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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 enumeration algorithm, algorithm that enumerates the members of ''S''. That means that its output is a list of all the members of ''S'': ''s''1, ''s''2, ''s''3, ... . If ''S'' is infinite, this algorithm will run forever, but each element of S will be returned after a finite amount of time. Note that these elements do not have to be listed in a particular way, say from smallest to largest. 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 can run forever, and no inf ... [...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] |
Recursive Enumeration Of All Halting Turing Machines
Recursion occurs when the definition of a concept or process depends on a simpler or previous version of itself. Recursion is used in a variety of disciplines ranging from linguistics to logic. The most common application of recursion is in mathematics and computer science, where a function being defined is applied within its own definition. While this apparently defines an infinite number of instances (function values), it is often done in such a way that no infinite loop or infinite chain of references can occur. A process that exhibits recursion is ''recursive''. Video feedback displays recursive images, as does an infinity mirror. Formal definitions In mathematics and computer science, a class of objects or methods exhibits recursive behavior when it can be defined by two properties: * A simple ''base case'' (or cases) — a terminating scenario that does not use recursion to produce an answer * A ''recursive step'' — a set of rules that reduces all successive cases ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complement (set Theory)
In set theory, the complement of a Set (mathematics), set , often denoted by A^c (or ), is the set of Element (mathematics), elements not in . When all elements in the Universe (set theory), universe, i.e. all elements under consideration, are considered to be Element (mathematics), members of a given set , the absolute complement of is the set of elements in that are not in . The relative complement of with respect to a set , also termed the set difference of and , written B \setminus A, is the set of elements in that are not in . Absolute complement Definition If is a set, then the absolute complement of (or simply the complement of ) is the set of elements not in (within a larger set that is implicitly defined). In other words, let be a set that contains all the elements under study; if there is no need to mention , either because it has been previously specified, or it is obvious and unique, then the absolute complement of is the relative complement of in : ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Preimage
In mathematics, for a function f: X \to Y, the image of an input value x is the single output value produced by f when passed x. The preimage of an output value y is the set of input values that produce y. More generally, evaluating f at each element of a given subset A of its domain X produces a set, called the "image of A under (or through) f". Similarly, the inverse image (or preimage) of a given subset B of the codomain Y is the set of all elements of X that map to a member of B. The image of the function f is the set of all output values it may produce, that is, the image of X. The preimage of f is the preimage of the codomain Y. Because it always equals X (the domain of f), it is rarely used. Image and inverse image may also be defined for general binary relations, not just functions. Definition The word "image" is used in three related ways. In these definitions, f : X \to Y is a function from the set X to the set Y. Image of an element If x is a member of X, ... [...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] |
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] |
Cantor Pairing Function
In mathematics, a pairing function is a process to uniquely encode two natural numbers into a single natural number. Any pairing function can be used in set theory to prove that integers and rational numbers have the same cardinality as natural numbers. Definition A pairing function is a bijection :\pi:\mathbb \times \mathbb \to \mathbb. Generalization More generally, a pairing function on a set A is a function that maps each pair of elements from A into an element of A, such that any two pairs of elements of A are associated with different elements of A, or a bijection from A^2 to A. Instead of abstracting from the domain, the arity of the pairing function can also be generalized: there exists an n-ary generalized Cantor pairing function on \mathbb. Cantor pairing function The Cantor pairing function is a primitive recursive pairing function :\pi:\mathbb \times \mathbb \to \mathbb defined by :\pi(k_1,k_2) := \frac(k_1 + k_2)(k_1 + k_2 + 1)+k_2 where k_1, k_2\in\. It ca ... [...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. Kurt Gödel developed the concept 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 that propert ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Productive 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] |
Simple Set
In computability theory, a subset of the natural numbers is called simple if it is computably enumerable (c.e.) and co-infinite (i.e. its complement is infinite), but every infinite subset of its complement is not c.e.. Simple sets are examples of c.e. sets that are not computable. Relation to Post's problem Simple sets were devised by Emil Leon Post in the search for a non-Turing-complete c.e. set. Whether such sets exist is known as Post's problem. Post had to prove two things in order to obtain his result: that the simple set ''A'' is not computable, and that the ''K'', the halting problem, does not Turing-reduce to ''A''. He succeeded in the first part (which is obvious by definition), but for the other part, he managed only to prove a many-one reduction In computability theory and computational complexity theory, a many-one reduction (also called mapping reduction) is a reduction that converts instances of one decision problem (whether an instance is in L_1) to anothe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Diophantine Set
In mathematics, a Diophantine equation is an equation of the form ''P''(''x''1, ..., ''x''''j'', ''y''1, ..., ''y''''k'') = 0 (usually abbreviated ''P''(', ') = 0) where ''P''(', ') is a polynomial with integer coefficients, where ''x''1, ..., ''x''''j'' indicate parameters and ''y''1, ..., ''y''''k'' indicate unknowns. A Diophantine set is a subset ''S'' of \mathbb^j, the set of all ''j''-tuples of natural numbers, so that for some Diophantine equation ''P''(', ') = 0, :\bar \in S \iff (\exists \bar \in \mathbb^)(P(\bar,\bar)=0) . That is, a parameter value is in the Diophantine set ''S'' if and only if the associated Diophantine equation is satisfiable under that parameter value. The use of natural numbers both in ''S'' and the existential quantification merely reflects the usual applications in computability theory and model theory. It does not matter whether natural numbers refer to the set of nonnegative integers or positive integers since the two definitions for Diophantine ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
