Turing completeness

Ability of a computing system to simulate Turing machines / From Wikipedia, the free encyclopedia

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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 (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.

A related concept is that of Turing equivalence  two computers P and Q are called equivalent if P can simulate Q and Q can simulate P. The Church–Turing thesis conjectures that any function whose values can be computed by an algorithm can be computed by a Turing machine, and therefore that if any real-world computer can simulate a Turing machine, it is Turing equivalent to a Turing machine. A universal Turing machine can be used to simulate any Turing machine and by extension the computational aspects of any possible real-world computer.[NB 1]

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.

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