In computability theory, a system of data-manipulation rules (such as a computer's instruction set, a programming language, or a cellular automaton) is said to be Turing-complete or computationally universal if it can be used to simulate any Turing machine (devised by English mathematician and computer scientist Alan Turing). This means that this system is able to recognize or decide other data-manipulation rule sets. Turing completeness is used as a way to express the power of such a data-manipulation rule set. Virtually all programming languages today are Turing-complete.
To show that something is Turing-complete, it is enough to show that it can be used to simulate some Turing-complete system. No physical system can have infinite memory, but if the limitation of finite memory is ignored, most programming languages are otherwise Turing-complete.
In colloquial usage, the terms "Turing-complete" and "Turing-equivalent" are used to mean that any real-world general-purpose computer or computer language can approximately simulate the computational aspects of any other real-world general-purpose computer or computer language. In real life, this leads to the practical concepts of computing virtualization and emulation.
Real computers constructed so far can be functionally analyzed like a single-tape Turing machine (the "tape" corresponding to their memory); thus the associated mathematics can apply by abstracting their operation far enough. However, real computers have limited physical resources, so they are only linear bounded automaton complete. In contrast, a universal computer is defined as a device with a Turing-complete instruction set, infinite memory, and infinite available time.
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. Babbage appreciated that the machine was capable of great feats of calculation, including primitive logical reasoning, but he did not appreciate that no other machine could do better. From the 1830s until the 1940s, mechanical calculating machines such as adders and multipliers were built and improved, but they could not perform a conditional branch and therefore were not Turing-complete.
The actual notion of computation was isolated soon after, starting with Gödel's incompleteness theorem. This theorem showed that axiom systems were limited when reasoning about the computation that deduces their theorems. Church and Turing independently demonstrated that Hilbert's Entscheidungsproblem (decision problem) was unsolvable, thus identifying the computational core of the incompleteness theorem. This work, along with Gödel's work on general recursive functions, established that there are sets of simple instructions, which, when put together, are able to produce any computation. The work of Gödel showed that the notion of computation is essentially unique.
In 1941 Konrad Zuse completed the Z3 computer. Zuse was not familiar with Turing's work on computability at the time. In particular, the Z3 lacked dedicated facilities for a conditional jump, thereby precluding it from being Turing complete. However, in 1998, it was shown by Rojas that the Z3 is capable of simulating conditional jumps, and therefore Turing complete in theory. To do this, its tape program would have to be long enough to execute every possible path through both sides of every branch.
The first computer capable of conditional branching in practice, and therefore Turing complete in practice, was the ENIAC in 1946. Zuse's Z4 computer was operational in 1945, but it did not support conditional branching until 1950.
Computability theory uses models of computation to analyze problems and determine whether they are computable and under what circumstances. The first result of computability theory is that there exist problems for which it is impossible to predict what a (Turing-complete) system will do over an arbitrarily long time.
The classic example is the halting problem: create an algorithm that takes as input a program in some Turing-complete language and some data to be fed to that program, and determines whether the program, operating on the input, will eventually stop or will continue forever. It is trivial to create an algorithm that can do this for some inputs, but impossible to do this in general. For any characteristic of the program's eventual output, it is impossible to determine whether this characteristic will hold.
One can instead limit a program to executing only for a fixed period of time (timeout) or limit the power of flow-control instructions (for example, providing only loops that iterate over the items of an existing array). However, another theorem shows that there are problems solvable by Turing-complete languages that cannot be solved by any language with only finite looping abilities (i.e., any language guaranteeing that every program will eventually finish to a halt). So any such language is not Turing-complete. For example, a language in which programs are guaranteed to complete and halt cannot compute the computable function produced by Cantor's diagonal argument on all computable functions in that language.
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:
Turing completeness is an abstract statement of ability, rather than a prescription of specific language features used to implement that ability. The 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. Most programming languages are describing computations on von Neumann architectures, which have memory (RAM and register) and a control unit. These two elements make this architecture Turing-complete. Even pure functional languages are Turing-complete.[NB 2]
Turing completeness in declarative SQL is implemented through recursive common table expressions. Unsurprisingly, procedural extensions to SQL (PLSQL, etc.) are also Turing-complete. This illustrates one reason 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. The value of typed systems is based in their ability to represent most typical computer programs while detecting more errors.
Many computational languages exist that are not Turing-complete. One such example is the set of regular languages, which are generated by regular expressions and which are recognized by finite automata. 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 compiling. Further examples include some of the early versions of the pixel shader languages embedded in Direct3D and OpenGL extensions.
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Programming languages are similar to those machines (although virtual). They take programs and run them. Now, a programing language is called "Turing complete", if it can run any program (irrespective of the language) that a Turing machine can run given enough time and memory.
For example: Let's say there is a program that takes 10 numbers and adds them. A Turing machine can easily run this program. But now imagine that for some reason your programming language can't perform the same addition. This would make it "Turing incomplete" (so to speak). On the other hand, if it can run any program that the universal Turing machine can run, then it's Turing complete.
A Turing complete language is one that can perform any computation. The Church-Turing Thesis states that any performable computation can be done by a Turing machine. A Turing machine is a machine with infinite random access memory and a finite 'program' that dictates when it should read, write, and move across that memory, when it should terminate with a certain result, and what it should do next. The input to a Turing machine is put in its memory before it starts.
A Turing machine can make decisions based on what it sees in memory - The 'language' that only supports +, -, *, and / on integers is not Turing complete because it can't make a choice based on its input, but a Turing machine can.
A Turing machine can use infinite memory - A language that was exactly like Java but would terminate once it used more than 4 Gigabytes of memory wouldn't be Turing complete, because a Turing machine can use infinite memory. This is why we can't actually build a Turing machine, but Java is still a Turing complete language because the Java language has no restriction preventing it from using infinite memory. This is one reason regular expressions aren't Turing complete. 041b061a72