Turing machine

[10] With this model, Turing was able to answer two questions in the negative: 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, or 'decision problem' (whether every mathematical statement is provable or disprovable).

Typically, the sequential memory is represented as a tape of infinite length on which the machine can perform read and write operations.

In the context of formal language theory, a Turing machine (automaton) is capable of enumerating some arbitrary subset of valid strings of an alphabet.

Given a Turing machine M and an arbitrary string s, it is generally not possible to decide whether M will eventually produce s. This is due to the fact that the halting problem is unsolvable, which has major implications for the theoretical limits of computing.

Another mathematical formalism, lambda calculus, with a similar "universal" nature was introduced by Alonzo Church.

This thesis states that Turing machines, lambda calculus, and other similar formalisms of computation do indeed capture the informal notion of effective methods in logic and mathematics and thus provide a model through which one can reason about an algorithm or "mechanical procedure" in a mathematically precise way without being tied to any particular formalism.

In his 1948 essay, "Intelligent Machinery", Turing wrote that his machine consists of: ...an unlimited memory capacity obtained in the form of an infinite tape marked out into squares, on each of which a symbol could be printed.

In the original article ("On Computable Numbers, with an Application to the Entscheidungsproblem", see also references below), Turing imagines not a mechanism, but a person whom he calls the "computer", who executes these deterministic mechanical rules slavishly (or as Turing puts it, "in a desultory manner").

More explicitly, a Turing machine consists of: In the 4-tuple models, erasing or writing a symbol (aj1) and moving the head left or right (dk) are specified as separate instructions.

In the words of van Emde Boas (1990), p. 6: "The set-theoretical object [his formal seven-tuple description similar to the above] provides only partial information on how the machine will behave and what its computations will look like."

For instance, Definitions in literature sometimes differ slightly, to make arguments or proofs easier or clearer, but this is always done in such a way that the resulting machine has the same computational power.

This expression is called the "state formula".Earlier in his paper Turing carried this even further: he gives an example where he placed a symbol of the current "m-configuration"—the instruction's label—beneath the scanned square, together with all the symbols on the tape (The Undecidable, p. 121); this he calls "the complete configuration" (The Undecidable, p. 118).

If the machine were to be stopped and cleared to blank both the "state register" and entire tape, these "configurations" could be used to rekindle a computation anywhere in its progress (cf.

A Turing machine is equivalent to a single-stack pushdown automaton (PDA) that has been made more flexible and concise by relaxing the last-in-first-out (LIFO) requirement of its stack.

For practical and didactic intentions, the equivalent register machine can be used as a usual assembly programming language.

A relevant question is whether or not the computation model represented by concrete programming languages is Turing equivalent.

This is because the size of memory reference data types, called pointers, is accessible inside the language.

[citation needed] The model of computation that Turing called his "universal machine"—"U" for short—is considered by some (cf.

Turing's paper ... contains, in essence, the invention of the modern computer and some of the programming techniques that accompanied it.In terms of computational complexity, a multi-tape universal Turing machine need only be slower by logarithmic factor compared to the machines it simulates.

He cites other proposals for "universal calculating machines" including those of Percy Ludgate (1909), Leonardo Torres Quevedo (1914),[21][22] Maurice d'Ocagne (1922), Louis Couffignal (1933), Vannevar Bush (1936), Howard Aiken (1937).

The fundamental importance of conditional iteration and conditional transfer for a general theory of calculating machines is not recognized…With regard to Hilbert's problems posed by the famous mathematician David Hilbert in 1900, an aspect of problem #10 had been floating about for almost 30 years before it was framed precisely.

The Entscheidungsproblem must be considered the main problem of mathematical logic.By 1922, this notion of "Entscheidungsproblem" had developed a bit, and H. Behmann stated that ... most general form of the Entscheidungsproblem [is] as follows: A quite definite generally applicable prescription is required which will allow one to decide in a finite number of steps the truth or falsity of a given purely logical assertion ...Behmann remarks that ... the general problem is equivalent to the problem of deciding which mathematical propositions are true.If one were able to solve the Entscheidungsproblem then one would have a "procedure for solving many (or even all) mathematical problems".By the 1928 international congress of mathematicians, Hilbert "made his questions quite precise.

The 'main idea' might have either been his analysis of computation or his realization that there was a universal machine, and so a diagonal argument to prove unsolvability.While Gandy believed that Newman's statement above is "misleading", this opinion is not shared by all.

It is not difficult, though somewhat laborious, to prove that these three definitions [the 3rd is the λ-calculus] are equivalent.Alan Turing invented the "a-machine" (automatic machine) in 1936.

[13] When Turing returned to the UK he ultimately became jointly responsible for breaking the German secret codes created by encryption machines called "The Enigma"; he also became involved in the design of the ACE (Automatic Computing Engine), "[Turing's] ACE proposal was effectively self-contained, and its roots lay not in the EDVAC [the USA's initiative], but in his own universal machine" (Hodges p. 318).

I propose, therefore to show that there can be no general process for determining whether a given formula U of the functional calculus K is provable, i.e. that there can be no machine which, supplied with any one U of these formulae, will eventually say whether U is provable.Turing's example (his second proof): If one is to ask for a general procedure to tell us: "Does this machine ever print 0", the question is "undecidable".

In 1937, while at Princeton working on his PhD thesis, Turing built a digital (Boolean-logic) multiplier from scratch, making his own electromechanical relays (Hodges p. 138).

"Alan's task was to embody the logical design of a Turing machine in a network of relay-operated switches ..." (Hodges p. 138).

While Turing might have been just initially curious and experimenting, quite-earnest work in the same direction was going in Germany (Konrad Zuse (1938)), and in the United States (Howard Aiken) and George Stibitz (1937); the fruits of their labors were used by both the Axis and Allied militaries in World War II (cf.

In particular, computational complexity theory makes use of the Turing machine: Depending on the objects one likes to manipulate in the computations (numbers like nonnegative integers or alphanumeric strings), two models have obtained a dominant position in machine-based complexity theory: the off-line multitape Turing machine..., which represents the standard model for string-oriented computation, and the random access machine (RAM) as introduced by Cook and Reckhow ..., which models the idealised Von Neumann-style computer.Only in the related area of analysis of algorithms this role is taken over by the RAM model.