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Strict Language Inequalities and Their Decision Problems

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Mathematical Foundations of Computer Science 2005 (MFCS 2005)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 3618))

Abstract

Systems of language equations of the form {ϕ(X 1, ..., X n ) = ∅, ψ(X 1, ..., X n )≠∅} are studied, where ϕ,ψ may contain set-theoretic operations and concatenation; they can be equivalently represented as strict inequalities ξ(X 1, ..., X n ) ⊂ L 0. It is proved that the problem whether such an inequality has a solution is Σ2-complete, the problem whether it has a unique solution is in (Σ3 ∩ Π3) ∖ (Σ2 ∪ Π2), the existence of a regular solution is a Σ1-complete problem, while testing whether there are finitely many solutions is Σ3-complete. The class of languages representable by their unique solutions is exactly the class of recursive sets, though a decision procedure cannot be algorithmically constructed out of an inequality, even if a proof of solution uniqueness is attached.

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Author information

Authors and Affiliations

  1. Department of Mathematics, University of Turku, Turku, FIN–20014, Finland

    Alexander Okhotin

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  1. Alexander Okhotin
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Editors and Affiliations

  1. Institute of Mathematics, Gdansk University, Wita Stwosza 57, 80 952, Gdansk, Poland

    Joanna Jȩdrzejowicz  & Andrzej Szepietowski  & 

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Okhotin, A. (2005). Strict Language Inequalities and Their Decision Problems. In: Jȩdrzejowicz, J., Szepietowski, A. (eds) Mathematical Foundations of Computer Science 2005. MFCS 2005. Lecture Notes in Computer Science, vol 3618. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11549345_61

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  • DOI: https://doi.org/10.1007/11549345_61

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-28702-5

  • Online ISBN: 978-3-540-31867-5

  • eBook Packages: Computer ScienceComputer Science (R0)

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