Unrestricted complementation in language equations over a one-letter alphabet

Abstract

Recently, the complete solution of systems of language equations over a one-letter alphabet has been derived where the operators are union, concatenation and star. This paper addresses the question of adding the complementation operator. It is known that if no stars are present and if the concatenation operator is restricted, such equations need not have solutions, but if they do, they can be solved explicitly if the constants are regular in which case the solutions are also regular. In this paper we study language equations over a one-letter alphabet where unrestricted concatenation interacts with complementation, union, and star. We define a procedure for determining a regular solution of the general type of equation assuming that all constants are regular, discuss its properties, and show examples. We then prove that in general equations with complementation and complementation and concatenation in which all constants are regular may have nonregular solutions. This shows that language equations can be used to obtain non-context-free languages from regular languages using only operators under which the regular languages are closed.