Random Access to Advice Strings and Collapsing Results

We propose a model of computation where a Turing machine is given random access to an advice string. With random access, an advice string of exponential length becomes meaningful for polynomially bounded complexity classes. We compare the power of complexity classes under this model. It gives a more stringent notion than the usual model of computation with relativization. Under this model of random access, we prove that there exist advice strings such that the Polynomial-time Hierarchy PH and Parity Polynomial-time all collapse to P. Our main proof technique uses the decision tree lower bounds for constant depth circuits [Yao85, Cai86, Has86], and the algebraic machinery of Razborov and Smolensky [Raz87, Smo87]