Explicit Relations with Union and Left-Concatenation

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. In this chapter, we examine relations where the operations involved are unrestricted union and left-concatenation; in other words, precisely the operations involved in classical equations. Note that for classical equations, each variable is equated to exactly one expression. In this chapter, relations also include the case of several equations for the same variable. First, we look at single explicit relations and resolve the question 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 equality (=), superrelation (⊒), and subrelation (⊑). We consider, first, decoupled systems, where for each variable, at most one of the three relation types may occur. We study methods to answer the questions of whether there exists a solution, whether there is more than one solution, and how to represent these solutions.