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In computability theory, a system of data-manipulation rules (such as a model of computation, 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 [1] [2] (devised by English mathematician and computer scientist Alan Turing).
There are no constraints on the productions rules. They generate exactly all languages that can be recognized by a Turing machine, thus any language that is possible to be generated can be generated by a Type-0 grammar. [8] These languages are also known as the recursively enumerable or Turing-recognizable languages. [8]
Turing completeness is the ability for a computational model or a system of instructions to simulate a Turing machine. A programming language that is Turing complete is theoretically capable of expressing all tasks accomplishable by computers; nearly all programming languages are Turing complete if the limitations of finite memory are ignored.
TeX82 also uses fixed-point arithmetic instead of floating-point, to ensure reproducibility of the results across different computer hardware, [9] and includes a real, Turing-complete programming language, following intense lobbying by Guy Steele. [10] In 1989, Donald Knuth released new versions of TeX and Metafont. [11]
Brainfuck is an example of a so-called Turing tarpit: it can be used to write any program, but it is not practical to do so because it provides so little abstraction that the programs get very long or complicated. While Brainfuck is fully Turing-complete, it is not intended for practical use but to challenge and amuse programmers.
Among the 88 possible unique elementary cellular automata, Rule 110 is the only one for which Turing completeness has been directly proven, although proofs for several similar rules follow as simple corollaries (e.g. Rule 124, which is the horizontal reflection of Rule 110). Rule 110 is arguably the simplest known Turing complete system. [2] [5]