Elsevier

Information and Computation

Volume 211, February 2012, Pages 138-159
Information and Computation

Low dimensional hybrid systems – decidable, undecidable, donʼt know

Dedicated to the memory of Amir Pnueli (1941–2009)
https://doi.org/10.1016/j.ic.2011.11.006Get rights and content
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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.

Keywords

Hybrid systems
Undecidability
Piece-wise affine maps
(Hierarchical) piece-wise constant derivative systems

Cited by (0)

1

Current affiliation: D. E. Shaw Research, New York, USA.

2

Affiliation at the moment of submission of this paper.