278 research outputs found
Algorithmic Verification of Continuous and Hybrid Systems
We provide a tutorial introduction to reachability computation, a class of
computational techniques that exports verification technology toward continuous
and hybrid systems. For open under-determined systems, this technique can
sometimes replace an infinite number of simulations.Comment: In Proceedings INFINITY 2013, arXiv:1402.661
Reachability analysis of linear hybrid systems via block decomposition
Reachability analysis aims at identifying states reachable by a system within
a given time horizon. This task is known to be computationally expensive for
linear hybrid systems. Reachability analysis works by iteratively applying
continuous and discrete post operators to compute states reachable according to
continuous and discrete dynamics, respectively. In this paper, we enhance both
of these operators and make sure that most of the involved computations are
performed in low-dimensional state space. In particular, we improve the
continuous-post operator by performing computations in high-dimensional state
space only for time intervals relevant for the subsequent application of the
discrete-post operator. Furthermore, the new discrete-post operator performs
low-dimensional computations by leveraging the structure of the guard and
assignment of a considered transition. We illustrate the potential of our
approach on a number of challenging benchmarks.Comment: Accepted at EMSOFT 202
Reach Set Approximation through Decomposition with Low-dimensional Sets and High-dimensional Matrices
Approximating the set of reachable states of a dynamical system is an
algorithmic yet mathematically rigorous way to reason about its safety.
Although progress has been made in the development of efficient algorithms for
affine dynamical systems, available algorithms still lack scalability to ensure
their wide adoption in the industrial setting. While modern linear algebra
packages are efficient for matrices with tens of thousands of dimensions,
set-based image computations are limited to a few hundred. We propose to
decompose reach set computations such that set operations are performed in low
dimensions, while matrix operations like exponentiation are carried out in the
full dimension. Our method is applicable both in dense- and discrete-time
settings. For a set of standard benchmarks, it shows a speed-up of up to two
orders of magnitude compared to the respective state-of-the art tools, with
only modest losses in accuracy. For the dense-time case, we show an experiment
with more than 10.000 variables, roughly two orders of magnitude higher than
possible with previous approaches
Failing with Grace: Learning Neural Network Controllers that are Boundedly Unsafe
In this work, we consider the problem of learning a feed-forward neural
network (NN) controller to safely steer an arbitrarily shaped planar robot in a
compact and obstacle-occluded workspace. Unlike existing methods that depend
strongly on the density of data points close to the boundary of the safe state
space to train NN controllers with closed-loop safety guarantees, we propose an
approach that lifts such assumptions on the data that are hard to satisfy in
practice and instead allows for graceful safety violations, i.e., of a bounded
magnitude that can be spatially controlled. To do so, we employ reachability
analysis methods to encapsulate safety constraints in the training process.
Specifically, to obtain a computationally efficient over-approximation of the
forward reachable set of the closed-loop system, we partition the robot's state
space into cells and adaptively subdivide the cells that contain states which
may escape the safe set under the trained control law. To do so, we first
design appropriate under- and over-approximations of the robot's footprint to
adaptively subdivide the configuration space into cells. Then, using the
overlap between each cell's forward reachable set and the set of infeasible
robot configurations as a measure for safety violations, we introduce penalty
terms into the loss function that penalize this overlap in the training
process. As a result, our method can learn a safe vector field for the
closed-loop system and, at the same time, provide numerical worst-case bounds
on safety violation over the whole configuration space, defined by the overlap
between the over-approximation of the forward reachable set of the closed-loop
system and the set of unsafe states. Moreover, it can control the tradeoff
between computational complexity and tightness of these bounds. Finally, we
provide a simulation study that verifies the efficacy of the proposed scheme
Reachability analysis of discrete-time systems with disturbances
Published versio
Probabilistic embeddings of the Fr\'echet distance
The Fr\'echet distance is a popular distance measure for curves which
naturally lends itself to fundamental computational tasks, such as clustering,
nearest-neighbor searching, and spherical range searching in the corresponding
metric space. However, its inherent complexity poses considerable computational
challenges in practice. To address this problem we study distortion of the
probabilistic embedding that results from projecting the curves to a randomly
chosen line. Such an embedding could be used in combination with, e.g.
locality-sensitive hashing. We show that in the worst case and under reasonable
assumptions, the discrete Fr\'echet distance between two polygonal curves of
complexity in , where , degrades
by a factor linear in with constant probability. We show upper and lower
bounds on the distortion. We also evaluate our findings empirically on a
benchmark data set. The preliminary experimental results stand in stark
contrast with our lower bounds. They indicate that highly distorted projections
happen very rarely in practice, and only for strongly conditioned input curves.
Keywords: Fr\'echet distance, metric embeddings, random projectionsComment: 27 pages, 11 figure
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