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

Abstract: To expand a fundamental theory of context-free languages, we equip nondeterministic one-way pushdown automata with additional oracle mechanisms, which naturally induce various nondeterministic reducibilities among formal languages. As a natural restriction of NP-reducibility, we introduce a notion of 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 CFL-reducibility, in particular, helps us introduce an exquisite hierarchy, called the CFL hierarchy, built over the family CFL of context-free languages. For each level of this hierarchy, its basic structural properties are proven and three alternative characterizations are presented. The second level is not included in NC(2) unless NP= NC(2). The first and second levels of the hierarchy are different. The rest of the hierarchy (more strongly, the Boolean hierarchy built over each level of the hierarchy) is also infinite unless the polynomial hierarchy over NP collapses. This follows from a characterization of the Boolean hierarchy over the k-th level of the polynomial hierarchy in terms of the Boolean hierarchy over the k+1st level of the CFL hierarchy using log-space many-one reductions. Similarly, the complexity class Theta(k) is related to the closure of the k-th level of the CFL hierarchy under log-space truth-table reductions. We also 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. Employing a known circuit lower bound and a switching lemma, we exhibit a relativized world where BPCFL is not located within the second level of the CFL hierarchy.