Boolean Language Equations

Abstract

We define boolean language equations and relate them to boolean automata. Boolean language equations are explicit equations over an arbitrary alphabet in which union, left-concatenation, and complementation may occur. Boolean automata are a generalization of nondeterministic finite automata that are able to accommodate, in a natural way, the complementation operation. We first show that not every boolean equation has a solution. Then, we give a constructive approach to solving any system of boolean equations that has a solution. We also give a complete characterization of the existence of solutions, decide whether more than one solution exists, give a representation of all solutions if more than one exists, and show that any boolean equation whose constant languages are regular must have a regular solution, if it has any.