Solving systems of explicit language relations

Abstract

While systems of language equations have been studied in various contexts, the corresponding problems for general relations between languages have not received much attention. This paper examines relations where the operations involved are unrestricted union and concatenation from the left by a constant. In the case of equations, this defines the classical ones provided each variable has exactly one equation. Since these equations express variables, they will be called explicit. For single explicit relations, we solve the problem of whether there exists a solution, study how to find all solutions, and investigate adequate representations for the solutions. Then we focus on systems of several explicit relations; the questions here are significantly more complicated since for a single variable X, there may be three types of relations, namely X = LX ∪ M , X ⊇ LX ∪ M , X⊆ LX ∪ M We concentrate on decoupled systems; these are systems where for each variable, at most one of the three types may occur (although different variables of the system may occur in relations of different types). Nevertheless, a single variable may have several relations of the same type. We give methods to answer the questions whether there exists a solution, whether there is more than one solution, and how to represent these solutions.