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Recursively Inseparable Sets
In computability theory, two disjoint sets of natural numbers are called computably inseparable or recursively inseparable if they cannot be "separated" with a computable set.Monk 1976, p. 100 These sets arise in the study of computability theory itself, particularly in relation to Π classes. Computably inseparable sets also arise in the study of Gödel's incompleteness theorem. Definition The natural numbers are the set \mathbb = \. Given disjoint subsets ''A'' and ''B'' of \mathbb, a separating set ''C'' is a subset of \mathbb such that ''A'' ⊆ ''C'' and ''B'' ∩ ''C'' = ∅ (or equivalently, ''A'' ⊆ ''C'' and ''B'' ⊆ ). For example, ''A'' itself is a separating set for the pair, as is '. If a pair of disjoint sets ''A'' and ''B'' has no computable separating set, then the two sets are computably inseparable. Examples If ''A'' is a non-computable set, then ''A'' and its complement are computably inseparable. However, there are many examples of sets ''A'' 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] |
Computable 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. E ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pi01 Class
The number (; spelled out as "pi") is a mathematical constant that is the ratio of a circle's circumference to its diameter, approximately equal to 3.14159. The number appears in many formulas across mathematics and physics. It is an irrational number, meaning that it cannot be expressed exactly as a ratio of two integers, although fractions such as \tfrac are commonly used to approximate it. Consequently, its decimal representation never ends, nor enters a permanently repeating pattern. It is a transcendental number, meaning that it cannot be a solution of an equation involving only sums, products, powers, and integers. The transcendence of implies that it is impossible to solve the ancient challenge of squaring the circle with a compass and straightedge. The decimal digits of appear to be randomly distributed, but no proof of this conjecture has been found. For thousands of years, mathematicians have attempted to extend their understanding of , sometimes by computing ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computable 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. E ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computably 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] |
Partial Computable Function
Computable functions are the basic objects of study in computability theory. Computable functions are the formalized analogue of the intuitive notion of algorithms, in the sense that a function is computable if there exists an algorithm that can do the job of the function, i.e. given an input of the function domain it can return the corresponding output. Computable functions are used to discuss computability without referring to any concrete model of computation such as Turing machines or register machines. Any definition, however, must make reference to some specific model of computation but all valid definitions yield the same class of functions. Particular models of computability that give rise to the set of computable functions are the Turing-computable functions and the general recursive functions. Before the precise definition of computable function, mathematicians often used the informal term ''effectively calculable''. This term has since come to be identified with the com ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
William Gasarch
William Ian Gasarch ( ; born 1959) is an American computer scientist known for his work in computational complexity theory, computability theory, computational learning theory, and Ramsey theory. He is currently a professor at the University of Maryland Department of Computer Science with an affiliate appointment in Mathematics. As of 2015 he has supervised over 40 high school students on research projects, including Jacob Lurie. He has co-blogged on computational complexity with Lance Fortnow since 2007. He was book review editor for ACM SIGACT NEWS from 1997 to 2015. Education Gasarch received his doctorate in computer science from Harvard in 1985, advised by Harry R. Lewis. His thesis was titled ''Recursion-Theoretic Techniques in Complexity Theory and Combinatorics''. He was hired into a tenure track professorial job at the University of Maryland in the Fall of 1985. He was promoted to Associate Professor with Tenure in 1991, and to Full Professor in 1998. Work Gasarch ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gödel Numbering
In mathematical logic, a Gödel numbering is a function that assigns to each symbol and well-formed formula of some formal language a unique natural number, called its Gödel number. The concept was developed by Kurt Gödel for the proof of his incompleteness theorems. () A Gödel numbering can be interpreted as an encoding in which a number is assigned to each symbol of a mathematical notation, after which a sequence of natural numbers can then represent a sequence of symbols. These sequences of natural numbers can again be represented by single natural numbers, facilitating their manipulation in formal theories of arithmetic. Since the publishing of Gödel's paper in 1931, the term "Gödel numbering" or "Gödel code" has been used to refer to more general assignments of natural numbers to mathematical objects. Simplified overview Gödel noted that each statement within a system can be represented by a natural number (its ''Gödel number''). The significance of this was th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Peano Arithmetic
In mathematical logic, the Peano axioms, also known as the Dedekind–Peano axioms or the Peano postulates, are axioms for the natural numbers presented by the 19th century Italian mathematician Giuseppe Peano. These axioms have been used nearly unchanged in a number of metamathematical investigations, including research into fundamental questions of whether number theory is consistent and complete. The need to formalize arithmetic was not well appreciated until the work of Hermann Grassmann, who showed in the 1860s that many facts in arithmetic could be derived from more basic facts about the successor operation and induction. In 1881, Charles Sanders Peirce provided an axiomatization of natural-number arithmetic. In 1888, Richard Dedekind proposed another axiomatization of natural-number arithmetic, and in 1889, Peano published a simplified version of them as a collection of axioms in his book, ''The principles of arithmetic presented by a new method'' ( la, Arithmetice ... [...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] |
