Regular Realizability Problems and Context-Free Languages

We investigate regular realizability (RR) problems, which are the problems of verifying whether the intersection of a regular language – the input of the problem – and a fixed language, called a filter, is non-empty. In this paper we focus on the case of context-free filters. The algorithmic complexity of the RR problem is a very coarse measure of the complexity of context-free languages. This characteristic respects the rational dominance relation. We show that a RR problem for a maximal filter under the rational dominance relation is P-complete. On the other hand, we present an example of a P-complete RR problem for a non-maximal filter. We show that RR problems for Greibach languages belong to the class NL. We also discuss RR problems with context-free filters that might have intermediate complexity. Possible candidates are the languages with polynomially-bounded rational indices. We show that RR problems for these filters lie in the class NSPACE(log 2 n).