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Index Sets as a Measure of Continuous Constraint Complexity

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Perspectives of System Informatics (PSI 2014)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 8974))

Abstract

We develop the theory of index sets in the context of computable analysis considering classes of effectively open sets and computable real-valued functions. First, we construct principal computable numberings for effectively open sets and computable real-valued functions. Then, we calculate the complexity of index sets for important problems such as root realisability, set equivalence and inclusion, function equivalence which naturally arise in continuous constraint solving. Using developed techniques we give a direct proof of the generalised Rice-Shapiro theorem for effectively open sets of Euclidean spaces and present an analogue of Rice’s theorem for computable real-valued functions. We illustrate how index sets can be used to estimate complexity of continuous constraints in the settings of the Kleene-Mostowski arithmetical hierarchy.

This research was partially supported by Marie Curie Int. Research Staff Scheme Fellowship project PIRSES-GA-2011-294962, DFG/RFBR grant CAVER BE 1267/14-1 and 14-01-91334.

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Authors and Affiliations

  1. A.P. Ershov Institute of Informatics Systems, Novosibirsk, Russia

    Margarita Korovina

  2. Sobolev Institute of Mathematics, Novosibirsk, Russia

    Oleg Kudinov

Authors
  1. Margarita Korovina
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  2. Oleg Kudinov
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Correspondence to Margarita Korovina .

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Editors and Affiliations

  1. School of Computer Science, University of Manchester, Manchester, United Kingdom

    Andrei Voronkov

  2. A.P. Ershov Institute of Informatics Systems, Novosibirsk, Russia

    Irina Virbitskaite

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Korovina, M., Kudinov, O. (2015). Index Sets as a Measure of Continuous Constraint Complexity. In: Voronkov, A., Virbitskaite, I. (eds) Perspectives of System Informatics. PSI 2014. Lecture Notes in Computer Science(), vol 8974. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-46823-4_17

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  • DOI: https://doi.org/10.1007/978-3-662-46823-4_17

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