We investigate regular realizability (RR) problems, which are the problems of verifying whether intersection of a regular language -the input of the problem -and fixed language called filter is non-empty. We consider two kind of problems depending on representation of regular language. If a regular language on input is represented by a DFA, then we obtain (deterministic) regular realizability problem and we show that in this case the complexity of regular realizability problem for an arbitrary regular filter is either L-complete or NL-complete. We also show that in case of representation regular language on input by NFA the problem is always NL-complete.

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We consider a computational model which is known as set automata. The set automata are one-way finite automata with an additional storage the set. There are two kinds of set automata the deterministic and the nondeterministic ones. We denote them as DSA and NSA respectively. The model was introduced by M. Kutrib, A. Malcher, M. Wendlandt in 2014 in [3] and [4]. It was shown that DSA-languages look similar to DCFL due to their closure properties and NSA-languages look similar to CFL due to their undecidability properties. In this paper, which is an extended version of the conference paper [10], we show that this similarity is natural: we prove that languages recognizable by NSA form a rational cone, so as CFL. The main topic of this paper is computational complexity: we prove that -languages recognizable by DSA belong to P and there are P-complete languages among them; -languages recognizable by NSA are in NP and there are NP-complete languages among them; -the word membership problem is P-complete for DSA without ε-loops and PSPACE-complete for general DSA; -the emptiness problem is in PSPACE for NSA and, moreover, it is PSPACE-complete for DSA.