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In computability theory, an undecidable problem is of a type of calculation which requires a yes/no answer, but where there can not possibly be any computer program that always gives the correct answer; that is any possible program would sometimes give the wrong answer or never give any answer at all. More formally, an undecidable problem is a problem whose language is not a recursive set; see decidability. There are uncountably many undecidable problems, so the list below is necessarily incomplete. Though undecidable languages are not recursive languages, they may be subsets of Turing recognizable languages i.e. such undecidable languages may be recursively enumerable. Many, if not most, undecidable problems in mathematics can be posed as word problems: determining when two distinct strings of symbols (encoding some mathematical concept or object) represent the same object or not. |
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