Bounding the Escape Time of a Linear Dynamical System over a Compact Semialgebraic Set

Julian d'Costa (1) , Engel Lefaucheux (2, 3) , Eike Neumann (4) , Joël Ouaknine (5) , James Worrell (1)

1 Department of Computer Science [Oxford]
2 LORIA - Laboratoire Lorrain de Recherche en Informatique et ses Applications
3 VERIDIS - Modeling and Verification of Distributed Algorithms and Systems
4 Swansea University
5 MPI-SWS - Max Planck Institute for Software Systems

We study the Escape Problem for discrete-time linear dynamical systems over compact semialgebraic sets. We establish a uniform upper bound on the number of iterations it takes for every orbit of a rational matrix to escape a compact semialgebraic set defined over rational data. Our bound is doubly exponential in the ambient dimension, singly exponential in the degrees of the polynomials used to define the semialgebraic set, and singly exponential in the bitsize of the coefficients of these polynomials and the bitsize of the matrix entries. We show that our bound is tight by providing a matching lower bound.

Mots clés: Program termination; Compact semialgebraic sets; Discrete linear dynamical systems; Uniform termination bounds

Domaines: Informatique [cs]

Julian d'Costa, Engel Lefaucheux, Eike Neumann, Joël Ouaknine, James Worrell. Bounding the Escape Time of a Linear Dynamical System over a Compact Semialgebraic Set. 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022), Aug 2022, Vienna, Austria. pp.14, ⟨10.4230/LIPIcs⟩. ⟨hal-03712991⟩