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For the usage of this term in Turing reductions, see Turing complete set. In Computability theory, a Turing reduction from a problem A to a problem B, named after Alan Turing, is a reduction which solves

In computability theory, several closely-related terms are used to describe the "computational power" of a computational system (such as an abstract machine or programming language):

Turing completeness
A computational system that can compute every Turing-computable function is called Turing-complete (or Turing-powerful). In Computer science, computability theory is the branch of the Theory of computation that studies which problems are computationally solvable using different 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 write programs which control the behavior of a machine particularly a Computer. Computable functions are the basic objects of study in computability theory. Alternatively, such a system is one that can simulate a universal Turing machine. This article is a supplement to the article Turing machine. Alan Turing 's "universal computing machine" (alternately "universal
Turing equivalence
A Turing-complete system is called Turing-equivalent if every function it can compute is also Turing-computable; i. e. , it computes precisely the same class of functions as do Turing machines. Turing machines are basic abstract symbol-manipulating devices which despite their simplicity can be adapted to simulate the logic of any Computer Algorithm Alternatively, a Turing-equivalent system is one that can simulate, and be simulated by, a universal Turing machine. (All known Turing-complete systems are Turing-equivalent, which adds support to the Church-Turing thesis. )
(Computational) universality
A system is called universal with respect to a class of systems if it can compute every function computable by systems in that class (or can simulate each of those systems). Typically, universality is tacitly with respect to a Turing-complete class of systems. The term weakly universal is sometimes used to distinguish a system (e. g. a cellular automaton) whose universality is achieved only by modifying the standard definition of Turing machine so as to include unbounded input. A cellular automaton (plural cellular automata) is a discrete model studied in computability theory, Mathematics, Theoretical biology Turing machines are basic abstract symbol-manipulating devices which despite their simplicity can be adapted to simulate the logic of any Computer Algorithm

Contents

Overview

Turing completeness, named after Alan Turing, 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. Alan Mathison Turing, OBE, FRS (ˈt(jʊ(ərɪŋ (23 June 1912 &ndash 7 June 1954 was an English Mathematician This article is a supplement to the article Turing machine. Alan Turing 's "universal computing machine" (alternately "universal 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. Computer programs (also software programs, or just programs) are instructions for 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 unlimited 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 been built at the time it was designed, but the first actual implementation appeared in 1941: the program-controlled Z3 of Konrad Zuse. The analytical engine, an important step in the History of computers, was the design of a mechanical general-purpose Computer by the British mathematician Charles Year 1941 ( MCMXLI) was a Common year starting on Wednesday (the link will display 1941 calendar of the Gregorian calendar. Konrad Zuse 's Konrad Zuse (ˈkɔnʁat ˈtsuːzə June 22, 1910 Berlin - December 18, 1995 Hünfeld) was a German The universality of the Z3 was shown by Raúl Rojas in 1998. Raúl Rojas (born 1955, in Mexico City) is a professor of informatics and mathematics at the Free University of Berlin and Year 1998 ( MCMXCVIII) was a Common year starting on Thursday (link will display full 1998 Gregorian calendar) Prior to Rojas' 1998 paper, the first machine known to be Turing-complete was ENIAC (1946). ENIAC, short for Electronic Numerical Integrator And Computer, was the first general-purpose electronic Computer.

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 were demonstrated in the 1930s and 1940s, justifying the "investment" on the modern Turing complete machines in the 1940s. A programming language is an Artificial language that can be used to write programs which control the behavior of a machine particularly a Computer. A computer is a Machine that manipulates data according to a list of instructions.

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). A hypothesis (from Greek) consists either of a suggested explanation for a phenomenon (an event that is observable or of a reasoned proposal suggesting a possible The Universe is defined as everything that Physically Exists: the entirety of Space and Time, all forms of Matter, Energy 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

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). In Computer science, computability theory is the branch of the Theory of computation that studies which problems are computationally solvable using different In computability theory, the halting problem is a Decision problem which can be stated as follows given a description of a program and a finite input 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.

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. In Computer science, computability theory is the branch of the Theory of computation that studies which problems are computationally solvable using different 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.

Examples

The computational systems (algebras, calculi) that are discussed as Turing complete systems are those intended for studying theoretical computer science. Theoretical computer science is the collection of topics of Computer science that focuses on the more abstract logical and mathematical aspects of Computing, such 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:

Most programming languages, conventional and unconventional, are Turing-complete. This article is a supplement to the article Turing machine. Alan Turing 's "universal computing machine" (alternately "universal In Mathematical logic and Computer science, lambda calculus, also written as λ-calculus, is a Formal system designed to investigate function Alonzo Church ( June 14, 1903 – August 11, 1995) was an American Mathematician and logician In Formal semantics, Computer science and Linguistics, a formal grammar (also called formation rules) is a precise description of a Formal A formal language is a set of words, ie finite strings of letters, or symbols. In Mathematics, Computer science and Logic, rewriting covers a wide range of potentially non-deterministic methods of replacing subterms of a The article Turing machine gives a general introduction to Turing machines while this article covers a specific class of Turing machines A programming language is an Artificial language that can be used to write programs which control the behavior of a machine particularly a Computer. This includes:

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. Fortran (previously FORTRAN) is a general-purpose, procedural, imperative Programming language that is especially suited to GOTO is a statement found in many computer Programming languages It is a combination of the English words go and to Recursion, in Mathematics and Computer science, is a method of defining functions in which the function being defined is applied within its own definition Turing-completeness is an abstract statement of capability, rather than a prescription of specific language features used to implement that capability.

It is difficult to find examples of non-Turing complete languages, as these languages are usually very limited (see, however, machine that always halts). Examples include some of the early versions of the pixel shader languages embedded in Direct3D and OpenGL extensions, or a series of mathematical formulae in a spreadsheet with no cycles. Direct3D is part of Microsoft 's DirectX API. Direct3D is only available for Microsoft's various Windows Operating systems ( Windows OpenGL ( Open G raphics L ibrary is a standard specification defining a cross-language Cross-platform API for writing applications that produce A spreadsheet is a Computer application that simulates a paper worksheet 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. Another example is the category of regular expressions, which are generated by finite automata. In Computing, regular expressions provide a concise and flexible means for identifying strings of text of interest such as particular characters words or patterns of characters A more powerful but still not Turing-complete extension of finite automata is the category of pushdown automata and context-free grammars, which are commonly used to generate parse trees in an initial stage of program compilation. In Automata theory, a pushdown automaton (PDA is a finite automaton that can make use of a stack containing data In Formal language theory, a context-free grammar ( CFG) is a grammar in which every production rule is of the form V &rarr A compiler is a Computer program (or set of programs that translates text written in a computer language (the source language) into another

There are some languages where all functions are total, and must terminate, such as Charity and Epigram. Charity is a Purely functional experimental Programming language, developed at the University of Calgary. Epigram is the name of a Functional programming language with Dependent types and of the IDE usually packaged with it Charity uses a type system and control constructs based on category theory, whereas Epigram uses dependent types. In Mathematics, category theory deals in an abstract way with mathematical Structures and relationships between them it abstracts from sets In Computer science and Logic, a dependent type is a Type which depends on a value

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.

The untyped lambda calculus is Turing-complete, but many typed lambda calculi, including System F, are not. In Mathematical logic and Computer science, lambda calculus, also written as λ-calculus, is a Formal system designed to investigate function System F, also known as the polymorphic lambda calculus or the second-order lambda calculus, is a Typed lambda calculus. The value of typed systems is based in their ability to represent most "typical" computer programs while detecting more errors.

Rule 110 and Conway's Game of Life, both cellular automata, are Turing-complete. The Rule 110 cellular automaton (often simply Rule 110) is a one-dimensional two-state Cellular automaton with the following rule table Interesting "Conway game" can refer to games as defined by Surreal numbers which Conway also developed A cellular automaton (plural cellular automata) is a discrete model studied in computability theory, Mathematics, Theoretical biology

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External links

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