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Regular Ideal Languages and Their Boolean Combinations

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Implementation and Application of Automata (CIAA 2012)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 7381))

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Abstract

We consider ideals and Boolean combinations of ideals. For the regular languages within these classes we give expressively complete automaton models. In addition, we consider general properties of regular ideals and their Boolean combinations. These properties include effective algebraic characterizations and lattice identities.

In the main part of this paper we consider the following deterministic one-way automaton models: unions of flip automata, weak automata, and Staiger-Wagner automata. We show that each of these models is expressively complete for regular Boolean combination of right ideals. Right ideals over finite words resemble the open sets in the Cantor topology over infinite words. An omega-regular language is a Boolean combination of open sets if and only if it is recognizable by a deterministic Staiger- Wagner automaton; and our result can be seen as a finitary version of this classical theorem. In addition, we also consider the canonical automaton models for right ideals, prefix-closed languages, and factorial languages.

In the last section, we consider a two-way automaton model which is known to be expressively complete for two-variable first-order logic. We show that the above concepts can be adapted to these two-way automata such that the resulting languages are the right ideals (resp. prefix-closed languages, resp. Boolean combinations of right ideals) definable in twovariable first-order logic.

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Author information

Authors and Affiliations

  1. FMI, University of Stuttgart, Germany

    Franz Jahn, Manfred Kufleitner & Alexander Lauser

Authors
  1. Franz Jahn
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  2. Manfred Kufleitner
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  3. Alexander Lauser
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Editors and Affiliations

  1. Departamento de Ciência de Computadores, Universidade do Porto, Faculdade de Ciências, Rua do Campo Alegre 1021-1055, 4169-007, Porto, Portugal

    Nelma Moreira  & Rogério Reis  & 

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Jahn, F., Kufleitner, M., Lauser, A. (2012). Regular Ideal Languages and Their Boolean Combinations. In: Moreira, N., Reis, R. (eds) Implementation and Application of Automata. CIAA 2012. Lecture Notes in Computer Science, vol 7381. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-31606-7_18

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  • DOI: https://doi.org/10.1007/978-3-642-31606-7_18

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-31605-0

  • Online ISBN: 978-3-642-31606-7

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