The Entscheidungsproblem (German: decision problem) is the challenge in symbolic logic to find a general algorithm which decides for given first-order statements whether they are universally valid or not. Alonzo Church and independently Alan Turing showed in 1936 that this is impossible. As a consequence, it is in particular impossible to algorithmically decide whether statements in arithmetic are true or false.
The question 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 Ackermann posed the question in the form outlined above.
A first-order statement is called "universally valid" or "logically valid" if it follows from the axioms of the first-order predicate calculus. Gödel's completeness theorem states that a statement is universally valid in this sense if and only if it is true in every interpretation of the formula in a model.
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 lambda calculus and by Alan Turing in the same year with his concept of Turing machines. The two approaches are equivalent, an instance of the Church-Turing thesis.
The negative answer to the Entscheidungsproblem was then given by Alonzo Church in 1936 and independently shortly thereafter by Alan Turing, also in 1936. Church proved that there is no algorithm (defined via recursive functions) which decides for two given lambda calculus expressions whether they are equivalent or not. He relied heavily on earlier work by Kleene. Turing reduced the problem to the halting problem for Turing machines and his paper is generally considered to be much more influential than Church's. 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 to logical formulas in order to reduce logic to arithmetic.
Turing's argument follows. Suppose we had a general decision algorithm for first order logic. 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 proved earlier that no general algorithm can decide whether a given Turing machine halts.
It is important to realize that if we restrict ourselves to a specific first-order theory with specified object constants, function constants, predicate constants and subject axioms, the truth of statements in that theory may very well be algorithmically decidable. An example of this is given by Presburger arithmetic.
- 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.
- Alan Turing, "On computable numbers, with an application to the Entscheidungsproblem", Proceedings of the London Mathematical Society, Series 2, 42 (1936), pp 230-265. Online version. Errata appeared in Series 2, 43 (1937), pp 544-546.