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Around Dot Depth Two

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Developments in Language Theory (DLT 2010)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 6224))

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Abstract

It is known that the languages definable by formulae of the logics \(FO^{2}[<,S], \Delta_{2}[<,S], LTL[F,P,X,Y]\) are exactly the variety DA*D. Automata for this class are not known, nor is its precise placement within the dot-depth hierarchy of starfree languages. It is easy to argue that Δ2[ < ,S] is included in Δ3[ < ]; in this paper we show that it is incomparable with B(Σ2)[ < ], the boolean combination of Σ2[ < ] formulae. Using ideas from Straubing’s “delay theorem”, we extend our earlier work [LPS08] to propose partially-ordered two-way deterministic finite automata with look-around (po2dla) and a new interval temporal logic called LITL and show that they also characterize the variety DA*D. We give effective reductions from LITL to equivalent po2dla and from po2dla to equivalent FO 2[ < ,S]. The po2dla automata admit efficient operations of boolean closure and the language non-emptiness of po2dla is NP-complete. Using this, we show that satisfiability of LITL remains NP-complete assuming a fixed look-around length. (Recall that for LTL[F,X], it is Pspace-hard.)

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Authors and Affiliations

  1. The Institute of Mathematical Sciences, CIT Campus, Chennai, 600113, India

    Kamal Lodaya

  2. Tata Institute of Fundamental Research, Colaba, Mumbai, 400005, India

    Paritosh K. Pandya & Simoni S. Shah

Authors
  1. Kamal Lodaya
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  2. Paritosh K. Pandya
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  3. Simoni S. Shah
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Editor information

Editors and Affiliations

  1. Dept. of Computer Science, University of Western Ontario, N6A 5B7, London, ON, Canada

    Yuan Gao , Hanlin Lu , Shinnosuke Seki  & Sheng Yu , ,  & 

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Lodaya, K., Pandya, P.K., Shah, S.S. (2010). Around Dot Depth Two. In: Gao, Y., Lu, H., Seki, S., Yu, S. (eds) Developments in Language Theory. DLT 2010. Lecture Notes in Computer Science, vol 6224. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-14455-4_28

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  • DOI: https://doi.org/10.1007/978-3-642-14455-4_28

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-14454-7

  • Online ISBN: 978-3-642-14455-4

  • eBook Packages: Computer ScienceComputer Science (R0)

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