In mathematics and computer science, the Entscheidungsproblem (pronounced [ɛntˈʃaɪ̯dʊŋspʁoˌbleːm], 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 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.“ As late as 1930, he 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 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.
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.
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 technique, 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 Tarski–Seidenberg theorem, which has been implemented in computers by using the cylindrical algebraic decomposition.