Turing machine

Classes of automata

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]

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. 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. 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. The choice of which replacement symbol to write, which direction to move the head, and whether to halt is based on a finite table that specifies what to do for each combination of the current state and the symbol that is read. Like a real computer program, it is possible for a Turing machine to go into an infinite loop which will never halt.

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.^[10] With this model, Turing was able to answer two questions in the negative:

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?^[11]^[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').^[13]

Turing machines proved the existence of fundamental limitations on the power of mechanical computation.^[14] While they can express arbitrary computations, their minimalist design makes them too slow for computation in practice: real-world computers are based on different designs that, unlike Turing machines, use random-access memory.

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.

^ Minsky 1967:107 "In his 1936 paper, A. M. Turing defined the class of abstract machines that now bear his name. A Turing machine is a finite-state machine associated with a special kind of environment -- its tape -- in which it can store (and later recover) sequences of symbols," also Stone 1972:8 where the word "machine" is in quotation marks.
^ Stone 1972:8 states "This "machine" is an abstract mathematical model", also cf. Sipser 2006:137ff that describes the "Turing machine model". Rogers 1987 (1967):13 refers to "Turing's characterization", Boolos Burgess and Jeffrey 2002:25 refers to a "specific kind of idealized machine".
^ Sipser 2006:137 "A Turing machine can do everything that a real computer can do".
^ Cf. Sipser 2002:137. Also, Rogers 1987 (1967):13 describes "a paper tape of infinite length in both directions". Minsky 1967:118 states "The tape is regarded as infinite in both directions". Boolos Burgess and Jeffrey 2002:25 include the possibility of "there is someone stationed at each end to add extra blank squares as needed".
^ Cf. Rogers 1987 (1967):13. Other authors use the word "square" e.g. Boolos Burgess Jeffrey 2002:35, Minsky 1967:117, Penrose 1989:37.
^ Boolos Burgess Jeffry 2002:25 illustrate the machine as moving along the tape. Penrose 1989:36-37 describes himself as "uncomfortable" with an infinite tape observing that it "might be hard to shift!"; he "prefer[s] to think of the tape as representing some external environment through which our finite device can move" and after observing that the " 'movement' is a convenient way of picturing things" and then suggests that "the device receives all its input from this environment. Some variations of the Turing machine model also allow the head to stay in the same position instead of moving or halting.
^ Hodges, Andrew (2012). Alan Turing: The Enigma (The Centenary ed.). Princeton University Press. ISBN 978-0-691-15564-7.
^ The idea came to him in mid-1935 (perhaps, see more in the History section) after a question posed by M. H. A. Newman in his lectures: "Was there a definite method, or as Newman put it, a "mechanical process" which could be applied to a mathematical statement, and which would come up with the answer as to whether it was provable" (Hodges 1983:93). Turing submitted his paper on 31 May 1936 to the London Mathematical Society for its Proceedings (cf. Hodges 1983:112), but it was published in early 1937 and offprints were available in February 1937 (cf. Hodges 1983:129).
^ See footnote in Davis 2000:151.
^ see note in forward to The Collected Works of Alonzo Church (Burge, Tyler; Enderton, Herbert, eds. (2019-04-23). The Collected Works of Alonzo Church. Cambridge, MA, US: MIT Press. ISBN 978-0-262-02564-5.)
^ Turing 1936 in The Undecidable 1965:132-134; Turing's definition of "circular" is found on page 119.
^ Turing, Alan Mathison (1937). "On Computable Numbers, with an Application to the Entscheidungsproblem". Proceedings of the London Mathematical Society. Series 2. 42 (1): 230–265. doi:10.1112/plms/s2-42.1.230. S2CID 73712. — Reprint at: Turing, Alan. "On computable numbers, with an application to the Entscheidungsproblem". The Turing Digital Archive. Retrieved 9 July 2020.^{[permanent dead link]}
^ Turing 1936 in The Undecidable 1965:145
^ Sipser 2006:137 observes that "A Turing machine can do everything that a real computer can do. Nevertheless, even a Turing machine cannot solve certain problems. In a very real sense, these problems are beyond the theoretical limits of computation."

[1] Minsky 1967:107 "In his 1936 paper, A. M. Turing defined the class of abstract machines that now bear his name. A Turing machine is a finite-state machine associated with a special kind of environment -- its tape -- in which it can store (and later recover) sequences of symbols," also Stone 1972:8 where the word "machine" is in quotation marks.

[2] Stone 1972:8 states "This "machine" is an abstract mathematical model", also cf. Sipser 2006:137ff that describes the "Turing machine model". Rogers 1987 (1967):13 refers to "Turing's characterization", Boolos Burgess and Jeffrey 2002:25 refers to a "specific kind of idealized machine".

[3] Sipser 2006:137 "A Turing machine can do everything that a real computer can do".

[4] Cf. Sipser 2002:137. Also, Rogers 1987 (1967):13 describes "a paper tape of infinite length in both directions". Minsky 1967:118 states "The tape is regarded as infinite in both directions". Boolos Burgess and Jeffrey 2002:25 include the possibility of "there is someone stationed at each end to add extra blank squares as needed".

[5] Cf. Rogers 1987 (1967):13. Other authors use the word "square" e.g. Boolos Burgess Jeffrey 2002:35, Minsky 1967:117, Penrose 1989:37.

[6] Boolos Burgess Jeffry 2002:25 illustrate the machine as moving along the tape. Penrose 1989:36-37 describes himself as "uncomfortable" with an infinite tape observing that it "might be hard to shift!"; he "prefer[s] to think of the tape as representing some external environment through which our finite device can move" and after observing that the " 'movement' is a convenient way of picturing things" and then suggests that "the device receives all its input from this environment. Some variations of the Turing machine model also allow the head to stay in the same position instead of moving or halting.

[Hodges-2012-7] Hodges, Andrew (2012). Alan Turing: The Enigma (The Centenary ed.). Princeton University Press. ISBN 978-0-691-15564-7.

[8] The idea came to him in mid-1935 (perhaps, see more in the History section) after a question posed by M. H. A. Newman in his lectures: "Was there a definite method, or as Newman put it, a "mechanical process" which could be applied to a mathematical statement, and which would come up with the answer as to whether it was provable" (Hodges 1983:93). Turing submitted his paper on 31 May 1936 to the London Mathematical Society for its Proceedings (cf. Hodges 1983:112), but it was published in early 1937 and offprints were available in February 1937 (cf. Hodges 1983:129).

[9] See footnote in Davis 2000:151.

[10] see note in forward to The Collected Works of Alonzo Church (Burge, Tyler; Enderton, Herbert, eds. (2019-04-23). The Collected Works of Alonzo Church. Cambridge, MA, US: MIT Press. ISBN 978-0-262-02564-5.)

[11] Turing 1936 in The Undecidable 1965:132-134; Turing's definition of "circular" is found on page 119.

[12] Turing, Alan Mathison (1937). "On Computable Numbers, with an Application to the Entscheidungsproblem". Proceedings of the London Mathematical Society. Series 2. 42 (1): 230–265. doi:10.1112/plms/s2-42.1.230. S2CID 73712. — Reprint at: Turing, Alan. "On computable numbers, with an application to the Entscheidungsproblem". The Turing Digital Archive. Retrieved 9 July 2020.^{[permanent dead link]}

[13] Turing 1936 in The Undecidable 1965:145

[14] Sipser 2006:137 observes that "A Turing machine can do everything that a real computer can do. Nevertheless, even a Turing machine cannot solve certain problems. In a very real sense, these problems are beyond the theoretical limits of computation."

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