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Kleene's Recursion Theorem
In computability theory, Kleene's recursion theorems are a pair of fundamental results about the application of computable functions to their own descriptions. The theorems were first proved by Stephen Kleene in 1938 and appear in his 1952 book ''Introduction to Metamathematics''. A related theorem, which constructs fixed points of a computable function, is known as Rogers's theorem and is due to Hartley Rogers, Jr. The recursion theorems can be applied to construct fixed points of certain operations on computable functions, to generate quines, and to construct functions defined via recursive definitions. Notation The statement of the theorems refers to an admissible numbering \varphi of the partial recursive functions, such that the function corresponding to index e is \varphi_e. If F and G are partial functions on the natural numbers, the notation F \simeq G indicates that, for each ''n'', either F(n) and G(n) are both defined and are equal, or else F(n) and G(n) are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kleene's Theorem
In theoretical computer science and formal language theory, a regular language (also called a rational language) is a formal language that can be defined by a regular expression, in the strict sense in theoretical computer science (as opposed to many modern regular expressions engines, which are Regular expression#Patterns for non-regular languages, augmented with features that allow recognition of non-regular languages). Alternatively, a regular language can be defined as a language recognized by a finite automaton. The equivalence of regular expressions and finite automata is known as Kleene's theorem (after American mathematician Stephen Cole Kleene). In the Chomsky hierarchy, regular languages are the languages generated by regular grammar, Type-3 grammars. Formal definition The collection of regular languages over an Alphabet (formal languages), alphabet Σ is defined recursively as follows: * The empty language Ø is a regular language. * For each ''a'' ∈ Σ (''a'' belong ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quine (computing)
A quine is a computer program which takes no input and produces a copy of its own source code as its only output. The standard terms for these programs in the computability theory and computer science literature are "self-replicating programs", "self-reproducing programs", and "self-copying programs". A quine is a fixed point of an execution environment, when the execution environment is viewed as a function transforming programs into their outputs. Quines are possible in any Turing-complete programming language, as a direct consequence of Kleene's recursion theorem. For amusement, programmers sometimes attempt to develop the shortest possible quine in any given programming language. The name "quine" was coined by Douglas Hofstadter, in his popular science book ''Gödel, Escher, Bach'', in honor of philosopher Willard Van Orman Quine (1908–2000), who made an extensive study of indirect self-reference, and in particular for the following paradox-producing expression, known as Q ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Denotational Semantics
In computer science, denotational semantics (initially known as mathematical semantics or Scott–Strachey semantics) is an approach of formalizing the meanings of programming languages by constructing mathematical objects (called ''denotations'') that describe the meanings of expressions from the languages. Other approaches providing formal semantics of programming languages include axiomatic semantics and operational semantics. Broadly speaking, denotational semantics is concerned with finding mathematical objects called domains that represent what programs do. For example, programs (or program phrases) might be represented by partial functionsDana S. ScottOutline of a mathematical theory of computation Technical Monograph PRG-2, Oxford University Computing Laboratory, Oxford, England, November 1970.Dana Scott and Christopher Strachey. ''Toward a mathematical semantics for computer languages'' Oxford Programming Research Group Technical Monograph. PRG-6. 1971. or by games ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Precomplete Numbering
In computability theory complete numberings are generalizations of Gödel numbering first introduced by A.I. Mal'tsev in 1963. They are studied because several important results like the Kleene's recursion theorem and Rice's theorem, which were originally proven for the Gödel-numbered set of computable functions, still hold for arbitrary sets with complete numberings. Definition A numbering \nu of a set A is called complete (with respect to an element a \in A) if for every partial computable function f there exists a total computable function h so that (Ershov 1999:482): : \nu \circ h(i) = \begin \nu \circ f(i) & \mbox ~ i \in \operatorname(f), \\ a & \mbox. \end Ershov refers to the element ''a'' as a "special" element for the numbering. A numbering \nu is called precomplete if the weaker property holds: : \nu \circ f(i) = \nu \circ h(i) \qquad i \in \operatorname(f). Examples * Any numbering of a singleton set is complete * The identity function on the natural nu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Yury Yershov
Yury Leonidovich Yershov (, born 1 May 194 is a Soviet and Russian mathematician. Yury Yershov was born in 1940 in Novosibirsk. In 1958 he entered the Tomsk State University and in 1963 graduated from the Mathematical Department of the Novosibirsk State University. In 1964 he successfully defended his PhD thesis "Decidable and Undecidable Theories" (advisor Anatoly Maltsev). In 1966 he successfully defended his DrSc thesis "Elementary Theory of Fields" (Элементарные теория полей). Apart from being a mathematician, Yershov was a member of the Communist Party and had different distinguished administrative duties in Novosibirsk State University. Yershov has been accused of antisemitic practices, and his visit to the U.S. in 1980 drew public protests by a number of U.S. mathematicians. Yershov himself denied the validity of these accusations. Yury Yershov is a member of the Russian Academy of Sciences, professor emeritus of Novosibirsk State University and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Order Theory
Order theory is a branch of mathematics that investigates the intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that". This article introduces the field and provides basic definitions. A list of order-theoretic terms can be found in the order theory glossary. Background and motivation Orders are everywhere in mathematics and related fields like computer science. The first order often discussed in primary school is the standard order on the natural numbers e.g. "2 is less than 3", "10 is greater than 5", or "Does Tom have fewer cookies than Sally?". This intuitive concept can be extended to orders on other sets of numbers, such as the integers and the reals. The idea of being greater than or less than another number is one of the basic intuitions of number systems (compare with numeral systems) in general (although one usually is also interested in the actual difference ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kleene Fixed-point Theorem
In the mathematical areas of order and lattice theory, the Kleene fixed-point theorem, named after American mathematician Stephen Cole Kleene, states the following: :Kleene Fixed-Point Theorem. Suppose (L, \sqsubseteq) is a directed-complete partial order (dcpo) with a least element, and let f: L \to L be a Scott-continuous (and therefore monotone) function. Then f has a least fixed point, which is the supremum of the ascending Kleene chain of f. The ascending Kleene chain of ''f'' is the chain :\bot \sqsubseteq f(\bot) \sqsubseteq f(f(\bot)) \sqsubseteq \cdots \sqsubseteq f^n(\bot) \sqsubseteq \cdots obtained by iterating ''f'' on the least element ⊥ of ''L''. Expressed in a formula, the theorem states that :\textrm(f) = \sup \left(\left\\right) where \textrm denotes the least fixed point. Although Tarski's fixed point theorem does not consider how fixed points can be computed by iterating ''f'' from some seed (also, it pertains to monotone functions on complete la ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Factorial
In mathematics, the factorial of a non-negative denoted is the product of all positive integers less than or equal The factorial also equals the product of n with the next smaller factorial: \begin n! &= n \times (n-1) \times (n-2) \times (n-3) \times \cdots \times 3 \times 2 \times 1 \\ &= n\times(n-1)!\\ \end For example, 5! = 5\times 4! = 5 \times 4 \times 3 \times 2 \times 1 = 120. The value of 0! is 1, according to the convention for an empty product. Factorials have been discovered in several ancient cultures, notably in Indian mathematics in the canonical works of Jain literature, and by Jewish mystics in the Talmudic book '' Sefer Yetzirah''. The factorial operation is encountered in many areas of mathematics, notably in combinatorics, where its most basic use counts the possible distinct sequences – the permutations – of n distinct objects: there In mathematical analysis, factorials are used in power series for the exponential function an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Enumeration Reducibility
In computability theory and computational complexity theory, enumeration reducibility is a method of reduction that determines if there is some effective procedure for determining enumerability between sets of natural numbers. An enumeration in the context of ''e''-reducibility is a listing of the elements in a particular set, or collection of items, though not necessarily ordered or complete. ''E''-reducibility is a form of positive reducibility, meaning that only positive information is processed. Positive information denotes the logic syntax for "and" (\land, \And) and "or" (\lor,\, ). The syntax for negation, "not" (\neg) is not included or used. According to Hartley Rogers Jr., an intuitive model that can be used to explain ''e''-reducibility is as follows:Let sets A and B be given. Consider a procedure that is determined by a finite set of instructions in the following way. A computation is begun. The computation proceeds algorithmically except that, from time to time, the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reflection (computer Programming)
In computer science, reflective programming or reflection is the ability of a process to examine, introspect, and modify its own structure and behavior. Historical background The earliest computers were programmed in their native assembly languages, which were inherently reflective, as these original architectures could be programmed by defining instructions as data and using self-modifying code. As the bulk of programming moved to higher-level compiled languages such as Algol, Cobol, Fortran, Pascal, and C, this reflective ability largely disappeared until new programming languages with reflection built into their type systems appeared. Brian Cantwell Smith's 1982 doctoral dissertation introduced the notion of computational reflection in procedural programming languages and the notion of the meta-circular interpreter as a component of 3-Lisp. Uses Reflection helps programmers make generic software libraries to display data, process different formats of data, perform serial ... [...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 cells, each of which can hold a single symbol drawn from a finite set of symbols called the 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 and which direction to move is based on a finite table that specifies what to do for each comb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
