithmetical Hierarchy'' (Studies in Logic and the Foundation of Mathematics, 2000), p. 22 * A set ''X'' is ''generalized low'' if it satisfies ''X''′ ≡T ''X'' + 0′, that is: if its jump has the lowest degree possible. * A degree d is ''generalized low n'' if its n'th jump is the (n-1)'st jump of the join of d with 0′. More generally, prop ... [...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] |
Turing Degree
In computer science and mathematical logic the Turing degree (named after Alan Turing) or degree of unsolvability of a set of natural numbers measures the level of algorithmic unsolvability of the set. Overview The concept of Turing degree is fundamental in computability theory, where sets of natural numbers are often regarded as decision problems. The Turing degree of a set is a measure of how difficult it is to solve the decision problem associated with the set, that is, to determine whether an arbitrary number is in the given set. Two sets are Turing equivalent if they have the same level of unsolvability; each Turing degree is a collection of Turing equivalent sets, so that two sets are in different Turing degrees exactly when they are not Turing equivalent. Furthermore, the Turing degrees are partially ordered, so that if the Turing degree of a set ''X'' is less than the Turing degree of a set ''Y'', then any (noncomputable) procedure that correctly decides whether numbers a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Turing Jump
In computability theory, the Turing jump or Turing jump operator, named for Alan Turing, is an operation that assigns to each decision problem a successively harder decision problem with the property that is not decidable by an oracle machine with an oracle for . The operator is called a ''jump operator'' because it increases the Turing degree of the problem . That is, the problem is not Turing-reducible to . Post's theorem establishes a relationship between the Turing jump operator and the arithmetical hierarchy of sets of natural numbers.. Informally, given a problem, the Turing jump returns the set of Turing machines that halt when given access to an oracle that solves that problem. Definition The Turing jump of ''X'' can be thought of as an oracle to the halting problem for oracle machines with an oracle for ''X''. Formally, given a set and a Gödel numbering of the -computable functions, the Turing jump of is defined as : X'= \. The th Turing jump is defined i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Turing Reducibility
In computability theory, a Turing reduction from a decision problem A to a decision problem B is an oracle machine which decides problem A given an oracle for B (Rogers 1967, Soare 1987). It can be understood as an algorithm that could be used to solve A if it had available to it a subroutine for solving ''B''. The concept can be analogously applied to function problems. If a Turing reduction from A to B exists, then every algorithm for B can be used to produce an algorithm for A, by inserting the algorithm for B at each place where the oracle machine computing A queries the oracle for B. However, because the oracle machine may query the oracle a large number of times, the resulting algorithm may require more time asymptotically than either the algorithm for B or the oracle machine computing A. A Turing reduction in which the oracle machine runs in polynomial time is known as a Cook reduction. The first formal definition of relative computability, then called relative reducibility, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Low Basis Theorem
The low basis theorem is one of several basis theorems in computability theory, each of which showing that, given an infinite subtree of the binary tree 2^, it is possible to find an infinite path through the tree with particular computability properties. The low basis theorem, in particular, shows that there must be a path which is low; that is, the Turing jump of the path is Turing equivalent to the halting problem \emptyset'. Statement and proof The low basis theorem states that every nonempty \Pi^0_1 class in 2^\omega (see arithmetical hierarchy In mathematical logic, the arithmetical hierarchy, arithmetic hierarchy or Kleene–Mostowski hierarchy (after mathematicians Stephen Cole Kleene and Andrzej Mostowski) classifies certain sets based on the complexity of formulas that define th ...) contains a set of low degree (Soare 1987:109). This is equivalent, by definition, to the statement that each infinite computable subtree of the binary tree 2^ has an infinite path ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
PA Degree
In the mathematical field of computability theory, a PA degree is a Turing degree that computes a complete extension of Peano arithmetic (Jockusch 1987). These degrees are closely related to fixed-point-free (DNR) functions, and have been thoroughly investigated in recursion theory. Background In recursion theory, \phi_e denotes the computable function with index (program) ''e'' in some standard numbering of computable functions, and \phi^B_e denotes the ''e''th computable function using a set ''B'' of natural numbers as an oracle. A set ''A'' of natural numbers is Turing reducible to a set ''B'' if there is a computable function that, given an oracle for set ''B'', computes the characteristic function χA of the set ''A''. That is, there is an ''e'' such that \chi_A = \phi^B_e. This relationship is denoted ''A'' ≤T ''B''; the relation ≤T is a preorder. Two sets of natural numbers are Turing equivalent if each is Turing reducible to the other. The notation ' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reverse Mathematics
Reverse mathematics is a program in mathematical logic that seeks to determine which axioms are required to prove theorems of mathematics. Its defining method can briefly be described as "going backwards from the theorems to the axioms", in contrast to the ordinary mathematical practice of deriving theorems from axioms. It can be conceptualized as sculpting out necessary conditions from sufficient ones. The reverse mathematics program was foreshadowed by results in set theory such as the classical theorem that the axiom of choice and Zorn's lemma are equivalent over ZF set theory. The goal of reverse mathematics, however, is to study possible axioms of ordinary theorems of mathematics rather than possible axioms for set theory. Reverse mathematics is usually carried out using subsystems of second-order arithmetic,Simpson, Stephen G. (2009), Subsystems of second-order arithmetic, Perspectives in Logic (2nd ed.), Cambridge University Press, doi:10.1017/CBO9780511581007, ISBN 978 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ramsey's Theorem
In combinatorics, Ramsey's theorem, in one of its graph-theoretic forms, states that one will find monochromatic cliques in any edge labelling (with colours) of a sufficiently large complete graph. To demonstrate the theorem for two colours (say, blue and red), let and be any two positive integers. Ramsey's theorem states that there exists a least positive integer for which every blue-red edge colouring of the complete graph on vertices contains a blue clique on vertices or a red clique on vertices. (Here signifies an integer that depends on both and .) Ramsey's theorem is a foundational result in combinatorics. The first version of this result was proved by F. P. Ramsey. This initiated the combinatorial theory now called Ramsey theory, that seeks regularity amid disorder: general conditions for the existence of substructures with regular properties. In this application it is a question of the existence of ''monochromatic subsets'', that is, subsets of connected edges of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
High (computability)
In computability theory, a Turing degree [''X''] is high if it is computable in 0′, and the Turing jump [''X''′] is 0′′, which is the greatest possible degree in terms of Turing reducibility for the jump of a set which is computable in 0′. Similarly, a degree is ''high n'' if its n'th jump is the (n+1)'st jump of 0. Even more generally, a degree d is ''generalized high n'' if its n'th jump is the n'th jump of the join of d with 0′. See also * Low (computability) References Computability theory {{mathlogic-stub, date=February 2009 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Low Basis Theorem
The low basis theorem is one of several basis theorems in computability theory, each of which showing that, given an infinite subtree of the binary tree 2^, it is possible to find an infinite path through the tree with particular computability properties. The low basis theorem, in particular, shows that there must be a path which is low; that is, the Turing jump of the path is Turing equivalent to the halting problem \emptyset'. Statement and proof The low basis theorem states that every nonempty \Pi^0_1 class in 2^\omega (see arithmetical hierarchy In mathematical logic, the arithmetical hierarchy, arithmetic hierarchy or Kleene–Mostowski hierarchy (after mathematicians Stephen Cole Kleene and Andrzej Mostowski) classifies certain sets based on the complexity of formulas that define th ...) contains a set of low degree (Soare 1987:109). This is equivalent, by definition, to the statement that each infinite computable subtree of the binary tree 2^ has an infinite path ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Springer-Verlag
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in Berlin, it expanded internationally in the 1960s, and through mergers in the 1990s and a sale to venture capitalists it fused with Wolters Kluwer and eventually became part of Springer Nature in 2015. Springer has major offices in Berlin, Heidelberg, Dordrecht, and New York City. History Julius Springer founded Springer-Verlag in Berlin in 1842 and his son Ferdinand Springer grew it from a small firm of 4 employees into Germany's then second largest academic publisher with 65 staff in 1872.Chronology ". Springer Science+Business Media. In 1964, Springer expanded its business internationally, o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
