The maximum value problem and NP real numbers

摘要

The concept of nondeterministic computation has been playing an important role in discrete complexity theory. In this paper the concept of nondeterminism is applied to a numerical problem—finding the maximum value of a polynomial time computable real function. The class of all these maximum values is characterized as the class of nondeterministic polynomial time (NP) computable real numbers. The completeness of real numbers is then investigated. The result that NP real numbers cannot be polynomial time many-one complete in NP (unless P = NP) shows a basic difference between the maximum value problem and many natural NP combinatorial problems. It is also shown that real numbers are not complete in r.e. sets or PSPACE (unless P = PSPACE) and this seems to be a general phenomenon. Finally, the relationship between NP real numbers and NP sets over a single-letter alphabet and the existence of hardest NP real numbers are also discussed.