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Turing machine
A Turing machine is a mathematical model of computation describing an abstract machine [1] that manipulates symbols on a strip of tape according to a table of rules. [2] Despite the model's simplicity, it is capable of implementing any computer algorithm. [3]
https://en.wikipedia.org/wiki/Mathematical_model_of_computation
https://en.wikipedia.org/wiki/Abstract_machine
https://en.wikipedia.org/wiki/Turing_machine#cite_note-1
https://en.wikipedia.org/wiki/Turing_machine#cite_note-2
https://en.wikipedia.org/wiki/Computer_algorithm
https://en.wikipedia.org/wiki/Turing_machine#cite_note-3
The machine operates on an infinite [4] memory tape divided into discrete cells, [5] each of which can hold a single symbol drawn from a finite set of symbols called the alphabet of the machine.
https://en.wikipedia.org/wiki/Turing_machine#cite_note-4
https://en.wikipedia.org/wiki/Discrete_mathematics
https://en.wikipedia.org/wiki/Turing_machine#cite_note-5
https://en.wikipedia.org/wiki/Finite_set
https://en.wikipedia.org/wiki/Alphabet_(formal_languages)
It has a "head" that, at any point in the machine's operation, is positioned over one of these cells, and a "state" selected from a finite set of states.
https://en.wikipedia.org/wiki/Finite_set
At each step of its operation, the head reads the symbol in its cell.
Then, based on the symbol and the machine's own present state, the machine writes a symbol into the same cell, and moves the head one step to the left or the right, [6] or halts the computation.
https://en.wikipedia.org/wiki/Turing_machine#cite_note-6
The choice of which replacement symbol to write and which direction to move is based on a finite table that specifies what to do for each combination of the current state and the symbol that is read.
The Turing machine was invented in 1936 by Alan Turing, [7] [8] who called it an "a-machine" (automatic machine). [9] It was Turing's Doctoral advisor, Alonzo Church, who later coined the term "Turing machine" in a review. With this model, Turing was able to answer two questions in the negative:
https://en.wikipedia.org/wiki/Alan_Turing
https://en.wikipedia.org/wiki/Turing_machine#cite_note-Hodges-2012-7
https://en.wikipedia.org/wiki/Turing_machine#cite_note-8
https://en.wikipedia.org/wiki/Turing_machine#cite_note-9
https://en.wikipedia.org/wiki/Alonzo_Church
https://en.wikipedia.org/wiki/Turing_machine#cite_note-10
10
Does a machine exist that can determine whether any arbitrary machine on its tape is "circular" (e.g., freezes, or fails to continue its computational task)?
Does a machine exist that can determine whether any arbitrary machine on its tape ever prints a given symbol?
https://en.wikipedia.org/wiki/Turing_machine#cite_note-11
11
https://en.wikipedia.org/wiki/Turing_machine#cite_note-12
12
Thus by providing a mathematical description of a very simple device capable of arbitrary computations, he was able to prove properties of computation in general—and in particular, the uncomputability of the Entscheidungsproblem ('decision problem').
https://en.wikipedia.org/wiki/Computability
https://en.wikipedia.org/wiki/Entscheidungsproblem
https://en.wikipedia.org/wiki/Turing_machine#cite_note-13
13
Turing machines proved the existence of fundamental limitations on the power of mechanical computation. While they can express arbitrary computations, their minimalist design makes them unsuitable for computation in practice: real-world computers are based on different designs that, unlike Turing machines, use random-access memory.
https://en.wikipedia.org/wiki/Turing_machine#cite_note-14
14
https://en.wikipedia.org/wiki/Computer
https://en.wikipedia.org/wiki/Random-access_memory
https://en.wikipedia.org/wiki/Turing_completeness
Turing completeness is the ability for 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