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Model checking linear dynamical systems under floating-point rounding

Lefaucheux, E ; Sharif University of Technology | 2023

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  1. Type of Document: Article
  2. DOI: 10.1007/978-3-031-30823-9_3
  3. Publisher: Springer Science and Business Media Deutschland GmbH , 2023
  4. Abstract:
  5. We consider linear dynamical systems under floating-point rounding. In these systems, a matrix is repeatedly applied to a vector, but the numbers are rounded into floating-point representation after each step (i.e., stored as a fixed-precision mantissa and an exponent). The approach more faithfully models realistic implementations of linear loops, compared to the exact arbitrary-precision setting often employed in the study of linear dynamical systems. Our results are twofold: We show that for non-negative matrices there is a special structure to the sequence of vectors generated by the system: the mantissas are periodic and the exponents grow linearly. We leverage this to show decidability of ω -regular temporal model checking against semialgebraic predicates. This contrasts with the unrounded setting, where even the non-negative case encompasses the long-standing open Skolem and Positivity problems. On the other hand, when negative numbers are allowed in the matrix, we show that the reachability problem is undecidable by encoding a two-counter machine. Again, this is in contrast with the unrounded setting where point-to-point reachability is known to be decidable in polynomial time. © 2023, The Author(s)
  6. Keywords:
  7. Dynamical systems ; Floating-point ; Model checking
  8. Source: Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) ; Volume 13993 LNCS , 2023 , Pages 47-65 ; 03029743 (ISSN); 978-303130822-2 (ISBN)
  9. URL: https://link.springer.com/chapter/10.1007/978-3-031-30823-9_3