Simplicity, Immunity, Relativizations and Nondeterminism

The relation between classes of languages recognized by resource bounded Turing machines remains an evasive subject in computational complexity theory. Especially results settling relationship between deterministic and nondeterministic complexity classes are rare. The interesting case of the relation between the classes of languages recognized by polynomial time bounded deterministic versus polynomial time bounded nondeterministic Turing machines, known as the P vs NP problem has been bugging mathematicians all over the world for a long period of time. Though special subcases of the problem have been solved (see, e.g., [7, 8, 1 I]), the complexity of the solutions and the amount of time and effort spent already to solve the general problem seem to indicate that this will be an open problem for a long time to come An attempt to gain more insight in the relation between P and NP is the study of relativized complexity classes. Instead of studying the power of nondeterminism itself, what is studied is the extra power nondeterminism gives over determinism if, during the computation, questions of membership may be asked to a fixed oracle set. Baker, Gill, and Solovay [ l] found a recursive set A such that NP(A)# P(A) and set the stage for numerous separation results. Bennet and Gill [4] showed that this separation result could be strengthened when they proved the existence of an oracle A such that NP(A) has a P(A)-immune set. That is a set which is infinite and has no infinite subset in P(A). &honing and Book [13] found that this oracle could be made recursive. Homer and Maass [6] showed a similar result for the relation between NP(A) and co-NP(A) when they constructed a simple set in NP(A). Later Baldzar [3] showed that there exists a technique for building a recursive oracle in this case also,