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Computational Aspects of Σ-Definability over the Real Numbers without the Equality Test

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Computer Science Logic (CSL 2003)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 2803))

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Abstract

In this paper we study the expressive power and algorithmic properties of the language of Σ-formulas intended to represent computability over the real numbers. In order to adequately represent computability, we extend the reals by the structure of hereditarily finite sets. In this setting it is crucial to consider the real numbers without equality since the equality test is undecidable over the reals. We prove Engeler’s Lemma for Σ-definability over the reals without the equality test which relates Σ-definability with definability in the constructive infinitary language \(L_{\omega_1\omega}\). Thus, a relation over the real numbers is Σ-definable if and only if it is definable by a disjunction of a recursively enumerable set of quantifier free formulas. This result reveals computational aspects of Σ-definability and also gives topological characterisation of Σ-definable relations over the reals without the equality test.

This research was partially supported by the Danish Natural Science Research Council, Grant no. 21-02-0474, RFFI-DFG Grant no. 01-01-04003 and Grant Scientific School-2112.2003.1.

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Author information

Authors and Affiliations

  1. BRICS, Department of Computer Science, University of Aarhus, Ny Munkegade, DK-8000, Aarhus C, Denmark

    Margarita Korovina

  2. A. P. Ershov Institute of Informatics Systems, Lavrent’ev ave., 6, 630090, Novosibirsk, Russia

    Margarita Korovina

Authors
  1. Margarita Korovina
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Editor information

Editors and Affiliations

  1. Institute of Discrete Mathematics and Geometry, Technical University Vienna, Wiedner Hauptstrasse 8-10, A-1040, Wien, Austria

    Matthias Baaz

  2. Members of EACSL Jury for the Ackermann Award,  

    Johann A. Makowsky

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Korovina, M. (2003). Computational Aspects of Σ-Definability over the Real Numbers without the Equality Test. In: Baaz, M., Makowsky, J.A. (eds) Computer Science Logic. CSL 2003. Lecture Notes in Computer Science, vol 2803. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-45220-1_27

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  • DOI: https://doi.org/10.1007/978-3-540-45220-1_27

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-40801-7

  • Online ISBN: 978-3-540-45220-1

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