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Myhill Isomorphism Theorem
In computability theory the Myhill isomorphism theorem, named after John Myhill, provides a characterization for two numberings to induce the same notion of computability on a set. Myhill isomorphism theorem Sets ''A'' and ''B'' of natural numbers are said to be recursively isomorphic if there is a total computable if there is a total computable bijective function ''f'' on the natural numbers such that for any x, x\in A\iff f(x)\in B.P. Odifreddi, Classical Recursion Theory: The theory of functions and sets of natural numbers (pp.324--325). Studies in Logic and the Foundations of Mathematics, vol. 125 (1989), Elsevier 0-444-87295-7. A set ''A'' of natural numbers is said to be many-one reduction">one-one reducible to a set ''B'' if there is a total computable injective function ''f'' on the natural numbers such that f(A) \subseteq B and f(\mathbb \setminus A) \subseteq \mathbb \setminus B. Myhill's isomorphism theorem states that two sets ''A'' and ''B'' of natural numbers ... [...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 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. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
John Myhill
John R. Myhill Sr. (11 August 1923 – 15 February 1987) was a British mathematician. Education Myhill received his Ph.D. from Harvard University under Willard Van Orman Quine in 1949. He was professor at SUNY Buffalo from 1966 until his death in 1987. He also taught at several other universities. His son, also called John Myhill, is a professor of linguistics in the English department of the University of Haifa in Israel. Contributions In the theory of formal languages, the Myhill–Nerode theorem, proven by Myhill with Anil Nerode, characterizes the regular languages as the languages that have only finitely many inequivalent prefixes. In computability theory, the Rice–Myhill–Shapiro theorem, more commonly known as Rice's theorem, states that, for any nontrivial property ''P'' of partial functions, it is undecidable to determine whether a given Turing machine computes a function with property ''P''. The Myhill isomorphism theorem is a computability-theoretic analogue ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Numbering (computability Theory)
In computability theory a numbering is the assignment of natural numbers to a set of objects such as functions, rational numbers, graphs, or words in some formal language. A numbering can be used to transfer the idea of computability and related concepts, which are originally defined on the natural numbers using computable functions, to these different types of objects. Common examples of numberings include Gödel numberings in first-order logic, the description numbers that arise from universal Turing machines and admissible numberings of the set of partial computable functions. Definition and examples A numbering of a set S is a surjective partial function from \mathbb to ''S'' (Ershov 1999:477). The value of a numbering ''ν'' at a number ''i'' (if defined) is often written ''ν''''i'' instead of the usual \nu(i) \!. Examples of numberings include: * The set of all finite subsets of \mathbb has a numbering \gamma , defined so that \gamma(0) = \emptyset and so that, for e ... [...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 numbers'', and numbers used for ordering are called ''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 jersey numbers). Some definitions, including the standard 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 numbers form a set. Many other number sets are built by succ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computable Isomorphism
In computability theory two sets A;B \subseteq \N of natural numbers are computably isomorphic or recursively isomorphic if there exists a total bijective computable function f \colon \N \to \N with f(A) = B. By the Myhill isomorphism theorem,Theorem 7.VI, Hartley Rogers, Jr., ''Theory of recursive functions and effective computability'' the relation of computable isomorphism coincides with the relation of mutual one-one reducibility. Two numbering There are many different numbering schemes for assigning nominal numbers to entities. These generally require an agreed set of rules, or a central coordinator. The schemes can be considered to be examples of a primary key of a database management ...s \nu and \mu are called computably isomorphic if there exists a computable bijection f so that \nu = \mu \circ f Computably isomorphic numberings induce the same notion of computability on a set. References *. Reduction (complexity) {{mathlogic-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Total Function
In mathematics, a partial function from a set to a set is a function from a subset of (possibly itself) to . The subset , that is, the domain of viewed as a function, is called the domain of definition of . If equals , that is, if is defined on every element in , then is said to be total. More technically, a partial function is a binary relation over two sets that associates every element of the first set to ''at most'' one element of the second set; it is thus a functional binary relation. It generalizes the concept of a (total) function by not requiring every element of the first set to be associated to ''exactly'' one element of the second set. A partial function is often used when its exact domain of definition is not known or difficult to specify. This is the case in calculus, where, for example, the quotient of two functions is a partial function whose domain of definition cannot contain the zeros of the denominator. For this reason, in calculus, and more gen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
