Abstract

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 \(\mathbf {P}\)-complete. On the other hand, we present an example of a \(\mathbf {P}\)-complete RR problem for a non-maximal filter. We show that RR problems for Greibach languages belong to the class \(\mathbf {NL}\). 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 \(\mathbf {NSPACE}(\log ^2 n)\).

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