Superpolynomial circuits, almost sparse oracles and the exponential hierarchy

Abstract

Several problems concerning superpolynomial size circuits and superpolynomial-time advice classes are investigated. First we consider the implications of PIP (and other fundamental complexity classes) having circuits slighter bigger than polynomial. We prove that if such circuits exist, for example if NP has nlogn size circuits, the exponential hierarchy collapses to the second level. Next we consider the consequences of the bottom levels of the exponential hierarchy being contained in small advice classes. Again various collapses result. For example, if EXPNP ⊆ EXP/poly then EXPNP = EXP.