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Concept
Undecidable problem
Summary
In computability theory and computational complexity theory, an undecidable problem is a decision problem for which it is proved to be impossible to construct an algorithm that always leads to a correct yes-or-no answer. The halting problem is an example: it can be proven that there is no algorithm that correctly determines whether arbitrary programs eventually halt when run.
A decision problem is a question which, for every input in some infinite set of inputs, answers "yes" or "no".. Those inputs can be numbers (for example, the decision problem "is the input a prime number?") or other values of some other kind, such as strings of a formal language.
The formal representation of a decision problem is a subset of the natural numbers. For decision problems on natural numbers, the set consists of those numbers that the decision problem answers "yes" to. For example, the decision problem "is the input even?" is formalized as the set of even numbers. A decision problem whose input consists of strings or more complex values is formalized as the set of numbers that, via a specific Gödel numbering, correspond to inputs that satisfy the decision problem's criteria.
A decision problem A is called decidable or effectively solvable if the formalized set of A is a recursive set. Otherwise, A is called undecidable. A problem is called partially decidable, semi-decidable, solvable, or provable if A is a recursively enumerable set.
In computability theory, the halting problem is a decision problem which can be stated as follows:
Given the description of an arbitrary program and a finite input, decide whether the program finishes running or will run forever.
Alan Turing proved in 1936 that a general algorithm running on a Turing machine that solves the halting problem for all possible program-input pairs necessarily cannot exist. Hence, the halting problem is undecidable for Turing machines.
The concepts raised by Gödel's incompleteness theorems are very similar to those raised by the halting problem, and the proofs are quite similar.
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