Reachability Problems for Hierarchical Piecewise Constant Derivative Systems

Abstract

Abstract. In this paper, we investigate the computability and complexity of reachability problems for two-dimensional hierarchical piecewise constant derivative (HPCD) systems. The main interest in HPCDs stems from the fact that their reachability problem is on the border between decidability and undecidability, since it is equivalent to that of reachability for one-dimensional piecewise affine maps (PAMs) which is a long standing open problem. Understanding the most expressive hybrid system models that retain decidability for reachability has generated a great deal of interest over the past few years. In this paper, we show a restriction of HPCDs (called RHPCDs) which leads to the reachability problem becoming decidable. We then study which additional powers we must add to the RHPCD model to render it 1D PAM-equivalent. Finally, we show NP-hardness of reachability for nondeterministic RHPCDs.

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

“…In the previous section, that we have defined f I over a subset of the entire space and it is clear that this subspace is not stable in any way 5 . In order to match the definition of a piecewise affine function, we need to define f over the entire space or a stable subspace (containing the initial region).…”

“…For example, it is known that reachability is undecidable for piecewise constant derivative systems of dimension 3, whereas it is decidable for dimension 2 [1]. It is known that piecewise affine maps of dimension 2 can simulate Turing machines [13], whereas the question for dimension 1 is still open and can be related to other natural problems [2,3,5]. Variations of such problems over the integers have recently been investigated [6].…”