BPP and the Polynomial Hierarchy

We start with some definitions about probabilistic Turing machines (for a detailed exposition of this subject the reader is referred to [2]). Our basic model of computation is the Turing machine with input alphabet S z (0, l}. All machines are assumed to have two terminal states, namely q0 (reject) and 9, (accept). A probabilistic Turing machine (PTM) is syntactically the same as a nondeterministic Turing machine. Without loss of generality we can assume that in each step the machine has exactly two (at least formally) different possible moves. To each computation of length t we assign the probability 2-l. Let M(x) = y (y E (0, 1)) denote the event that, on input x, M eventually halts in state qy. The set decided by M, L,, is then defined by