HIERARCHIES OVER THE CONTEXT-FREE LANGUAGES

Alternation is a generalized principle of nondeterminism. The alternating turing machine is used to characterize the polynomial hierarchy. In this paper we show, that a hierarchy can be characterized with alternating pushdown automata, which we expect to be strict in contrast to a hierarchy with alternating finite automata or alternating space bounded automata. We describe a similar oracle hierarchy over the context-free languages, for which we construct complete languages. We show, that each level of the hierarchy with alternating pushdown automata is included in the corresponding level of the oracle hierarchy and that the logarithmic closure over both levels is the corresponding level of the polynomial hierarchy with one alternation less. The principle of the alternation is also transfered to grammars. Hereby we prove, that the hierarchy with alternating context-free grammars is identical with the oracle hierarchy over the context-free languages and that in case of unbounded alternation context-free and context-sensitive grammars have the same power.