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Robust Positivity Problems for Linear Recurrence Sequences: The Frontiers of Decidability for Explicitly Given Neighbourhoods

Author Mihir Vahanwala

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Subject Classification

ACM Subject Classification
  • Theory of computation → Logic and verification
Keywords
  • Dynamical Systems
  • Verification
  • Robustness
  • Linear Recurrence Sequences
  • Positivity
  • Ultimate Positivity

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Abstract

Linear Recurrence Sequences (LRS) are a fundamental mathematical primitive for a plethora of applications such as the verification of probabilistic systems, model checking, computational biology, and economics. Positivity (are all terms of the given LRS non-negative?) and Ultimate Positivity (are all but finitely many terms of the given LRS non-negative?) are important open number-theoretic decision problems. Recently, the robust versions of these problems, that ask whether the LRS is (Ultimately) Positive despite small perturbations to its initialisation, have gained attention as a means to model the imprecision that arises in practical settings. However, the state of the art is ill-equipped to reason about imprecision when its extent is explicitly specified. In this paper, we consider Robust Positivity and Ultimate Positivity problems where the neighbourhood of the initialisation, expressed in a natural and general format, is also part of the input. We contribute by proving sharp decidability results: decision procedures at orders our techniques are unable to handle for general LRS would entail significant number-theoretic breakthroughs.

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Mihir Vahanwala. Robust Positivity Problems for Linear Recurrence Sequences: The Frontiers of Decidability for Explicitly Given Neighbourhoods. In 43rd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 284, pp. 17:1-17:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023) https://doi.org/10.4230/LIPIcs.FSTTCS.2023.17

Author Details

Mihir Vahanwala
  • Max Planck Institute for Software Systems, Saarland Informatics Campus, Saarbrücken, Germany

Funding

The author is partially funded by DFG grant 389792660 as part of TRR 248 (see https://perspicuous-computing.science).

Acknowledgements

This work is an extension of a recent collaboration with S Akshay, Hugo Bazille, and Blaise Genest. I am especially grateful to Valérie Berthé for her help in number-theoretic aspects of the paper. Our discussions took place during the Bellairs Dynamical Systems workshop organised by my advisor, Joël Ouaknine. I thank Bellairs Research Institute of McGill University, Barbados, for the hospitality and peaceful environment conducive to scientific discussion. Finally, I thank all the anonymous reviewers for their astute observations and constructive suggestions.

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