Language equations with complementation: Decision problems

Systems of language equations of the form are studied. Here every φi may contain the operations of concatenation and complementation. The properties of having solutions and of having a unique solution are given mathematical characterizations. As decision problems, the former is NP-complete, while the latter is PSPACE-hard and is in co-RE, and its decidability remains, in general, open. Uniqueness becomes decidable in the case of a unary alphabet, where it is US-complete, and in the case of linear concatenation, where it is L-complete.