On the power of polynomial time bit-reductions

For a nondeterministic polynomial-time Turing machine M and an input string x, the leaf string of M on x is the 0-1-sequence of leaf-values (0~ reject, 1~ accept) of the computation tree of M with input x. The set A is said to be bit-reducible to B if there exists and M as above such that every input x is in A if and only if the leaf string of M on x is in B. A class C is definable via leaf language B, if C is the class of all languages that are bit-reducible to B. The question of how complex a leaf language must be in order to characterize some given class C is investigated. This question leads to the examination of the closure of different language classes under bit-reducibility. The question is settled for subclasses of regular languages, context free languages, and a number of time and space bounded classes, resulting in a number of surprising characterizations for PSPACE