| 31. | Hence, the halting problem is " undecidable " for Turing machines.
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| 32. | There are two distinct senses of the word " undecidable " in contemporary use.
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| 33. | It is even algorithmically undecidable whether a given finite presentation defines a nontrivial group.
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| 34. | :It's an undecidable problem, so no such algorithm can exist.
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| 35. | As with many undecidable questions, one can still attempt to give useful approximate solutions.
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| 36. | However the halting problem is provably undecidable and so such an algorithm does not exist.
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| 37. | An application of proof theory is to show that certain undecidable statements are not provable.
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| 38. | Later showed that the Whitehead problem remains undecidable even if one assumes the Continuum hypothesis.
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| 39. | However, type inference in System F ( without explicit type annotations ) is undecidable.
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| 40. | Rice's theorem says " any nontrivial property of partial recursive functions is undecidable"
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