On complexity of regular realizability problems

A regular realizability (RR) problem is testing nonemptiness of intersection of some fixed language (filter) with a regular language. We show that RR problems are universal in the following sense. For any language L there exists RR problem equivalent to L under disjunctive reductions on nondeterministic log space. We conclude from this result an existence of complete problems under polynomial reductions for many complexity classes including all classes of the polynomial hierarchy.