The Pseudo-Reachability Problem for Diagonalisable Linear Dynamical Systems

TL;DR

This paper investigates pseudo-reachability in diagonalisable linear dynamical systems, establishing decidability results for discrete and continuous-time cases using o-minimality and reductions to classical problems.

Contribution

It proves decidability of discrete-time pseudo-reachability with semialgebraic targets and reduces continuous-time pseudo-reachability to classical time-bounded reachability.

Findings

Decidability of discrete-time pseudo-reachability for diagonalisable systems.

Reduction of continuous-time pseudo-reachability to classical time-bounded reachability.

Application of o-minimality to reachability problems.

Abstract

We study fundamental reachability problems on pseudo-orbits of linear dynamical systems. Pseudo-orbits can be viewed as a model of computation with limited precision and pseudo-reachability can be thought of as a robust version of classical reachability. Using an approach based on $o$-minimality of $\reals_{\exp}$ we prove decidability of the discrete-time pseudo-reachability problem with arbitrary semialgebraic targets for diagonalisable linear dynamical systems. We also show that our method can be used to reduce the continuous-time pseudo-reachability problem to the (classical) time-bounded reachability problem, which is known to be conditionally decidable.