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A Quest for Algorithmically Random Infinite Structures, II

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Logical Foundations of Computer Science (LFCS 2016)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9537))

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Abstract

We present an axiomatic approach that introduces algorithmic randomness into various classes of structures. The central concept is the notion of a branching class. Through this technical yet simple notion we define measure, metric, and topology in many classes of graphs, trees, relational structures, and algebras. As a consequence we define algorithmically random structures. We prove the existence of algorithmically random structures with various computability-theoretic properties. We show that any nontrivial variety of algebras has an effective measure 0. We also prove a counter-intuitive result that there are algorithmically random yet computable structures. This establishes a connection between algorithmic randomness and computable model theory.

B. Khoussainov— The author also acknowledges support of Marsden Fund of Royal Society of New Zealand.

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Notes

  1. 1.

    The next subsection provides many examples of classes with height function. For now, for the reader a good example of a class with a height function is the class of rooted finite binary trees.

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Author information

Authors and Affiliations

  1. The University of Auckland, Auckland, New Zealand

    Bakhadyr Khoussainov

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  1. Bakhadyr Khoussainov
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Correspondence to Bakhadyr Khoussainov .

Editor information

Editors and Affiliations

  1. City University of New York , New York, New York, USA

    Sergei Artemov

  2. Cornell University , Ithaca, New York, USA

    Anil Nerode

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Khoussainov, B. (2016). A Quest for Algorithmically Random Infinite Structures, II. In: Artemov, S., Nerode, A. (eds) Logical Foundations of Computer Science. LFCS 2016. Lecture Notes in Computer Science(), vol 9537. Springer, Cham. https://doi.org/10.1007/978-3-319-27683-0_12

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  • DOI: https://doi.org/10.1007/978-3-319-27683-0_12

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-27682-3

  • Online ISBN: 978-3-319-27683-0

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