Explicit Equations over a One-Letter Alphabet

Abstract

We restrict the underlying alphabet to one letter, denoted by a. Languages over {a} have interesting properties, some of which we derive. Then, we use them to show that any system of explicit equations over the alphabet {a} where the operators are union, unrestricted concatenation, and star have solutions that are expressible as regular expressions in terms of the constant languages only. Most importantly, we provide a complete solution of the problem, including a parametric representation of all solutions if there is more than one. We also study the case (in Section 7.7) where complementation is added to this catalog of operators and show that such equations need not have context-free solutions even if the constants are single letters.