In mathematics and computer science, the Entscheidungsproblem (pronounced [ɛntˈʃaɪdʊŋspʁoˌbleːm], German for 'decision problem') is a challenge posed by David Hilbert in 1928. The Entscheidungsproblem asks for an algorithm that takes as input a statement of a first-order logic (possibly with a finite number of axioms beyond the usual axioms of first-order logic) and answers "Yes" or "No" according to whether the statement is universally valid, i.e., valid in every structure satisfying the axioms. By the completeness theorem of first-order logic, a statement is universally valid if and only if it can be deduced from the axioms, so the Entscheidungsproblem can also be viewed as asking for an algorithm to decide whether a given statement is provable from the axioms using the rules of logic.
In 1936, Alonzo Church and Alan Turing, respectively, published independent papers showing that a general solution to the Entscheidungsproblem is impossible. This result is now known as Church's Theorem or the Church–Turing Theorem (not to be confused with the Church–Turing thesis).
History of the problem 
The origin of the Entscheidungsproblem goes back to Gottfried Leibniz, who in the seventeenth century, after having constructed a successful mechanical calculating machine, dreamt of building a machine that could manipulate symbols in order to determine the truth values of mathematical statements. He realized that the first step would have to be a clean formal language, and much of his subsequent work was directed towards that goal. In 1928, David Hilbert and Wilhelm Ackermann posed the question in the form outlined above.
In continuation of his "program" with which he challenged the mathematics community in 1900, at a 1928 international conference David Hilbert asked three questions, the third of which became known as "Hilbert's Entscheidungsproblem". As late as 1930 he believed that there would be no such thing as an unsolvable problem.
Negative answer 
Before the question could be answered, the notion of "algorithm" had to be formally defined. This was done by Alonzo Church in 1936 with the concept of "effective calculability" based on his λ calculus and by Alan Turing in the same year with his concept of Turing machines. It was recognized immediately by Turing that these are equivalent models of computation.
The negative answer to the Entscheidungsproblem was then given by Alonzo Church in 1935–36 and independently shortly thereafter by Alan Turing in 1936. Church proved that there is no computable function which decides for two given λ calculus expressions whether they are equivalent or not. He relied heavily on earlier work by Stephen Kleene. Turing reduced the halting problem for Turing machines to the Entscheidungsproblem. The work of both authors was heavily influenced by Kurt Gödel's earlier work on his incompleteness theorem, especially by the method of assigning numbers (a Gödel numbering) to logical formulas in order to reduce logic to arithmetic.
Turing's argument is as follows. Suppose that we had a general decision algorithm for statements in a first-order language. The question whether a given Turing machine halts or not can be formulated as a first-order statement, which would then be susceptible to the decision algorithm. But Turing had proven earlier that no general algorithm can decide whether a given Turing machine halts.
The Entscheidungsproblem is related to Hilbert's tenth problem, which asks for an algorithm to decide whether Diophantine equations have a solution. The non-existence of such an algorithm, established by Yuri Matiyasevich in 1970, also implies a negative answer to the Entscheidungsproblem.
Some first-order theories are algorithmically decidable; examples of this include Presburger arithmetic, real closed fields and static type systems of many programming languages. The general first-order theory of the natural numbers expressed in Peano's axioms cannot be decided with such an algorithm, however.
See also 
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- Church's paper was presented to the American Mathematical Society on 19 April 1935 and published on 15 April 1936. Turing, who had made substantial progress in writing up his own results, was disappointed to learn of Church's proof upon its publication (see correspondence between Max Newman and Church in Alonzo Church papers). Turing quickly completed his paper and rushed it to publication; it was received by the Proceedings of the London Mathematical Society on 28 May 1936, read on 12 November 1936, and published in series 2, volume 42 (1936-7); it appeared in two sections: in Part 3 (pages 230-240), issued on Nov 30, 1936 and in Part 4 (pages 241-265), issued on Dec 23, 1936; Turing added corrections in volume 43(1937) pp. 544–546. See the footnote at the end of Soare:1996.
- Davis 2000: pp. 3–20
- Hodges p. 91
- Hodges p. 92, quoting from Hilbert
- Alonzo Church, "An unsolvable problem of elementary number theory", American Journal of Mathematics, 58 (1936), pp 345–363
- Alonzo Church, "A note on the Entscheidungsproblem", Journal of Symbolic Logic, 1 (1936), pp 40–41.
- Martin Davis, 2000, Engines of Logic, W.W. Norton & Company, London, ISBN 0-393-32229-7 pbk.
- Alan Turing, "On computable numbers, with an application to the Entscheidungsproblem", Proceedings of the London Mathematical Society, Series 2, 42 (1936-7), pp 230–265. Online versions: from journal website, from Turing Digital Archive, from abelard.org. Errata appeared in Series 2, 43 (1937), pp 544–546.
- Martin Davis, "The Undecidable, Basic Papers on Undecidable Propositions, Unsolvable Problems And Computable Functions", Raven Press, New York, 1965. Turing's paper is #3 in this volume. Papers include those by Godel, Church, Rosser, Kleene, and Post.
- Andrew Hodges, Alan Turing: The Enigma, Simon and Schuster, New York, 1983. Alan M. Turing's biography. Cf Chapter "The Spirit of Truth" for a history leading to, and a discussion of, his proof.
- Robert Soare, "Computability and recursion", Bull. Symbolic Logic 2 (1996), no. 3, 284–321.
- Toulmin, Stephen, "Fall of a Genius", a book review of "Alan Turing: The Enigma by Andrew Hodges", in The New York Review of Books, 19 January 1984, p. 3ff.
- Alfred North Whitehead and Bertrand Russell, Principia Mathematica to *56, Cambridge at the University Press, 1962. Re: the problem of paradoxes, the authors discuss the problem of a set not be an object in any of its "determining functions", in particular "Introduction, Chap. 1 p. 24 "...difficulties which arise in formal logic", and Chap. 2.I. "The Vicious-Circle Principle" p. 37ff, and Chap. 2.VIII. "The Contradictions" p. 60 ff.