Complexity classes of alternating machines with oracles

Abstract

Relativized complexity theory based on alternating Turing machines is considered. Alternating complexity classes are shown to provide natural counterexamples to the longstanding conjecture that known proofs of complexity class inclusion results relativize. In particular, there exist oracle sets separating classes APSPACE and E (of languages recognizable in alternating polynomial space and deterministic exponential time, respectively), although the classes are known to be equal in the unrelativized case. Classes APSPACE and E may even be shown to differ for almost all oracles, thus providing a counterexample also to the so called random oracle hypothesis. A complexity hierarchy built by quantifying over oracle sets for alternating machines is also defined, with its first ⌆-level coinciding with class NE (nondeterministic exponential time). This representation of NE is noted not to relativize. Problems about the structure of this second-order hierarchy are shown to be related to open problems concerning the polynomial-time hierarchy.