research-article

On linear recurrence sequences and loop termination

Authors:

Joël Ouaknine,

James WorrellAuthors Info & Claims

ACM SIGLOG News, Volume 2, Issue 2

Pages 4 - 13

https://doi.org/10.1145/2766189.2766191

Published: 22 April 2015 Publication History

Abstract

A sequence of real numbers is said to be positive if all terms are positive, and ultimately positive if all but finitely many terms are positive. In this article we survey recent progress on long-standing open problems concerning deciding the positivity and ultimate positivity of integer linear recurrence sequences. We briefly discuss some of the many contexts in which these problems arise, and relate them to the well-known Skolem Problem, which asks whether a given linear recurrence sequence has a zero term. We also highlight some of the mathematical techniques that have been used to obtain decision procedures for these problems, pointing out obstacles to further progress.

In the second half of this survey we move on to closely related questions concerning the termination of linear while loops. This is a well-studied subject in software verification and by now there is rich toolkit of techniques to prove termination in practice. However the decidability of termination for some of the most basic types of loop remains open. Here again we discuss recent progress and remaining open problems.

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Cited By

Ait El Manssour RKenison GShirmohammadi MVaronka A(2025)Simple Linear Loops: Algebraic Invariants and ApplicationsProceedings of the ACM on Programming Languages10.1145/37048629:POPL(745-771)Online publication date: 9-Jan-2025
https://doi.org/10.1145/3704862
Teguia Tabuguia BWorrell J(2024)On Rational Recursion for Holonomic SequencesComputer Algebra in Scientific Computing10.1007/978-3-031-69070-9_18(314-327)Online publication date: 1-Sep-2024
https://dl.acm.org/doi/10.1007/978-3-031-69070-9_18
Dong R(2023)Recent Advances in Algorithmic Problems for SemigroupsACM SIGLOG News10.1145/3636362.363636510:4(3-23)Online publication date: 4-Dec-2023
https://dl.acm.org/doi/10.1145/3636362.3636365
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Index Terms

On linear recurrence sequences and loop termination
1. Theory of computation
  1. Semantics and reasoning
    1. Program reasoning
      1. Program analysis
    2. Program semantics

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Published In

cover image ACM SIGLOG News

ACM SIGLOG News Volume 2, Issue 2

April 2015

36 pages

EISSN:2372-3491

DOI:10.1145/2766189

Editor:
Andrzej Murawski
University of Warwick

Issue’s Table of Contents

Copyright © 2015 Authors.

Publisher

Association for Computing Machinery

New York, NY, United States

Publication History

Published: 22 April 2015

Published in SIGLOG Volume 2, Issue 2

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Cited By

Ait El Manssour RKenison GShirmohammadi MVaronka A(2025)Simple Linear Loops: Algebraic Invariants and ApplicationsProceedings of the ACM on Programming Languages10.1145/37048629:POPL(745-771)Online publication date: 9-Jan-2025
https://doi.org/10.1145/3704862
Teguia Tabuguia BWorrell J(2024)On Rational Recursion for Holonomic SequencesComputer Algebra in Scientific Computing10.1007/978-3-031-69070-9_18(314-327)Online publication date: 1-Sep-2024
https://dl.acm.org/doi/10.1007/978-3-031-69070-9_18
Dong R(2023)Recent Advances in Algorithmic Problems for SemigroupsACM SIGLOG News10.1145/3636362.363636510:4(3-23)Online publication date: 4-Dec-2023
https://dl.acm.org/doi/10.1145/3636362.3636365
Kenison GKovács LVaronka A(2023)From Polynomial Invariants to Linear LoopsProceedings of the 2023 International Symposium on Symbolic and Algebraic Computation10.1145/3597066.3597109(398-406)Online publication date: 24-Jul-2023
https://dl.acm.org/doi/10.1145/3597066.3597109
Karimov TKelmendi ENieuwveld JOuaknine JWorrell J(2023)The Power of Positivity2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)10.1109/LICS56636.2023.10175758(1-11)Online publication date: 26-Jun-2023
https://doi.org/10.1109/LICS56636.2023.10175758
Neumann E(2023)On the Complexity of Robust Eventual Inequality Testing for C-Finite FunctionsReachability Problems10.1007/978-3-031-45286-4_8(98-112)Online publication date: 11-Oct-2023
https://dl.acm.org/doi/10.1007/978-3-031-45286-4_8
Akshay SChatterjee KMeggendorfer TŽikelić Đ(2023)MDPs as Distribution Transformers: Affine Invariant Synthesis for Safety ObjectivesComputer Aided Verification10.1007/978-3-031-37709-9_5(86-112)Online publication date: 17-Jul-2023
https://dl.acm.org/doi/10.1007/978-3-031-37709-9_5
Kovács LVaronka A(2023)What Else is Undecidable About Loops?Relational and Algebraic Methods in Computer Science10.1007/978-3-031-28083-2_11(176-193)Online publication date: 3-Apr-2023
https://dl.acm.org/doi/10.1007/978-3-031-28083-2_11
Lipton RLuca FNieuwveld JOuaknine JPurser DWorrell J(2022)On the Skolem Problem and the Skolem ConjectureProceedings of the 37th Annual ACM/IEEE Symposium on Logic in Computer Science10.1145/3531130.3533328(1-9)Online publication date: 2-Aug-2022
https://dl.acm.org/doi/10.1145/3531130.3533328
Almagor SChistikov DOuaknine JWorrell J(2022)O-Minimal Invariants for Discrete-Time Dynamical SystemsACM Transactions on Computational Logic10.1145/350129923:2(1-20)Online publication date: 14-Jan-2022
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