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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 o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Basis Theorem (computability)
In computability theory, there are a number of basis theorems. These theorems show that particular kinds of sets always must have some members that are, in terms of Turing degree, not too complicated. One family of basis theorems concern nonempty effectively closed sets (that is, nonempty \Pi^0_1 sets in the arithmetical hierarchy); these theorems are studied as part of classical computability theory. Another family of basis theorems concern nonempty lightface analytic sets (that is, \Sigma^1_1 in the analytical hierarchy); these theorems are studied as part of hyperarithmetical theory. Effectively closed sets Effectively closed sets are a topic of study in classical computability theory. An effectively closed set is the set of all paths through some computable subtree of the binary tree 2^. These sets are closed, in the topological sense, as subsets of the Cantor space 2^\omega, and the complement of an effective closed set is an effective open set in the sense of effective Poli ... [...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 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] |
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 recursively enumerable 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 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 can be thought of as an oracle to the halting problem for oracle machines with an oracle for . 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 inductive ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Halting Problem
In computability theory (computer science), computability theory, the halting problem is the problem of determining, from a description of an arbitrary computer program and an input, whether the program will finish running, or continue to run forever. The halting problem is ''Undecidable problem, undecidable'', meaning that no general algorithm exists that solves the halting problem for all possible program–input pairs. The problem comes up often in discussions of computability since it demonstrates that some functions are mathematically Definable set, definable but not Computable function, computable. A key part of the formal statement of the problem is a mathematical definition of a computer and program, usually via a Turing machine. The proof then shows, for any program that might determine whether programs halt, that a "pathological" program exists for which makes an incorrect determination. Specifically, is the program that, when called with some input, passes its own s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Î 01 Class
Pi () is a mathematical constant equal to a circle's circumference divided by its diameter. Pi, π or Πmay also refer to: Language and typography * Pi (letter), the 17th letter of the Greek alphabet * Pi characters, uncommon characters in typesetting * Pi font, a term for some kinds of dingbat fonts * Pali, an Indo-Aryan language (ISO 639-1 code: pi) Places * Pi (state), an ancient state during China's Zhou dynasty * Pi, Catalonia, a village of Bellver de Cerdanya, Spain * Pi County, Sichuan province, China * Pi Islands, Palmer Archipelago, Antarctica People Given name * Pi de Bruijn (born 1942), Dutch architect * Pi O or П. O. (born 1951), Greek-Australian anarchist poet * Pi Schwert (1892–1941), American baseball player and politician * Pi Vèriss, Dutch songwriter and composer Stage name * Pi (entertainer), American drag performer Surname * Bi (surname) (), romanized as Pi in Wade–Giles * Pi (surname) (), a Chinese surname * Enric Pi (born 1983), Catal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 them. Any set that receives a classification is called arithmetical. The arithmetical hierarchy was invented independently by Kleene (1943) and Mostowski (1946).P. G. Hinman, ''Recursion-Theoretic Hierarchies'' (p.89), Perspectives in Logic, 1978. Springer-Verlag Berlin Heidelberg, ISBN 3-540-07904-1. The arithmetical hierarchy is important in computability theory, effective descriptive set theory, and the study of formal theories such as Peano arithmetic. The Tarski–Kuratowski algorithm provides an easy way to get an upper bound on the classifications assigned to a formula and the set it defines. The hyperarithmetical hierarchy and the analytical hierarchy extend the arithmetical hierarchy to classify additional formulas and set ... [...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] |
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, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
