A note on algebras of languages

• Published in: Theoretical computer science Vol. 412; no. 46; pp. 6531 - 6536
• Main Authors: Marini, Claudio, Simi, Giulia, Sorbi, Andrea, Sorrentino, Marianna
• Format: Journal Article
• Language: English
• Published: Oxford Elsevier B.V 28.10.2011; Elsevier
• Subjects: Algebra; Algebras of languages; Alphabets; Applied sciences; Boolean algebra; Boolean algebras; Combinatorics. Ordered structures; Computer science; control theory; systems; Context-sensitive languages; Decidable languages; Exact sciences and technology; Fréchet ideal; Immunity; Inclusions; Mathematical analysis; Mathematics; Miscellaneous; Order, lattices, ordered algebraic structures; Regular languages; Sciences and techniques of general use; Theoretical computing; Regular languages; Algebras of languages; Fréchet ideal; Decidable languages; Context-sensitive languages; Boolean algebras; Alphabet; Context sensitive language; Inclusion; Computer theory; Fréchet differentiability; Ideal; Regular language; Boolean algebra; Immunity
• ISSN: 0304-3975; 1879-2294
• DOI: 10.1016/j.tcs.2011.08.022

We study the Boolean algebras R,CS,D of regular languages, context-sensitive languages and decidable languages, respectively, over any alphabet. It is well known that R⊂CS⊂D, with proper inclusions. After observing that these Boolean algebras are all isomorphic, we study some immunity properties: for instance we prove that for every coinfinite decidable language L there exists a decidable language L′ such that L⊆L′, L′−L is infinite, and there is no context-sensitive language L″, with L″⊆L′ unless L″−L is finite; similarly, for every coinfinite regular language L there exists a context-sensitive language L′ such that L⊆L′, L′−L is infinite, and there is no regular language L″ such that L″⊆L′, unless L″−L is finite.