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Encyclopedia > Turing completeness

In computability theory, an abstract machine or programming language is called Turing complete, Turing equivalent, or (computationally) universal if it has a computational power equivalent to (i.e., capable of emulating) a simplified model of a programmable computer known as the universal Turing machine. Being equivalent to the universal Turing machine essentially means being able to perform any computational task—though it does not mean being able to perform such tasks efficiently, quickly, or easily. In computability theory, a Turing reduction from a problem A to a problem B is, intuitively, a reduction which easily solves A, assuming B is easy to solve. ... In computer science, computability theory is the branch of the theory of computation that studies which problems are computationally solvable using different models of computation. ... An abstract machine, also called an abstract computer, is a theoretical model of a computer hardware or software system used in Automata theory. ... A programming language is an artificial language that can be used to control the behavior of a machine, particularly a computer. ... This article is about emulators in computer science. ... This article is about the machine. ... The Turing machine is an abstract machine introduced in 1936 by Alan Turing to give a mathematically precise definition of algorithm or mechanical procedure. As such it is still widely used in theoretical computer science, especially in complexity theory and the theory of computation. ...


The term derives from the name of mathematician Alan Turing who introduced the model of the universal Turing machine. Alan Mathison Turing, OBE, FRS (23 June 1912 – 7 June 1954) was an English mathematician, logician, and cryptographer. ...

Contents

Overview

Turing completeness is significant in that every plausible design for a computing device so far advanced can be emulated by a universal Turing machine - an observation (it is not and cannot be mathematically proven) that has become known as the Church-Turing thesis. Thus, a machine that can act as a universal Turing machine can, in principle, perform any calculation that any other computer is capable of (that is, if it is programmable). Note, however, that this says nothing about the effort required to write a program for the machine, the time it may take for the machine to perform the calculation, or any abilities unrelated to computation (such as communication or randomness) which the machine may possess. The Turing machine is an abstract machine introduced in 1936 by Alan Turing to give a mathematically precise definition of algorithm or mechanical procedure. As such it is still widely used in theoretical computer science, especially in complexity theory and the theory of computation. ... In computability theory the Church-Turing thesis, Churchs thesis, Churchs conjecture or Turings thesis, named after Alonzo Church and Alan Turing, is a hypothesis about the nature of mechanical calculation devices, such as electronic computers. ... A computer program is a collection of instructions that describe a task, or set of tasks, to be carried out by a computer. ...


While truly Turing-complete machines are very likely physically impossible as they require unlimited storage, Turing completeness is often loosely attributed to physical machines or programming languages that would be universal if they had indefinitely enlargeable storage. All modern computers are Turing-complete in this sense.


Charles Babbage's analytical engine (1830s) would have been the first Turing-complete machine if it had ever been built, but the first actual implementation appeared in 1941: the program-controlled Z3 of Konrad Zuse. The universality of the Z3 was shown by Raúl Rojas in 1998. Prior to Rojas' 1998 paper, the first machine known to be Turing-complete was ENIAC (1946). Babbage redirects here. ... The analytical engine, an important step in the history of computers, was the design of a mechanical general-purpose computer by the British professor of mathematics Charles Babbage. ... For other uses, see 1941 (disambiguation). ... Konrad Zuses Z3 was the first working programmable, fully automatic machine, whose attributes, with the addition of conditional branching, have often been the ones used as criteria in defining a computer. ... Statue in Bad Hersfeld Konrad Zuse (June 22, 1910 Berlin - December 18, 1995 Hünfeld) was a German engineer and computer pioneer. ... Year 1998 (MCMXCVIII) was a common year starting on Thursday (link will display full 1998 Gregorian calendar). ... ENIAC ENIAC, short for Electronic Numerical Integrator And Computer,[1] was the first large-scale, electronic, digital computer capable of being reprogrammed to solve a full range of computing problems,[2] although earlier computers had been built with some of these properties. ...


The gap from the 1830s to the 1940s was not a period of continuous "mechanical computer" development. A mathematical demonstration of the computational resolution of problems remains with the first formal programming languages (1930s), and a wide range of solutions was demonstrated in the 1930s and 1940s, justifying the "investment" on the modern Turing complete machines in the 1940s. Other listings of programming languages are: Categorical list of programming languages Generational list of programming languages Chronological list of programming languages Note: Esoteric programming languages have been moved to the separate List of esoteric programming languages. ... This article is about the machine. ...


The hypothesis exists that the universe is computable on a universal Turing machine, which would imply that no computer more powerful than a universal Turing machine can be physically built (see philosophical implications in the Church-Turing thesis and digital physics). Look up Hypothesis in Wiktionary, the free dictionary. ... For other uses, see Universe (disambiguation). ... In computability theory the Church-Turing thesis, Churchs thesis, Churchs conjecture or Turings thesis, named after Alonzo Church and Alan Turing, is a hypothesis about the nature of mechanical calculation devices, such as electronic computers. ... Digital physics holds the basic premise that the entire history of our universe is computable, that is, the output of a (presumably short) computer program. ...


See the article on computability theory for a long list of systems that are Turing-complete, as well as several systems that are less powerful, and several theoretical systems that are even more powerful than a universal Turing machine. In computer science, computability theory is the branch of the theory of computation that studies which problems are computationally solvable using different models of computation. ...


Related work

One important result from computability theory is that it is impossible in general to determine whether a program written in a Turing-complete language will continue executing forever or will stop within a finite period of time (see halting problem). One method of commonly getting around this is to cause programs to stop executing after a fixed period of time, or to limit the power of flow control instructions. Such systems are strictly not Turing-complete by design. In computer science, computability theory is the branch of the theory of computation that studies which problems are computationally solvable using different models of computation. ... In computability theory the halting problem is a decision problem which can be informally stated as follows: Given a description of a program and a finite input, decide whether the program finishes running or will run forever, given that input. ...


Another curious theorem from computability theory is that there are problems solvable by Turing-complete languages that cannot be solved by languages with finite looping capabilities (i.e. languages that guarantee any program will halt). This result is derived by, for example, Brainerd and Landweber using the PL and PL-{GOTO} languages. In computer science, computability theory is the branch of the theory of computation that studies which problems are computationally solvable using different models of computation. ...


Examples

The computational systems (algebras, calculi) that are discussed as Turing complete systems are those intended for studying theoretical computer science. They are intended to be as simple as possible, so that it would be easier to understand the limits of computation. Here are a few: Computer science (informally, CS or compsci) is, in its most general sense, the study of computation and information processing, both in hardware and in software. ...

Most programming languages, conventional and unconventional, are Turing-complete. This includes: “Automata” redirects here. ... The Turing machine is an abstract machine introduced in 1936 by Alan Turing to give a mathematically precise definition of algorithm or mechanical procedure. As such it is still widely used in theoretical computer science, especially in complexity theory and the theory of computation. ... The lambda calculus is a formal system designed to investigate function definition, function application, and recursion. ... ‹ The template below (Expand) is being considered for deletion. ... In computer science and linguistics, a formal grammar, or sometimes simply grammar, is a precise description of a formal language — that is, of a set of strings. ... In mathematics, logic, and computer science, a formal language is a language that is defined by precise mathematical or machine processable formulas. ... Rewriting in mathematics, computer science and logic covers a wide range of methods of transforming strings, written in some fixed alphabet, that are not deterministic but are governed by explicit rules. ... A Post-Turing Machine is a simple model of computation that has been shown to be equivalent in computational power to a Turing Machine. ... A programming language is an artificial language that can be used to control the behavior of a machine, particularly a computer. ...

The specific language features used to achieve Turing-completeness can be quite different; FORTRAN systems would use loop constructs or possibly even GOTO statements to achieve repetition; Haskell and Prolog, lacking looping almost entirely, would use recursion. Turing-completeness is an abstract statement of capability, rather than a prescription of specific language features used to implement that capability. This does not adequately cite its references or sources. ... C is a general-purpose, block structured, procedural, imperative computer programming language developed in 1972 by Dennis Ritchie at the Bell Telephone Laboratories for use with the Unix operating system. ... An object-oriented programming language (also called an OO language) is one that allows or encourages, to some degree, object-oriented programming techniques such as encapsulation, inheritance, interfaces, and polymorphism. ... “Java language” redirects here. ... This does not cite any references or sources. ... Ada is a structured, statically typed imperative computer programming language designed by a team led by Jean Ichbiah of CII Honeywell Bull during 1977–1983. ... C++ (pronounced see plus plus, IPA: /siː pləs pləs/) is a general-purpose computer programming language. ... Common Lisp, commonly abbreviated CL, is a dialect of the Lisp programming language, published in ANSI standard X3. ... Functional programming is a programming paradigm that treats computation as the evaluation of mathematical functions and avoids state and mutable data. ... For the programming language, see Lisp (programming language). ... Haskell is a standardized purely functional programming language with non-strict semantics, named after the logician Haskell Curry. ... Logic programming (which might better be called logical programming by analogy with mathematical programming and linear programming) is, in its broadest sense, the use of mathematical logic for computer programming. ... Prolog is a logic programming language. ... Declarative programming is a term with two distinct meanings, both of which are in current use. ... Diagram of the basic elements and process flow of Extensible Stylesheet Language Transformations. ... An esoteric programming language (sometimes shortened to esolang[1]) is a programming language designed as a test of the boundaries of computer programming language design, as a proof of concept, or as a joke. ... The brainfuck language is an esoteric programming language noted for its extreme minimalism. ... Fortran (previously FORTRAN[1]) is a general-purpose[2], procedural,[3] imperative programming language that is especially suited to numeric computation and scientific computing. ... GOTO is a statement found in many computer programming languages. ... This article is about the concept of recursion. ...


It is difficult to find examples of non-Turing complete languages, as these languages are very limited (see, however, machines that always halt). Examples include some of the early versions of the pixel shader languages embedded in Direct3D and OpenGL extensions. Another example is a series of mathematical formulae in a spreadsheet with no cycles. While it is possible to perform many interesting operations in such a system, this fails to be Turing-complete as it is impossible to form loops; BASIC languages associated with common spreadsheet programs such as Excel and OpenOffice Calc are however Turing-complete. Another famous example is the category of regular expressions, which are generated by finite automata. A more powerful but still not Turing-complete extension of finite automata is the category of pushdown automata. In computability theory, a machine that always halts — also called a decider (Sipser, 1996) — is any abstract machine or model of computation that, contrary to the most general Turing machines, is guaranteed to halt for any particular description and input (see halting problem). ... Direct3D is part of Microsofts DirectX API. Direct3D is only available for Microsofts various Windows operating systems (Windows 95 and above) and is the base for the graphics API on the Xbox and Xbox 360 console systems. ... OpenGL (Open Graphics Library) is a standard specification defining a cross-language cross-platform API for writing applications that produce 2D and 3D computer graphics. ... A spreadsheet is a rectangular table (or grid) of information, often financial information. ... BASIC (Beginners All-purpose Symbolic Instruction Code) is a family of high-level programming languages. ... Microsoft Excel (full name Microsoft Office Excel) is a spreadsheet application written and distributed by Microsoft for Microsoft Windows and Mac OS. It features calculation and graphing tools which, along with aggressive marketing, have made Excel one of the most popular microcomputer applications to date. ... OpenOffice. ... A regular expression (abbreviated as regexp, regex or regxp) is a string that describes or matches a set of strings, according to certain syntax rules. ... Fig. ... Fig. ... In automata theory, a pushdown automaton (PDA) is a finite automaton that can make use of a stack containing data. ...


Turing-completeness in SQL is implemented through proprietary extensions, illustrating one of the reasons why relatively powerful non-Turing-complete languages are rare: the more powerful the language is initially, the more complex are the tasks to which it is applied and the sooner its lack of completeness becomes perceived as a drawback, encouraging its extension until it is Turing-complete. SQL (IPA: or IPA: ), commonly expanded as Structured Query Language, is a computer language designed for the retrieval and management of data in relational database management systems, database schema creation and modification, and database object access control management. ...


The untyped lambda calculus is Turing-complete, but many typed lambda calculi, including System F, are not. The value of typed systems is based in their ability to represent most "typical" computer programs while detecting more errors. The lambda calculus is a formal system designed to investigate function definition, function application, and recursion. ... System F is a typed lambda calculus. ...


Rule 110 and Conway's Game of Life, both cellular automata, are Turing-complete. The rule 110 cellular automaton is a one-dimensional two-state cellular automaton with the following rule table: Rule 110, like the Game of Life, exhibits what Stephen Wolfram calls Class 4 behavior, which is neither completely random nor completely repetitive. ... Gospers Glider Gun creating gliders. The Game of Life is a cellular automaton devised by the British mathematician John Horton Conway in 1970. ... A cellular automaton (plural: cellular automata) is a discrete model studied in computability theory, mathematics, and theoretical biology. ...


See also

In computability theory the Church-Turing thesis, Churchs thesis, Churchs conjecture or Turings thesis, named after Alonzo Church and Alan Turing, is a hypothesis about the nature of mechanical calculation devices, such as electronic computers. ... In computer science, computability theory is the branch of the theory of computation that studies which problems are computationally solvable using different models of computation. ... This article or section is in need of attention from an expert on the subject. ... The Chomsky hierarchy is a containment hierarchy of classes of formal grammars that generate formal languages. ... In computability theory, a machine that always halts — also called a decider (Sipser, 1996) — is any abstract machine or model of computation that, contrary to the most general Turing machines, is guaranteed to halt for any particular description and input (see halting problem). ... Stephen Wolfram (born August 29, 1959 in London) is a scientist known for his work in theoretical particle physics, cellular automata, complexity theory, and computer algebra, and is the creator of the computer program Mathematica. ... A New Kind of Science is a controversial book by Stephen Wolfram, published in 2002. ... The Principle of computational equivalence is one of the main ideas proposed by Stephen Wolfram in his book A New Kind of Science. ... A Turing tarpit is a programming language designed to be Turing-complete while minimizing the number of distinct instructions. ...

References

  • Brainerd, W.S., Landweber, L.H. (1974), Theory of Computation, Wiley.

External links

  • c2.com

  Results from FactBites:
 
Turing completeness - Wikipedia, the free encyclopedia (900 words)
In computability theory, an abstract machine or programming language is called Turing complete, Turing equivalent, or (computationally) universal if it has a computational power equivalent to a universal Turing machine (a simplified model of a programmable computer).
Turing completeness is significant in that every plausible design for a computing device so far advanced can be emulated by a universal Turing machine.
The computational systems (algebras, calculi) that are discussed as Turing complete systems are those intended for studying theoretical computer science.
Turing machine - Wikipedia, the free encyclopedia (3442 words)
Turing machines are extremely basic symbol-manipulating devices which — despite their simplicity — can be adapted to simulate the logic of any computer that could possibly be constructed.
Turing completeness, an attribute used in computability theory to describe computing systems with power equivalent to a universal Turing machine.
Turing tarpit, any computing system or language which, despite possessing Turing completeness, is generally considered useless for practical computing.
  More results at FactBites »

 
 

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