On Piecewise Affine Reachability with Bellman Operators

The paper proves BOR decidable (i) for all dimensions when t ≠ μΦ via an explicit finite bound derived from interval iteration, and (ii) when t = μΦ with s comparable to μΦ by showing eventual tightness of actions and a finite-state sign abstraction; in d=2 it further gives a complete decision procedure for the remaining incomparable case (Theorem 29). These claims are supported by Proposition 9, Lemma 16 and Corollary 17, Propositions 13–14 (combined as Theorem 19), and the d=2 analysis built via a matrix-semigroup rephrasing and the F-operator, culminating in Lemma 28 and Theorem 29 . By contrast, the model’s solution contains a critical flaw in the “s ≥ μΦ” comparable case: it assumes a symmetric closure argument that incorrectly treats leaking actions as always below μΦ, contradicting the paper’s Example 15 and subsequent analysis ; and its 2D algorithm lacks a proved loop-elimination bound, leaving termination unestablished.