In our study of unsolvable problems, we associated a set with each problem. In essence, we discussed only problems of the form “does x have a certain property.” By coding programs onto IN, many natural problems in computer science can be represented as sets. From our study of abstract complexity we know that there are sets (problems) for which deciding membership is arbitrarily difficult. A problem is “complete” for some class if it is the “hardest” problem in the class. If you have some class of sets and a complete problem for the set, then that complete problem embodies all that is difficult about any problem in the class. For example, we will show that K is complete for r.e. sets.
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