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Composition Over the Natural Number Ordering with an Extra Binary Relation

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Fields of Logic and Computation II

Part of the book series: Lecture Notes in Computer Science ((LNPSE,volume 9300))

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Abstract

The composition method was developed in the 1970’s and 1980’s by Shelah and Gurevich as a powerful tool in the study of monadic second-order theories of labelled orderings and trees. In this paper, we use a variant of the technique for first-order theories of structures \((\mathbb {N}, <, R)\) where R is binary. For the case that R is of “finite valency” (where each element has only finitely many neighbors in the symmetric closure of R), we show results on (non-) definability, on decidability, and on the recursion theoretic complexity of such theories.

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Notes

  1. 1.

    When the present author took his first steps in the study of the composition method, he met Yuri Gurevich and gained a lot by discussions with him, also by his kind encouragement. The results mentioned above were then included in the author’s habilitation thesis [Th80], which however was not published due to the author’s move to computer science. Now, 35 years later, at Yuri’s 75th birthday, it seems fitting to come back to this outgrowth of the author’s first contact with Yuri and to explain these results.

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Authors and Affiliations

  1. RWTH Aachen University, Aachen, Germany

    Wolfgang Thomas

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  1. Wolfgang Thomas
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Correspondence to Wolfgang Thomas .

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Editors and Affiliations

  1. Steklov Mathematical Institute, Moscow, Russia

    Lev D. Beklemishev

  2. University of Michigan, Ann Arbor, Michigan, USA

    Andreas Blass

  3. Tel Aviv University, Tel Aviv, Israel

    Nachum Dershowitz

  4. Universität des Saarlandes, Saarbrücken, Germany

    Bernd Finkbeiner

  5. Microsoft Research, REDMOND, Washington, USA

    Wolfram Schulte

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Thomas, W. (2015). Composition Over the Natural Number Ordering with an Extra Binary Relation. In: Beklemishev, L., Blass, A., Dershowitz, N., Finkbeiner, B., Schulte, W. (eds) Fields of Logic and Computation II. Lecture Notes in Computer Science(), vol 9300. Springer, Cham. https://doi.org/10.1007/978-3-319-23534-9_19

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  • DOI: https://doi.org/10.1007/978-3-319-23534-9_19

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-23533-2

  • Online ISBN: 978-3-319-23534-9

  • eBook Packages: Computer ScienceComputer Science (R0)

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