Implicit language equations: existence and uniqueness of solutions

AbstractWe introduce a new type of language equation, namely implicit equations, and derive a number of properties which clearly demonstrate that the behavior of implicit equations differs substantially from that of conventional or explicit ones. Explicit language equations express each variable as a regular expression in terms of their variables and constant languages; implicit language equations express constant languages as regular expressions in terms of their variables and constant languages. An algorithm is given for determining a solution of a given system of implicit language equations, if one exists, provided all constant languages are regular and the operations permitted in the regular expressions are unrestricted union and concatenation from the left by a constant. A characterization of the uniqueness of a solution of such a system and a method for determining all solutions are also provided