Theory of Computing Systems

, Volume 60, Issue 3, pp 438–472 | Cite as

On Boolean Closed Full Trios and Rational Kripke Frames

  • Georg ZetzscheEmail author
  • Dietrich Kuske
  • Markus Lohrey


We study what languages can be constructed from a non-regular language L using Boolean operations and synchronous or non-synchronous rational transductions. If all rational transductions are allowed, one can construct the whole arithmetical hierarchy relative to L. In the case of synchronous rational transductions, we present non-regular languages that allow constructing languages arbitrarily high in the arithmetical hierarchy and we present non-regular languages that allow constructing only recursive languages. A consequence of the results is that aside from the regular languages, no full trio generated by a single language is closed under complementation. Another consequence is that there is a fixed rational Kripke frame such that assigning an arbitrary non-regular language to some variable allows the definition of any language from the arithmetical hierarchy in the corresponding Kripke structure using multimodal logic.


Regular languages Closure properties Trio operations Boolean operations Synchronous rational transductions Arithmetical hierarchy Rational Kripke structures 


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Authors and Affiliations

  1. 1.Fachbereich InformatikTechnische Universität KaiserslauternKaiserslauternGermany
  2. 2.Institut für Theoretische InformatikTechnische Universität IlmenauIlmenauGermany
  3. 3.Department für Elektrotechnik und InformatikUniversität SiegenSiegenGermany

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