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While such machines may be physically impossible as they require unlimited storage and zero crashing probability, Turing completeness is often loosely attributed to physical machines or programming languages that would be universal if they had indefinitely enlargeable storage and were absolutely reliable. Charles Babbage's Analytical Engine would have been Turing-complete if it had ever been built, but the first actual implementation appeared in 1941: the program-controlled Z3 of Konrad Zuse. Its universality, however, was shown only much later, namely, by Raśl Rojas in 1998. The first machine known to be Turing-complete was ENIAC. All modern computers are also Turing-complete in this lax sense.
Turing completeness is significant in that every plausible design for a computing device so far advanced (even quantum computers) can be emulated by a universal Turing machine. Thus, a machine that can act as a universal Turing machine can, in principle, perform any calculation that any other computer is capable of (in other words, it is programmable). Note, however, that this says nothing about the effort to write a program for the machine or the time it may take to do the calculation.
It is hypothesized by some that the universe is Turing-complete (see philosophical implications in the Church-Turing thesis and digital physics).
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.
It is difficult to find examples of non-turing complete languages, as these languages are very limited. An example would be a series of mathematical formulas 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. The macro language within Excel however is turing complete. Another famous example is regular expressions, as contained in languages such as perlImage:Programming-republic-of-perl. gif|right|framed|Programming Republic of logo]] Perl also Practical Extraction and Report Language (a backronym, see below), is a programming language released by Larry Wall on December 18, 1987 that borrows features fr. A list of Turing-complete languages is contained under computability theory.
One important result from computability theory is that it is impossible in general to find if a program writen in a Turing-complete language will continue executing forever or will stop within a finite period of time. One method of commonly getting around this is to cause programs to stop executing after a fixed period of time. Such systems are strictly not Turing-complete.
The untyped lambda calculusThe lambda calculus is a formal system designed to investigate function definition, function application, and recursion. It was introduced by Alonzo Church and Stephen Cole Kleene in the 1930s; Church used the lambda calculus in 1936 to give a negative an is Turing-complete, but many typed lambda calculi, including System FSystem F is a typed lambda calculus. It is also known as the second-order or polymorphic lambda calculus . It was discovered independently by the logician Jean-Yves Girard and the computer scientist John C. System F formalizes the notion of parametric pol, are not. The value of typed systems is based in their ability to represent most "typical" computer programs while detecting more errors.
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