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Turing Reduction
In computability theory, a Turing reduction from a decision problem A to a decision problem B is an oracle machine that decides problem A given an oracle for B (Rogers 1967, Soare 1987) in finitely many steps. It can be understood as an algorithm that could be used to solve A if it had access to a subroutine for solving B. The concept can be analogously applied to function problems. If a Turing reduction from A to B exists, then every algorithm for B can be used to produce an algorithm for A, by inserting the algorithm for B at each place where the oracle machine computing A queries the oracle for B. However, because the oracle machine may query the oracle a large number of times, the resulting algorithm may require more time asymptotically than either the algorithm for B or the oracle machine computing A. A Turing reduction in which the oracle machine runs in polynomial time is known as a Cook reduction. The first formal definition of relative computability, then called relative ... [...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] |
Equivalence Class
In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements a and b belong to the same equivalence class if, and only if, they are equivalent. Formally, given a set S and an equivalence relation \sim on S, the of an element a in S is denoted /math> or, equivalently, to emphasize its equivalence relation \sim, and is defined as the set of all elements in S with which a is \sim-related. The definition of equivalence relations implies that the equivalence classes form a partition of S, meaning, that every element of the set belongs to exactly one equivalence class. The set of the equivalence classes is sometimes called the quotient set or the quotient space of S by \sim, and is denoted by S /. When the set S has some structure (such as a group operation or a topology) and the equivalence re ... [...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] |
Well-founded
In mathematics, a binary relation is called well-founded (or wellfounded or foundational) on a set (mathematics), set or, more generally, a Class (set theory), class if every non-empty subset has a minimal element with respect to ; that is, there exists an such that, for every , one does not have . In other words, a relation is well-founded if: (\forall S \subseteq X)\; [S \neq \varnothing \implies (\exists m \in S) (\forall s \in S) \lnot(s \mathrel m)]. Some authors include an extra condition that is Set-like relation, set-like, i.e., that the elements less than any given element form a set. Equivalently, assuming the axiom of dependent choice, a relation is well-founded when it contains no infinite descending chains, meaning there is no infinite sequence of elements of such that for every natural number . In order theory, a partial order is called well-founded if the corresponding strict order is a well-founded relation. If the order is a total order then it is called ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Total Order
In mathematics, a total order or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some set X, which satisfies the following for all a, b and c in X: # a \leq a ( reflexive). # If a \leq b and b \leq c then a \leq c ( transitive). # If a \leq b and b \leq a then a = b ( antisymmetric). # a \leq b or b \leq a ( strongly connected, formerly called totality). Requirements 1. to 3. just make up the definition of a partial order. Reflexivity (1.) already follows from strong connectedness (4.), but is required explicitly by many authors nevertheless, to indicate the kinship to partial orders. Total orders are sometimes also called simple, connex, or full orders. A set equipped with a total order is a totally ordered set; the terms simply ordered set, linearly ordered set, toset and loset are also used. The term ''chain'' is sometimes defined as a synonym of ''totally ordered set'', but generally refers to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partial Order
In mathematics, especially order theory, a partial order on a set is an arrangement such that, for certain pairs of elements, one precedes the other. The word ''partial'' is used to indicate that not every pair of elements needs to be comparable; that is, there may be pairs for which neither element precedes the other. Partial orders thus generalize total orders, in which every pair is comparable. Formally, a partial order is a homogeneous binary relation that is reflexive, antisymmetric, and transitive. A partially ordered set (poset for short) is an ordered pair P=(X,\leq) consisting of a set X (called the ''ground set'' of P) and a partial order \leq on X. When the meaning is clear from context and there is no ambiguity about the partial order, the set X itself is sometimes called a poset. Partial order relations The term ''partial order'' usually refers to the reflexive partial order relations, referred to in this article as ''non-strict'' partial orders. However som ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Preorder
In mathematics, especially in order theory, a preorder or quasiorder is a binary relation that is reflexive relation, reflexive and Transitive relation, transitive. The name is meant to suggest that preorders are ''almost'' partial orders, but not quite, as they are not necessarily Antisymmetric relation, antisymmetric. A natural example of a preorder is the Divisor#Definition, divides relation "x divides y" between integers, polynomials, or elements of a commutative ring. For example, the divides relation is reflexive as every integer divides itself. But the divides relation is not antisymmetric, because 1 divides -1 and -1 divides 1. It is to this preorder that "greatest" and "lowest" refer in the phrases "greatest common divisor" and "lowest common multiple" (except that, for integers, the greatest common divisor is also the greatest for the natural order of the integers). Preorders are closely related to equivalence relations and (non-strict) partial orders. Both of th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computable Set
In computability theory, a set of natural numbers is computable (or decidable or recursive) if there is an algorithm that computes the membership of every natural number in a finite number of steps. A set is noncomputable (or undecidable) if it is not computable. Definition A subset S of the natural numbers is computable if there exists a total computable function f such that: :f(x)=1 if x\in S :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. Examples *Every recursive language is a computable. *Every finite or cofinite subset of the natural numbers is computable. **The empty set is computable. **The entire set of natural numbers is computable. **Every natural number is computable. *The subset of prime numbers is computable. *The set of Gödel numbers is computable. Non-examples *The set of Turing machines that halt is not computable. *The set of pairs of homeomorphic finite simplicial complexes ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Smn Theorem
In computability theory the ' theorem, written also as "smn-theorem" or "s-m-n theorem" (also called the translation lemma, parameter theorem, and the parameterization theorem) is a basic result about programming languages (and, more generally, Gödel numberings of the computable functions) (Soare 1987, Rogers 1967). It was first proved by Stephen Cole Kleene (1943). The name ' comes from the occurrence of an ''S'' with subscript ''n'' and superscript ''m'' in the original formulation of the theorem (see below). In practical terms, the theorem says that for a given programming language and positive integers ''m'' and ''n'', there exists a particular algorithm that accepts as input the source code of a program with free variables, together with ''m'' values. This algorithm generates source code that effectively substitutes the values for the first ''m'' free variables, leaving the rest of the variables free. The smn-theorem states that given a function of two arguments g(x,y) wh ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
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 another decision problem (whether an instance is in L_2) using a computable function. The reduced instance is in the language L_2 if and only if the initial instance is in its language L_1. Thus if we can decide whether L_2 instances are in the language L_2, we can decide whether L_1 instances are in the language L_1 by applying the reduction and solving for L_2. Thus, reductions can be used to measure the relative computational difficulty of two problems. It is said that L_1 reduces to L_2 if, in layman's terms L_2 is at least as hard to solve as L_1. This means that any algorithm that solves L_2 can also be used as part of a (otherwise relatively simple) program that solves L_1. Many-one reductions are a special case and stronger form of Turing reductions. With many-one ... [...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] |
