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Creative Set
In computability theory, productive sets and creative sets are types of sets of natural numbers that have important applications in mathematical logic. They are a standard topic in mathematical logic textbooks such as and . Definition and example For the remainder of this article, assume that \varphi_i is an admissible numbering of the computable functions and ''W''''i'' the corresponding numbering of the recursively enumerable sets. A set ''A'' of natural numbers is called productive if there exists a total recursive (computable) function f so that for all i \in \mathbb, if W_i \subseteq A then f(i) \in A \setminus W_i. The function f is called the productive function for A. A set ''A'' of natural numbers is called creative if ''A'' is recursively enumerable and its complement \mathbb\setminus A is productive. Not every productive set has a recursively enumerable complement, however, as illustrated below. The archetypal creative set is K = \, the set representing the halting ... [...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 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] |
First-order Arithmetic
In first-order logic, a first-order theory is given by a set of axioms in some language. This entry lists some of the more common examples used in model theory and some of their properties. Preliminaries For every natural mathematical structure there is a signature σ listing the constants, functions, and relations of the theory together with their arities, so that the object is naturally a σ-structure. Given a signature σ there is a unique first-order language ''L''σ that can be used to capture the first-order expressible facts about the σ-structure. There are two common ways to specify theories: #List or describe a set of sentences in the language ''L''σ, called the axioms of the theory. #Give a set of σ-structures, and define a theory to be the set of sentences in ''L''σ holding in all these models. For example, the "theory of finite fields" consists of all sentences in the language of fields that are true in all finite ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
NP-complete
In computational complexity theory, NP-complete problems are the hardest of the problems to which ''solutions'' can be verified ''quickly''. Somewhat more precisely, a problem is NP-complete when: # It is a decision problem, meaning that for any input to the problem, the output is either "yes" or "no". # When the answer is "yes", this can be demonstrated through the existence of a short (polynomial length) ''solution''. # The correctness of each solution can be verified quickly (namely, in polynomial time) and a brute-force search algorithm can find a solution by trying all possible solutions. # The problem can be used to simulate every other problem for which we can verify quickly that a solution is correct. Hence, if we could find solutions of some NP-complete problem quickly, we could quickly find the solutions of every other problem to which a given solution can be easily verified. The name "NP-complete" is short for "nondeterministic polynomial-time complete". In this name, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Berman–Hartmanis Conjecture
In structural complexity theory, the Berman–Hartmanis conjecture is an unsolved conjecture named after Leonard C. Berman and Juris Hartmanis.. Informally, it states that all NP-complete languages look alike, in the sense that they can be related to each other by polynomial time isomorphisms. Statements Statement using p-isomorphism An isomorphism between formal languages ''L''1 and ''L''2 is a bijective map ''f'' from strings in the alphabet of ''L''1 to strings in the alphabet of ''L''2, with the property that a string ''x'' belongs to ''L''1 if and only if ''f''(''x'') belongs to ''L''2. A polynomial-time isomorphism, or ''p''-isomorphism for short, is an isomorphism ''f'' where both ''f'' and its inverse function can be computed in an amount of time polynomial in the lengths of their arguments. Berman and Hartmanis conjectured that all NP-complete languages are p-isomorphic to each other. Statement using paddable languages A formal language ''L'' is paddable if there ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computational Complexity Theory
In theoretical computer science and mathematics, computational complexity theory focuses on classifying computational problems according to their resource usage, and explores the relationships between these classifications. A computational problem is a task solved by a computer. A computation problem is solvable by mechanical application of mathematical steps, such as an algorithm. A problem is regarded as inherently difficult if its solution requires significant resources, whatever the algorithm used. The theory formalizes this intuition, by introducing mathematical models of computation to study these problems and quantifying their computational complexity, i.e., the amount of resources needed to solve them, such as time and storage. Other measures of complexity are also used, such as the amount of communication (used in communication complexity), the number of logic gate, gates in a circuit (used in circuit complexity) and the number of processors (used in parallel computing). O ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polynomial Creativity
In computational complexity theory, polynomial creativity is a theory analogous to the theory of creative sets in recursion theory and mathematical logic. The are a family of formal languages in the complexity class NP whose complements certifiably do not have nondeterministic recognition algorithms. It is generally believed that NP is unequal to co-NP (the class of complements of languages in NP), which would imply more strongly that the complements of all NP-complete languages do not have polynomial-time nondeterministic recognition algorithms. However, for the sets, the lack of a (more restricted) recognition algorithm can be proven, whereas a proof that remains elusive. The sets are conjectured to form counterexamples to the Berman–Hartmanis conjecture on isomorphism of NP-complete sets. It is NP-complete to test whether an input string belongs to any one of these languages, but no polynomial time isomorphisms between all such languages and other NP-complete languages ar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lambda Calculus
In mathematical logic, the lambda calculus (also written as ''λ''-calculus) is a formal system for expressing computability, computation based on function Abstraction (computer science), abstraction and function application, application using variable Name binding, binding and Substitution (algebra), substitution. Untyped lambda calculus, the topic of this article, is a universal machine, a model of computation that can be used to simulate any Turing machine (and vice versa). It was introduced by the mathematician Alonzo Church in the 1930s as part of his research into the foundations of mathematics. In 1936, Church found a formulation which was #History, logically consistent, and documented it in 1940. Lambda calculus consists of constructing #Lambda terms, lambda terms and performing #Reduction, reduction operations on them. A term is defined as any valid lambda calculus expression. In the simplest form of lambda calculus, terms are built using only the following rules: # x: A ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Church's Thesis
Church & Co Limited, branded Church's, is a luxury footwear manufacturer that was founded in 1873 by Thomas Church in Northampton, England. In 1999 the company was bought by Italian luxury fashion house Prada. Family Three brothers Alfred, (John) William, and (Thomas) Dudley formed the company; their father Thomas died on 23 March 1905. The granddaughter of Dudley was the Olympic swimmer Elizabeth Church. (Thomas) Dudley married Rhoda Wooding, daughter of Henry Wooding of Billing Road, at Victoria Road church on 3 January 1893. Alfred died on Saturday 29 September 1928 aged 77. He lived on Cheyne Walk, and attended Abington Avenue Congregational Church, where his funeral was held. Alfred had attended the church with Walter Drawbridge Crick, grandfather of Francis Crick. (John) William lived at Nine Springs Villa on Billing Road in Cliftonville, the former house of Walter Drawbridge Crick, until around 1928, when he moved to Leicester, where he died aged 76 on Thursday ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Turing Machine
A Turing machine is a mathematical model of computation describing an abstract machine that manipulates symbols on a strip of tape according to a table of rules. Despite the model's simplicity, it is capable of implementing any computer algorithm. The machine operates on an infinite memory tape divided into discrete mathematics, discrete cells, each of which can hold a single symbol drawn from a finite set of symbols called the Alphabet (formal languages), alphabet of the machine. It has a "head" that, at any point in the machine's operation, is positioned over one of these cells, and a "state" selected from a finite set of states. At each step of its operation, the head reads the symbol in its cell. Then, based on the symbol and the machine's own present state, the machine writes a symbol into the same cell, and moves the head one step to the left or the right, or halts the computation. The choice of which replacement symbol to write, which direction to move the head, and whet ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alan Turing
Alan Mathison Turing (; 23 June 1912 – 7 June 1954) was an English mathematician, computer scientist, logician, cryptanalyst, philosopher and theoretical biologist. He was highly influential in the development of theoretical computer science, providing a formalisation of the concepts of algorithm and computation with the Turing machine, which can be considered a model of a general-purpose computer. Turing is widely considered to be the father of theoretical computer science. Born in London, Turing was raised in southern England. He graduated from University of Cambridge, King's College, Cambridge, and in 1938, earned a doctorate degree from Princeton University. During World War II, Turing worked for the Government Code and Cypher School at Bletchley Park, Britain's codebreaking centre that produced Ultra (cryptography), Ultra intelligence. He led Hut 8, the section responsible for German naval cryptanalysis. Turing devised techniques for speeding the breaking of Germ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Second Incompleteness Theorem
The second (symbol: s) is a unit of time derived from the division of the day first into 24 hours, then to 60 minutes, and finally to 60 seconds each (24 × 60 × 60 = 86400). The current and formal definition in the International System of Units (SI) is more precise: The second ..is defined by taking the fixed numerical value of the caesium frequency, Δ''ν''Cs, the unperturbed ground-state hyperfine transition frequency of the caesium 133 atom, to be when expressed in the unit Hz, which is equal to s−1. This current definition was adopted in 1967 when it became feasible to define the second based on fundamental properties of nature with caesium clocks. As the speed of Earth's rotation varies and is slowing ever so slightly, a leap second is added at irregular intervals to civil time to keep clocks in sync with Earth's rotation. The definition that is based on of a rotation of the earth is still used by the Universal Time 1 (UT1) system. Etymology "Minute" comes ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
