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Least and Greatest Solutions of Equations over Sets of Integers

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Mathematical Foundations of Computer Science 2010 (MFCS 2010)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 6281))

Abstract

Systems of equations with sets of integers as unknowns are considered, with the operations of union, intersection and addition of sets, \(S+T=\{m+n \mid m \in S, \: n \in T\}\). These equations were recently studied by the authors (“On equations over sets of integers”, STACS 2010), and it was shown that their unique solutions represent exactly the hyperarithmetical sets. In this paper it is demonstrated that greatest solutions of such equations represent exactly the \(\Sigma^1_1\) sets in the analytical hierarchy, and these sets can already be represented by systems in the resolved form X i  = ϕ i (X 1, ..., X n ). Least solutions of such resolved systems represent exactly the recursively enumerable sets.

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Author information

Authors and Affiliations

  1. Institute of Computer Science, University of Wrocław, Poland

    Artur Jeż

  2. Department of Mathematics, University of Turku, Finland

    Alexander Okhotin

  3. Academy of Finland,  

    Alexander Okhotin

Authors
  1. Artur Jeż
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  2. Alexander Okhotin
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Editor information

Editors and Affiliations

  1. Falculty of Informatics, Masaryk University, Botanicka 68a, 602 00, Brno, Czech Republic

    Petr Hliněný

  2. Faculty of Informatics, Masaryk University, Botanicka 68a, 602 00, Brno, Czech Republic

    Antonín Kučera

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Jeż, A., Okhotin, A. (2010). Least and Greatest Solutions of Equations over Sets of Integers. In: Hliněný, P., Kučera, A. (eds) Mathematical Foundations of Computer Science 2010. MFCS 2010. Lecture Notes in Computer Science, vol 6281. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-15155-2_39

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  • DOI: https://doi.org/10.1007/978-3-642-15155-2_39

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-15154-5

  • Online ISBN: 978-3-642-15155-2

  • eBook Packages: Computer ScienceComputer Science (R0)

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