Regular realizability problems and models of a generalized nondeterminism

Models of a generalized nondeterminism are defined by limitations on nondeterministic behavior of a computing device. A regular realizability problem is a problem of verifying existence of a special sort word in a regular language. These notions are closely connected. In this paper we consider regular realizability problems for languages consisting of all prefixes of an infinite word. These problems are related to the automata on infinite words and to the decidability of monadic second-order theories. The main contribution is a new decidability condition for regular realizability problems and for monadic-second order theories. We also show that decidability of a regular realizability problem is equivalent to decidability of some prefix realizability problem.