Implicit Equations with Union and Left-Concatenation

Abstract

We begin our coverage of implicit equations by considering the exact analogue to the classical (explicit) equations solved in Chapter 3. Specifically, the operations are union and left-concatenation and the constants are over an arbitrary alphabet. We first establish that, in contrast to explicit equations, implicit ones need not have any solutions. Furthermore, there exist implicit equations with context-free constants whose only solutions are non-context-free. Then we show a general criterion for the existence of a solution. This criterion is constructive if the constants are regular. Furthermore, the criterion gives rise to a test whether there exist finitely or infinitely many solutions. If the constants are regular, solutions can be constructed effectively; moreover, if there are finitely many solutions, all solutions can be constructed, and if there are infinitely many, an arbitrarily large number of them can be constructed.