|
Specker Sequence
In computability theory, a Specker sequence is a computable, monotonically increasing, bounded sequence of rational numbers whose supremum is not a computable real number. The first example of such a sequence was constructed by Ernst Specker (1949). The existence of Specker sequences has consequences for computable analysis. The fact that such sequences exist means that the collection of all computable real numbers does not satisfy the least upper bound principle of real analysis, even when considering only computable sequences. A common way to resolve this difficulty is to consider only sequences that are accompanied by a modulus of convergence; no Specker sequence has a computable modulus of convergence. More generally, a Specker sequence is called a ''recursive counterexample'' to the least upper bound principle, i.e. a construction that shows that this theorem is false when restricted to computable reals. The least upper bound principle has also been analyzed in the progr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Least Upper Bound Principle
In mathematics, the least-upper-bound property (sometimes called completeness or supremum property or l.u.b. property) is a fundamental property of the real numbers. More generally, a partially ordered set has the least-upper-bound property if every non-empty set (mathematics), subset of with an upper bound has a least upper bound, ''least'' upper bound (supremum) in . Not every (partially) ordered set has the least upper bound property. For example, the set \mathbb of all rational numbers with its natural order does ''not'' have the least upper bound property. The least-upper-bound property is one form of the Completeness of the real numbers, completeness axiom for the real numbers, and is sometimes referred to as Dedekind completeness.Willard says that an ordered space "X is Dedekind complete if every subset of X having an upper bound has a least upper bound." (pp. 124-5, Problem 17E.) It can be used to prove many of the fundamental results of real analysis, such as the intermed ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computation In The Limit
In computability theory, a function is called limit computable if it is the limit of a uniformly computable sequence of functions. The terms computable in the limit, limit recursive and recursively approximable are also used. One can think of limit computable functions as those admitting an eventually correct computable guessing procedure at their true value. A set is limit computable just when its characteristic function is limit computable. If the sequence is uniformly computable relative to ''D'', then the function is limit computable in ''D''. Formal definition A total function r(x) is limit computable if there is a total computable function \hat(x,s) such that : \displaystyle r(x) = \lim_ \hat(x,s) The total function r(x) is limit computable in ''D'' if there is a total function \hat(x,s) computable in ''D'' also satisfying : \displaystyle r(x) = \lim_ \hat(x,s) A set of natural numbers is defined to be computable in the limit if and only if its characteristic functi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Decidable Set
In computability theory, a set of natural numbers is called computable, recursive, or decidable if there is an algorithm which takes a number as input, terminates after a finite amount of time (possibly depending on the given number) and correctly decides whether the number belongs to the set or not. A set which is not computable is called noncomputable or undecidable. A more general class of sets than the computable ones consists of the computably enumerable (c.e.) sets, also called semidecidable sets. For these sets, it is only required that there is an algorithm that correctly decides when a number ''is'' in the set; the algorithm may give no answer (but not the wrong answer) for numbers not in the set. Formal definition A subset S of the natural numbers is called computable if there exists a total computable function f such that f(x)=1 if x\in S and f(x)=0 if x\notin S. In other words, the set S is computable if and only if the indicator function \mathbb_ is computable. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Natural Number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''Cardinal number, cardinal numbers'', and numbers used for ordering are called ''Ordinal number, ordinal numbers''. Natural numbers are sometimes used as labels, known as ''nominal numbers'', having none of the properties of numbers in a mathematical sense (e.g. sports Number (sports), jersey numbers). Some definitions, including the standard ISO/IEC 80000, ISO 80000-2, begin the natural numbers with , corresponding to the non-negative integers , whereas others start with , corresponding to the positive integers Texts that exclude zero from the natural numbers sometimes refer to the natural numbers together with zero as the whole numbers, while in other writings, that term is used instead for the integers (including negative integers). The natural ... [...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] |
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] |
Modulus Of Convergence
In real analysis, a branch of mathematics, a modulus of convergence is a function that tells how quickly a convergent sequence converges. These moduli are often employed in the study of computable analysis and constructive mathematics. If a sequence of real numbers x_i converges to a real number x, then by definition, for every real \varepsilon > 0 there is a natural number N such that if i > N then \left, x - x_i\ f(n) then \left, x - x_i\ g(n) then \left, x_i - x_j\ < 1/n. The latter definition is often employed in constructive settings, where the limit may actually be identified with the convergent sequence. Some authors use an alternate definition that replaces with . See also *Modulus of continuity In mathematical analysis, a modulus of ...
[...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] |
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] |
Ernst Specker
Ernst Paul Specker (11 February 1920, Zurich – 10 December 2011, Zurich) was a Swiss mathematician. Much of his most influential work was on Quine's New Foundations, a set theory with a universal set, but he is most famous for the Kochen–Specker theorem in quantum mechanics, showing that certain types of hidden variable theories are impossible. He also proved the ordinal partition relation ω2 → (ω2,3)2, thereby solving a problem of Erdős. Specker received his Ph.D. in 1949 from ETH Zurich, where he remained throughout his professional career. See also * Specker sequence * Baer-Specker group References External links Biography at the University of St. Andrews (Aien aristeuein) , motto_lang = grc , mottoeng = Ever to ExcelorEver to be the Best , established = , type = Public research university Ancient university , endowment ... Ernst Specker (1920-2011) Martin Fürer, Janu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computable Real Number
In mathematics, computable numbers are the real numbers that can be computed to within any desired precision by a finite, terminating algorithm. They are also known as the recursive numbers, effective numbers or the computable reals or recursive reals. Equivalent definitions can be given using μ-recursive functions, Turing machines, or λ-calculus as the formal representation of algorithms. The computable numbers form a real closed field and can be used in the place of real numbers for many, but not all, mathematical purposes. Informal definition using a Turing machine as example In the following, Marvin Minsky defines the numbers to be computed in a manner similar to those defined by Alan Turing in 1936; i.e., as "sequences of digits interpreted as decimal fractions" between 0 and 1: The key notions in the definition are (1) that some ''n'' is specified at the start, (2) for any ''n'' the computation only takes a finite number of steps, after which the machine produces the d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
