Regular realizability problems and models of a generalized   nondeterminism
  Models of a generalized nondeterminism are defined by limitations on nonde- terministic 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 consist- ing 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. 
A. Rubtsov, M. Vyalyi
cs.FL cs.DM