Regulated nondeterminism in pushdown automata

A generalization of pushdown automata towards regulated nondeterminism is studied. The nondeterminism is governed in such a way that the decision, whether or not a nondeterministic rule is applied, depends on the whole content of the stack. More precisely, the content of the stack is considered as a word over the stack alphabet, and the pushdown automaton is allowed to act nondeterministically, if this word belongs to some given set R of control words. Otherwise its behavior is deterministic. The computational capacity of such R-PDAs depends on the complexity of R. It turns out that non-context-free languages are accepted even if R is a linear, deterministic context-free language. On the other hand, regular control sets R do not increase the computational capacity of nondeterministic pushdown automata. This raises the natural question for the relations between the structure and complexity of regular sets R on one hand and the computational capacity of the corresponding R-PDA on the other hand. Clearly, if R is empty, the deterministic context-free languages are characterized. For R = {a,b}* one obtains all context-free languages. Furthermore, if R is finite, then the regular closure of the deterministic context-free languages is described. We investigate these questions, and discuss closure properties of the language classes in question under AFL operations.