Oracle Pushdown Automata, Nondeterministic Reducibilities, and the Hierarchy over the Family of Context-Free Languages

We impose various oracle mechanisms on nondeterministic pushdown automata, which naturally induce nondeterministic reducibilities among formal languages in a theory of context-free languages. In particular, we examine a notion of nondeterministic many-one CFL reducibility and conduct a ground work to formulate a coherent framework for further expositions. Two more powerful reducibilities— bounded truth-table and Turing CFL-reducibilities—are also discussed in comparison. The Turing CFLreducibility, in particular, makes it possible to induce a useful hierarchy built over the family CFL of context-free languages. Basic structural properties are proven for each level of this CFL hierarchy. The first and second levels of the hierarchy are proven to be different. The rest of the hierarchy (more strongly, the Boolean hierarchy built over each level of the CFL hierarchy) is also infinite unless the polynomial hierarchy over NP collapses. This follows from a characterization of the Boolean hierarchy over the kth level of the polynomial hierarchy in terms of the Boolean hierarchy over the k+1st level of the CFL hierarchy. Similarly, the complexity class ΘPk is related to the kth level of the CFL hierarchy. We argue that the CFL hierarchy coincides with a hierarchy over CFL built by application of many-one CFL-reductions. We show that BPCFL—a bounded-error probabilistic version of CFL—is not included in CFL even in the presence of advice. Moreover, we exhibit a relativized world where BPCFL is not located within the second level of the CFL hierarchy.