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A decidable or recursive language is a formal language that is a recursive set, i.e., for which there exists an algorithm to solve the following decision problem: Given string w, does w belong to the language? The algorithm is not allowed to run into an infinite loop and has to produce a YES/NO answer for any input string after a finite number of steps. To formalize the rather vague term "algorithm", one usually employs Turing machines, but several other equivalent approaches are possible.
All regular, context-free and context-sensitive languages are recursive, but there exist recursively enumerable languages that are not recursive; one example is given by the halting problem.
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