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In mathematics and computer science, the *Entscheidungsproblem* (pronounced [ɛntˈʃaɪ̯dʊŋspʁoˌbleːm], German for "decision problem") is a challenge posed by David Hilbert and Wilhelm Ackermann in 1928.

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 published independent papers^{[2]} showing that a general solution to the * Entscheidungsproblem* is impossible, assuming that the intuitive notion of "effectively calculable" is captured by the functions computable by a Turing machine (or equivalently, by those expressible in the lambda calculus). This assumption is now known as the Church–Turing thesis.

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.

In continuation of his "program", Hilbert posed three questions at an international conference in 1928, the third of which became known as "Hilbert's * Entscheidungsproblem*."

As late as 1930, Hilbert believed that there would be no such thing as an unsolvable problem.^{[6]}

Before the question could be answered, the notion of "algorithm" had to be formally defined. This was done by Alonzo Church in 1935 with the concept of "effective calculability" based on his λ-calculus and by Alan Turing the next year with his concept of Turing machines. Turing immediately recognized that these are equivalent models of computation.

The negative answer to the * Entscheidungsproblem* was then given by Alonzo Church in 1935–36 (

The work of both Church and Turing 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.

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 the work of Yuri Matiyasevich, Julia Robinson, Martin Davis, and Hilary Putnam, with the final piece of the proof 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 an algorithm, however.

Having practical decision procedures for classes of logical formulas is of considerable interest for program verification and circuit verification. Pure Boolean logical formulas are usually decided using SAT-solving techniques based on the DPLL algorithm. Conjunctive formulas over linear real or rational arithmetic can be decided using the simplex algorithm, formulas in linear integer arithmetic (Presburger arithmetic) can be decided using Cooper's algorithm or William Pugh's Omega test. Formulas with negations, conjunctions and disjunctions combine the difficulties of satisfiability testing with that of decision of conjunctions; they are generally decided nowadays using SMT-solving techniques, which combine SAT-solving with decision procedures for conjunctions and propagation techniques. Real polynomial arithmetic, also known as the theory of real closed fields, is decidable; this is the Tarski–Seidenberg theorem, which has been implemented in computers by using the cylindrical algebraic decomposition.

**^**David Hilbert and Wilhelm Ackermann. Grundzüge der Theoretischen Logik. Springer, Berlin, Germany, 1928. English translation: David Hilbert and Wilhelm Ackermann. Principles of Mathematical Logic. AMS Chelsea Publishing, Providence, Rhode Island, USA, 1950**^**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 30 Nov 1936 and in Part 4 (pages 241–265), issued on 23 Dec 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**^**Kline, G. L.; Anovskaa, S. A. (1951), "Review of Foundations of mathematics and mathematical logic by S. A. Yanovskaya",*Journal of Symbolic Logic*,**16**(1): 46–48, doi:10.2307/2268665, JSTOR 2268665**^**Hodges p. 92, quoting from Hilbert

- David Hilbert and Wilhelm Ackermann (1928).
*Grundzüge der theoretischen Logik*(*Principles of Mathematical Logic*). Springer-Verlag, ISBN 0-8218-2024-9. - 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 Gödel, 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.
- Stephen Toulmin, "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.

- The dictionary definition of
*entscheidungsproblem*at Wiktionary