Entscheidungsproblem

In mathematics and computer science, the Entscheidungsproblem (pronounced , German for 'decision problem') is a challenge posed by David Hilbert and Wilhelm Ackermann in 1928. The problem 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 mathematics and computer science, the Entscheidungsproblem (pronounced , German for 'decision problem') is a challenge posed by David Hilbert and Wilhelm Ackermann in 1928. The problem 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 published independent papers 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. 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', Hilbert posed three questions at an international conference in 1928, the third of which became known as 'Hilbert's Entscheidungsproblem.' In 1929, Moses Schönfinkel published one paper on special cases of the decision problem, that was prepared by Paul Bernays. As late as 1930, Hilbert believed that there would be no such thing as an unsolvable problem. 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. 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 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 question of the existence of a 'general method' which decides whether any given Turing Machine halts or not (the halting problem) to the question of the existence of an 'algorithm' or 'general method' able to solve the Entscheidungsproblem. If 'Algorithm' is understood as being equivalent to a Turing Machine, and with the answer to the latter question negative (in general), the question about the existence of an Algorithm for the Entscheidungsproblem also must be negative (in general). In his 1936 paper, Turing says: 'Corresponding to each computing machine 'it' we construct a formula 'Un(it)' and we show that, if there is a general method for determining whether 'Un(it)' is provable, then there is a general method for determining whether 'it' ever prints 0'. 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 Yuri Matiyasevich in 1970, also implies a negative answer to the Entscheidungsproblem.