The Boolean Closure of Linear Context-Free Languages

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Abstract

Closures of linear context-free languages under Boolean operations are investigated. The intersection closure and the complementation closure are incomparable. By closing these closures under further Boolean operations we obtain several new language families. The hierarchy obtained by such closures of closures is proper up to level four, where it collapses to the Boolean closure which, in turn, is incomparable with several closures of the family of context-free languages. The Boolean closure of the linear context-free languages is properly contained in the Boolean closure of the context-free languages. A characterization of a class of non-unary languages that cannot be expressed as a Boolean formula over the linear context-free languages is presented.