On the Complexity of the Escape Problem for Linear Dynamical Systems over Compact Semialgebraic Sets

TL;DR

This paper investigates the computational complexity of the Escape Problem for linear dynamical systems over compact semialgebraic sets, establishing its completeness for a specific complexity class within the theory of the reals.

Contribution

It characterizes the complexity of the Escape Problem, showing its completeness for a particular class defined by negation-free $orall eg$-sentences, revealing its expressive power.

Findings

The Escape Problem is complete for a specific complexity class.

Equivalent characterizations of the complexity class are provided.

The results demonstrate the robustness of the complexity class.

Abstract

We study the computational complexity of the Escape Problem for discrete-time linear dynamical systems over compact semialgebraic sets, or equivalently the Termination Problem for affine loops with compact semialgebraic guard sets. Consider the fragment of the theory of the reals consisting of negation-free $\exists \forall$-sentences without strict inequalities. We derive several equivalent characterisations of the associated complexity class which demonstrate its robustness and illustrate its expressive power. We show that the Compact Escape Problem is complete for this class.