On the complexity of bounded time and precision reachability for piecewise affine systems

TL;DR

This paper analyzes the computational complexity of various reachability problems in piecewise affine systems, revealing their NP-complete, co-NP-complete, and PSPACE-complete classifications depending on the variant.

Contribution

It provides the first detailed complexity classifications for several decidable reachability variants in piecewise affine systems.

Findings

Region-to-region bounded time reachability is NP-complete or co-NP-complete from dimension 2.

Bounded precision reachability is PSPACE-complete.

Undecidability starts from dimension 2 for general reachability.

Abstract

Reachability for piecewise affine systems is known to be undecidable, starting from dimension $2$. In this paper we investigate the exact complexity of several decidable variants of reachability and control questions for piecewise affine systems. We show in particular that the region to region bounded time versions leads to $NP$-complete or co-$NP$-complete problems, starting from dimension $2$. We also prove that a bounded precision version leads to $PSPACE$-complete problems.