Complement (complexity)

Article summary

In computational complexity theory, the complement of a decision problem is the decision problem resulting from reversing the yes and no answers. Equivalently, if we define decision problems as sets of finite strings, then the complement of this set over some fixed domain is its complement problem. For example, one important problem is whether a number is a prime number. Its complement is to determine whether a number is a composite number (a number that is not prime). Here the domain of both the original problem and of the complement is the set of all natural numbers. There is a Turing reduction from every problem to its complement problem. The complement operation is an involution, meaning it "undoes itself", or the complement of the complement is the original problem. One can generalize this to the complement of a complexity class, called the complement class, which is the set of complements of every problem in the class. If a class is called C, its complement is conventionally labelled co-C. Notice that this is not the complement of the complexity class itself as a set of problems, which would generally contain a great deal more problems. A class is said to be closed under complement if the complement of any problem in the class is still in the class.…

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