Certain problems that occur in the theory of computation and elsewhere do not have solutions. In those cases it is not that a solution cannot be found easily, nor even that a solution, if found, would be too hard, or perhaps take too long, to implement on a real existing computer. It is simply that there is no well-defined procedure guaranteed to answer the question "yes" or "no" in all cases. In logical terms, neither a well-formed statement nor its negation may be logically implied by a logical system, so it is possible that the algorithm may not halt on either a "yes" or a "no." Problems for which a solution exists, but where the solution cannot be efficiently implemented on existing computers, are called intractable problems. Those for which no general solution exists at all, are called undecidable problems. The study of undecidable problems--or, more generally, the...