Elements of the Theory of Computation
Elements of the Theory of Computation
Programming Approach to Computability
Programming Approach to Computability
Theory of Computation
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One of this century's major intellectual discoveries is that there are some perfectly precise problems that can never be solved. This is a technical result, not a mystical one. There is nothing ineffable about these problems; indeed, one can give specific examples of them. Nor should this result be confused with the claim that some problems (e.g. what is the meaning of life?) are too imprecise to be solved computationally. Actually, these unsolvable problems are just as precise as the problem of computing a sum. Also, the result is not based on the fact that some problems are just too large to solve in practical terms. Rather, these problems are unsolvable even under the assumption that wholly unreasonable amounts of time and space are available for their solution. By way of contrast, it is a trivial matter under such an assumption to write a program that plays a perfect game of chess: just explore all of the possibilities systematically. (There are only a finite number of possibilities, but that number is so astronomically large that this exhaustive approach will never be of practical use.)