If one knew the first n bits of Chaitin's problem, then one can solve this problem for all inputs of length less than n. The problem itself is recognizable, but its complement is not.
If one knew the first n bits of Chaitin's problem, then one can solve this problem for all inputs of length less than n. The problem itself is recognizable, but its complement is not. A corollary of Rice's theorem concerning proper, non-empty subsets of partial functions can be used to prove that it's undecidable, but the usual argument is to consider a candidate Turing machine that computes the problem M and that, when called with the input M goes into an infinite loop if M halts. For ten points, name this first known undecidable problem, the question of whether, for given input, a given algorithm will complete or run forever.
If one knew the first n bits of Chaitin's problem, then one can solve this problem for all inputs of length less than n. The problem itself is recognizable, but its complement is not.
Halting problem