However: ===The Entscheidungsproblem (the "decision problem"): Hilbert's tenth question of 1900=== 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.

Hilbert's original expression for #10 is as follows: By 1922, this notion of "Entscheidungsproblem" had developed a bit, and H.

Behmann stated that By the 1928 international congress of mathematicians, Hilbert "made his questions quite precise.

The first two questions were answered in 1930 by Kurt Gödel at the very same meeting where Hilbert delivered his retirement speech (much to the chagrin of Hilbert); the third—the Entscheidungsproblem—had to wait until the mid-1930s. The problem was that an answer first required a precise definition of "definite general applicable prescription", which Princeton professor Alonzo Church would come to call "effective calculability", and in 1928 no such definition existed.

The first two questions were answered in 1930 by Kurt Gödel at the very same meeting where Hilbert delivered his retirement speech (much to the chagrin of Hilbert); the third—the Entscheidungsproblem—had to wait until the mid-1930s. The problem was that an answer first required a precise definition of "definite general applicable prescription", which Princeton professor Alonzo Church would come to call "effective calculability", and in 1928 no such definition existed.

Subsequent to Turing's original paper in 1936–1937, machine-models have allowed all nine possible types of five-tuples: Any Turing table (list of instructions) can be constructed from the above nine 5-tuples.

But over the next 6–7 years Emil Post developed his definition of a worker moving from room to room writing and erasing marks per a list of instructions (Post 1936), as did Church and his two students Stephen Kleene and J.

Church's paper (published 15 April 1936) showed that the Entscheidungsproblem was indeed "undecidable" and beat Turing to the punch by almost a year (Turing's paper submitted 28 May 1936, published January 1937).

In the meantime, Emil Post submitted a brief paper in the fall of 1936, so Turing at least had priority over Post.

Davies' Corrections to Turing's Universal Computing Machine Martin Davis (ed.) (1965), The Undecidable, Raven Press, Hewlett, NY. Emil Post (1936), "Finite Combinatory Processes—Formulation 1", Journal of Symbolic Logic, 1, 103–105, 1936.

In the Appendix of this paper Post comments on and gives corrections to Turing's paper of 1936–1937.

Includes Turing's 1936–1937 paper, with brief commentary and biography of Turing as written by Hawking. Andrew Hodges, The Enigma, Simon and Schuster, New York.

Church's paper (published 15 April 1936) showed that the Entscheidungsproblem was indeed "undecidable" and beat Turing to the punch by almost a year (Turing's paper submitted 28 May 1936, published January 1937).

Studying their abstract properties yields many insights into computer science and complexity theory. ===Physical description=== In his 1948 essay, "Intelligent Machinery", Turing wrote that his machine consisted of: ==Description== The Turing machine mathematically models a machine that mechanically operates on a tape.

in The Undecidable, pp. 115–154; available on the web in many places. Alan Turing, 1948, "Intelligent Machinery." Reprinted in "Cybernetics: Key Papers." Ed.

In his obituary of Turing 1955 Newman writes: Gandy states that: While Gandy believed that Newman's statement above is "misleading", this opinion is not shared by all.

This result was obtained in 1966 by F.

JACM, 13(4):533–546, 1966. ===Computability theory=== Some parts have been significantly rewritten by Burgess.

Marvin Minsky, Computation: Finite and Infinite Machines, Prentice–Hall, Inc., N.J., 1967.

Baltimore: University Park Press, 1968.

Computability theory, which studies computability of functions from inputs to outputs, and for which Turing machines were invented, reflects this practice. Since the 1970s, interactive use of computers became much more common.

On pages 90–103 Hennie discusses the UTM with examples and flow-charts, but no actual 'code'. Centered around the issues of machine-interpretation of "languages", NP-completeness, etc. Distinctly different and less intimidating than the first edition. Stephen Kleene (1952), Introduction to Metamathematics, North–Holland Publishing Company, Amsterdam Netherlands, 10th impression (with corrections of 6th reprint 1971).

With reference to the role of Turing machines in the development of computation (both hardware and software) see 1.4.5 History and Bibliography pp. 225ff and 2.6 History and Bibliographypp. 456ff. Zohar Manna, 1974, Mathematical Theory of Computation.

Valuable survey, with 141 references. ===Church's thesis=== ===Small Turing machines=== Rogozhin, Yurii, 1998, "A Universal Turing Machine with 22 States and 2 Symbols", Romanian Journal of Information Science and Technology, 1(3), 259–265, 1998.

Post (1947), Boolos & Jeffrey (1974, 1999), Davis-Sigal-Weyuker (1994)); also see more at Post–Turing machine. ===The "state"=== The word "state" used in context of Turing machines can be a source of confusion, as it can mean two things.

(surveys known results about small universal Turing machines) Stephen Wolfram, 2002, A New Kind of Science, Wolfram Media, Brunfiel, Geoff, Student snags maths prize, Nature, October 24.

Reprinted, Dover, 2003.

2007. Jim Giles (2007), Simplest 'universal computer' wins student $25,000, New Scientist, October 24, 2007. Alex Smith, Universality of Wolfram’s 2, 3 Turing Machine, Submission for the Wolfram 2, 3 Turing Machine Research Prize. Vaughan Pratt, 2007, "Simple Turing machines, Universality, Encodings, etc.", FOM email list.

October 29, 2007. Martin Davis, 2007, "Smallest universal machine", and Definition of universal Turing machine FOM email list.

October 26–27, 2007. Alasdair Urquhart, 2007 "Smallest universal machine", FOM email list.

October 26, 2007. Hector Zenil (Wolfram Research), 2007 "smallest universal machine", FOM email list.

Kirner et al., 2009 have shown that among the general-purpose programming languages some are Turing complete while others are not.

(Arora and Barak, 2009, theorem 1.9) ==Comparison with real machines== It is often said that Turing machines, unlike simpler automata, are as powerful as real machines, and are able to execute any operation that a real program can.

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