Discuss at length the assertion that Turing machines define the limits of computability.
Fill in the blank(s) with the appropriate word(s).
Turing machines define the limits of computability—that which can be done by symbol manipulation algorithms. What can be done by an algorithm is doable by a Turing machine, and what is not doable by a Turing machine cannot be done by an algorithm. In particular, if we find a symbol manipulation task that no Turing machine can perform (in its elementary way of moving around over a tape of 0s and 1s), then there is no algorithm for this task, and no real computer, no matter how sophisticated, will ever be able to do it either. That’s why the Turing machine is so important. In the search for a problem that has no algorithmic solution, the implications are that if we can find a (symbol manipulation) problem for which we can prove that no Turing machine exists to solve it, then, because of the Church–Turing thesis, no algorithm exists to solve it either. The problem is an uncomputable or unsolvable problem.
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Fill in the blank(s) with the appropriate word(s).
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