Low dimensional hybrid systems – decidable, undecidable, donʼt know

Abstract

Even though many attempts have been made to define the boundary between decidable and undecidable hybrid systems, the affair is far from being resolved. More and more low dimensional systems are being shown to be undecidable with respect to reachability, and many open problems in between are being discovered. In this paper, we present various two dimensional hybrid systems for which the reachability problem is undecidable. We show their undecidability by simulating Minsky machines. Their proximity to the decidability frontier is understood by inspecting the most parsimonious constraints necessary to make reachability over these automata decidable. We also show that for other two dimensional systems, the reachability question remains unanswered, by proving that it is as hard as the reachability problem for piecewise affine maps on the real line, which is a well known open problem.

Reachability Problems for Hierarchical Piecewise Constant Derivative Systems

“…Similar definitions exist for half-open and closed intervals. We use similar definitions as [3] for the following.…”

“…A Hierarchical Piecewise Constant Derivative (HPCD) system [3] is a hybrid automaton H = (X , Q, f , l 0 , Inv, δ) such that Q and l 0 are defined as in Definition 1, with the dynamics at each l ∈ Q given by a 2-PCD and each transition tr = (l, g, γ, l ) is such that: (1) Its guard g is a line segment in R 2 ; and (2) The reset relation γ is an affine function of the form:…”

“…It was shown in [3] that reachability for an HPCD is equivalent to reachability for a 1-PAM. An HPCD system has elementary flows if the derivatives of all variables in each location are in {0, ±1}, otherwise it has arbitrary constant flows.…”

“…If each atomic formula contains only one variable (x or y), then the guard is called non-comparative. An HPCD has non-comparative guards if all guards are non-comparative, e.g., 3 2 ≤ x ≤ 7 ∧ y = −1 is non-comparative, but 0 ≤ x ≤ 1 ∧ 0 ≤ y ≤ 1 2 ∧ x = 2y is a comparative guard. We define an affine reset of variables z = (x, y) as γ(z) = Az + b where A ∈ Q 2×2 and b ∈ Q 2 , a linear reset is of the form γ(z) = Az and an identity reset is γ(z) = z.…”

“…Clearly then, the model of HPCD seems more powerful than that of a 2-PCD. Indeed, the reachability problem for a one-dimensional Piecewise Affine Map (1-PAM) was shown to be equivalent to that of reachability for a bounded HPCD with either: i) comparative guards, identity resets and elementary flows in Proposition 3.20 of [3] or else ii) affine resets, non-comparative guards and elementary flows in Lemma 3.4 of [3] (See Section 2 for definitions).…”