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Random Subgroups of Rationals

Authors Ziyuan Gao, Sanjay Jain, Bakhadyr Khoussainov, Wei Li, Alexander Melnikov, Karen Seidel, Frank Stephan

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Author Details

Ziyuan Gao
  • Department of Mathematics, National University of Singapore, Singapore
Sanjay Jain
  • School of Computing, National University of Singapore, Singapore
Bakhadyr Khoussainov
  • Department of Computer Science, University of Auckland, New Zealand
Wei Li
  • Department of Mathematics, National University of Singapore, Singapore
Alexander Melnikov
  • Institute of Natural and Mathematical Sciences, Massey University, New Zealand
Karen Seidel
  • Hasso Plattner Institute, University of Potsdam, Germany
Frank Stephan
  • Department of Mathematics, National University of Singapore, Singapore

Funding

Sanjay Jain was supported in part by NUS grant C252-000-087-001. Furthermore, Ziyuan Gao, Sanjay Jain and Frank Stephan have been supported in part by the Singapore Ministry of Education Academic Research Fund Tier 2 grant MOE2016-T2-1-019 / R146-000-234-112. Bakhadyr Khoussainov was supported by the Marsden fund of Royal Society of New Zealand. Karen Seidel was supported by the German Research Foundation (DFG) under Grant KO 4635/1-1 (SCL) and by the Marsden fund of Royal Society of New Zealand.

Acknowledgements

The authors would like to thank Philipp Schlicht and Tin Lok Wong for helpful discussions, as well as thank Timo Kötzing for pointers to the literature.

Cite AsGet BibTex

Ziyuan Gao, Sanjay Jain, Bakhadyr Khoussainov, Wei Li, Alexander Melnikov, Karen Seidel, and Frank Stephan. Random Subgroups of Rationals. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 25:1-25:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
https://doi.org/10.4230/LIPIcs.MFCS.2019.25

Abstract

This paper introduces and studies a notion of algorithmic randomness for subgroups of rationals. Given a randomly generated additive subgroup (G,+) of rationals, two main questions are addressed: first, what are the model-theoretic and recursion-theoretic properties of (G,+); second, what learnability properties can one extract from G and its subclass of finitely generated subgroups? For the first question, it is shown that the theory of (G,+) coincides with that of the additive group of integers and is therefore decidable; furthermore, while the word problem for G with respect to any generating sequence for G is not even semi-decidable, one can build a generating sequence beta such that the word problem for G with respect to beta is co-recursively enumerable (assuming that the set of generators of G is limit-recursive). In regard to the second question, it is proven that there is a generating sequence beta for G such that every non-trivial finitely generated subgroup of G is recursively enumerable and the class of all such subgroups of G is behaviourally correctly learnable, that is, every non-trivial finitely generated subgroup can be semantically identified in the limit (again assuming that the set of generators of G is limit-recursive). On the other hand, the class of non-trivial finitely generated subgroups of G cannot be syntactically identified in the limit with respect to any generating sequence for G. The present work thus contributes to a recent line of research studying algorithmically random infinite structures and uncovers an interesting connection between the arithmetical complexity of the set of generators of a randomly generated subgroup of rationals and the learnability of its finitely generated subgroups.

Subject Classification

ACM Subject Classification
  • Theory of computation → Pseudorandomness and derandomization
  • Theory of computation → Inductive inference
Keywords
  • Martin-Löf randomness
  • subgroups of rationals
  • finitely generated subgroups of rationals
  • learning in the limit
  • behaviourally correct learning

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