In
mathematics and
computer science, the (,
German for 'decision problem') is a challenge posed by
David Hilbert in 1928. The 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 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.