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In computability theory and computational complexity theory, an undecidable problem is a decision problem for which it is known to be impossible to construct a single algorithm that always leads to a correct yes-or-no answer. A decision problem is any arbitrary yes-or-no question on an infinite set of inputs. Because of this, it is traditional to define the decision problem equivalently as the set of inputs for which the problem returns ''yes''. These inputs can be natural numbers, but also other values of some other kind, such as strings of a formal language. Using some encoding, such as a Gödel numbering, the strings can be encoded as natural numbers. Thus, a decision problem informally phrased in terms of a formal language is also equivalent to a set of natural numbers. To keep the formal definition simple, it is phrased in terms of subsets of the natural numbers. Formally, a decision problem is a subset of the natural numbers. The corresponding informal problem is that of deciding whether a given number is in the set. A decision problem ''A'' is called decidable or effectively solvable if ''A'' is a recursive set. A problem is called partially decidable, semi-decidable, solvable, or provable if ''A'' is a recursively enumerable set. This means that there exists an algorithm that halts eventually when the answer is ''yes'' but may run for ever if the answer is ''no''. Partially decidable problems and any other problems that are not decidable are called undecidable. ==In computability theory== 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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「undecidable problem」の詳細全文を読む スポンサード リンク
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