On The Complexity of Bounded Time Reachability for Piecewise Affine Systems

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

“…and at the moment we have only defined f over R i,1 = R i,0 and R i,4 = R i,1 . Also note that 4 and we define f separately on those two subregions. In order to correctly and continuously extend f , we will need to further split the region R i,3 into linear and saturated parts R lin i,3 and R sat i,3 : see Figure 2.…”

“…In other words, we have: . Furthermore, thinking about the future a bit, on R i,3 ∩R i, 4 we have that f (a, b) = (a + 2 −p , 0). In other words, somewhere on R i,2 ∪ R i,3 , the second coordinate has to go from β −i to 0 in a continuous fashion.…”

“…A short version of this paper has been presented at the conference "Reachability Problems 2014" [4]. The current journal version contains full proofs for all statements, and is also providing new results: bounded precision variants (problems REACH-REGION-PRECISION and CONTROL-REGION-PRECISION) were not considered in short version [4].…”

Reachability Problems for One-Dimensional Piecewise Affine Maps

“…The computations in PAMs have a natural geometrical interpretation as pseudo-billiard system [21] and Hierarchical Piecewise Constant Derivative (HPCD) system [3]. The reachability problem for one-dimensional PAMs is still open even if restricted to a map with only two intervals [2,3,6,13].…”

“…Later the result was improved by constructing undecidability proof for a map with a finite number of bounded intervals and only one unbounded interval (l, + 8 ) in [5]. On the other hand some simplifications or additional constrains on affine maps can make the system to be more predictable with decidable reachability problems, see [2,3,6,13].…”