Language Equations with Complementation
Systems of language equations of the form X i =ϕ i (X 1, ..., X n ) (\(1 \leqslant i \leqslant n\)) are studied, in which every ϕ i 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 in co-RE and its decidability remains, in general, open. Uniqueness becomes decidable for a unary alphabet, where it is US-complete, and in the case of linear concatenation, where it is L-complete. The position of the languages defined by these equations in the hierarchy of language families is established.
KeywordsUnique Solution Boolean Operation Regular Language Language Family Solution Existence
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