On regular realizability problems

We consider the class of regular realizability problems. Any of such problems is specified by some language (filter) and consists in verifying that the intersection of a given regular language and the filter is nonempty. The main question is diversity of the computational complexity of such problems. We show that any regular realizability problem with an infinite filter is hard for a class of problems decidable in logarithmic space with respect to logarithmic reductions. We give examples of NP-complete and PSPACE-complete regular realizability problems.