321 research outputs found
Formal Verification of Neural Network Controlled Autonomous Systems
In this paper, we consider the problem of formally verifying the safety of an
autonomous robot equipped with a Neural Network (NN) controller that processes
LiDAR images to produce control actions. Given a workspace that is
characterized by a set of polytopic obstacles, our objective is to compute the
set of safe initial conditions such that a robot trajectory starting from these
initial conditions is guaranteed to avoid the obstacles. Our approach is to
construct a finite state abstraction of the system and use standard
reachability analysis over the finite state abstraction to compute the set of
the safe initial states. The first technical problem in computing the finite
state abstraction is to mathematically model the imaging function that maps the
robot position to the LiDAR image. To that end, we introduce the notion of
imaging-adapted sets as partitions of the workspace in which the imaging
function is guaranteed to be affine. We develop a polynomial-time algorithm to
partition the workspace into imaging-adapted sets along with computing the
corresponding affine imaging functions. Given this workspace partitioning, a
discrete-time linear dynamics of the robot, and a pre-trained NN controller
with Rectified Linear Unit (ReLU) nonlinearity, the second technical challenge
is to analyze the behavior of the neural network. To that end, we utilize a
Satisfiability Modulo Convex (SMC) encoding to enumerate all the possible
segments of different ReLUs. SMC solvers then use a Boolean satisfiability
solver and a convex programming solver and decompose the problem into smaller
subproblems. To accelerate this process, we develop a pre-processing algorithm
that could rapidly prune the space feasible ReLU segments. Finally, we
demonstrate the efficiency of the proposed algorithms using numerical
simulations with increasing complexity of the neural network controller
Bounded Verification with On-the-Fly Discrepancy Computation
Simulation-based verification algorithms can provide formal safety guarantees
for nonlinear and hybrid systems. The previous algorithms rely on user provided
model annotations called discrepancy function, which are crucial for computing
reachtubes from simulations. In this paper, we eliminate this requirement by
presenting an algorithm for computing piece-wise exponential discrepancy
functions. The algorithm relies on computing local convergence or divergence
rates of trajectories along a simulation using a coarse over-approximation of
the reach set and bounding the maximal eigenvalue of the Jacobian over this
over-approximation. The resulting discrepancy function preserves the soundness
and the relative completeness of the verification algorithm. We also provide a
coordinate transformation method to improve the local estimates for the
convergence or divergence rates in practical examples. We extend the method to
get the input-to-state discrepancy of nonlinear dynamical systems which can be
used for compositional analysis. Our experiments show that the approach is
effective in terms of running time for several benchmark problems, scales
reasonably to larger dimensional systems, and compares favorably with respect
to available tools for nonlinear models.Comment: 24 page
Diffeomorphism-invariant properties for quasi-linear elliptic operators
For quasi-linear elliptic equations we detect relevant properties which
remain invariant under the action of a suitable class of diffeomorphisms. This
yields a connection between existence theories for equations with degenerate
and non-degenerate coerciveness.Comment: 16 page
Approximate probabilistic verification of hybrid systems
Hybrid systems whose mode dynamics are governed by non-linear ordinary
differential equations (ODEs) are often a natural model for biological
processes. However such models are difficult to analyze. To address this, we
develop a probabilistic analysis method by approximating the mode transitions
as stochastic events. We assume that the probability of making a mode
transition is proportional to the measure of the set of pairs of time points
and value states at which the mode transition is enabled. To ensure a sound
mathematical basis, we impose a natural continuity property on the non-linear
ODEs. We also assume that the states of the system are observed at discrete
time points but that the mode transitions may take place at any time between
two successive discrete time points. This leads to a discrete time Markov chain
as a probabilistic approximation of the hybrid system. We then show that for
BLTL (bounded linear time temporal logic) specifications the hybrid system
meets a specification iff its Markov chain approximation meets the same
specification with probability . Based on this, we formulate a sequential
hypothesis testing procedure for verifying -approximately- that the Markov
chain meets a BLTL specification with high probability. Our case studies on
cardiac cell dynamics and the circadian rhythm indicate that our scheme can be
applied in a number of realistic settings
\epsilon-regularity for systems involving non-local, antisymmetric operators
We prove an epsilon-regularity theorem for critical and super-critical
systems with a non-local antisymmetric operator on the right-hand side.
These systems contain as special cases, Euler-Lagrange equations of
conformally invariant variational functionals as Rivi\`ere treated them, and
also Euler-Lagrange equations of fractional harmonic maps introduced by Da
Lio-Rivi\`ere.
In particular, the arguments presented here give new and uniform proofs of
the regularity results by Rivi\`ere, Rivi\`ere-Struwe, Da-Lio-Rivi\`ere, and
also the integrability results by Sharp-Topping and Sharp, not discriminating
between the classical local, and the non-local situations
LNCS
Reachability analysis is difficult for hybrid automata with affine differential equations, because the reach set needs to be approximated. Promising abstraction techniques usually employ interval methods or template polyhedra. Interval methods account for dense time and guarantee soundness, and there are interval-based tools that overapproximate affine flowpipes. But interval methods impose bounded and rigid shapes, which make refinement expensive and fixpoint detection difficult. Template polyhedra, on the other hand, can be adapted flexibly and can be unbounded, but sound template refinement for unbounded reachability analysis has been implemented only for systems with piecewise constant dynamics. We capitalize on the advantages of both techniques, combining interval arithmetic and template polyhedra, using the former to abstract time and the latter to abstract space. During a CEGAR loop, whenever a spurious error trajectory is found, we compute additional space constraints and split time intervals, and use these space-time interpolants to eliminate the counterexample. Space-time interpolation offers a lazy, flexible framework for increasing precision while guaranteeing soundness, both for error avoidance and fixpoint detection. To the best of out knowledge, this is the first abstraction refinement scheme for the reachability analysis over unbounded and dense time of affine hybrid systems, which is both sound and automatic. We demonstrate the effectiveness of our algorithm with several benchmark examples, which cannot be handled by other tools
LNCS
We address the problem of analyzing the reachable set of a polynomial nonlinear continuous system by over-approximating the flowpipe of its dynamics. The common approach to tackle this problem is to perform a numerical integration over a given time horizon based on Taylor expansion and interval arithmetic. However, this method results to be very conservative when there is a large difference in speed between trajectories as time progresses. In this paper, we propose to use combinations of barrier functions, which we call piecewise barrier tube (PBT), to over-approximate flowpipe. The basic idea of PBT is that for each segment of a flowpipe, a coarse box which is big enough to contain the segment is constructed using sampled simulation and then in the box we compute by linear programming a set of barrier functions (called barrier tube or BT for short) which work together to form a tube surrounding the flowpipe. The benefit of using PBT is that (1) BT is independent of time and hence can avoid being stretched and deformed by time; and (2) a small number of BTs can form a tight over-approximation for the flowpipe, which means that the computation required to decide whether the BTs intersect the unsafe set can be reduced significantly. We implemented a prototype called PBTS in C++. Experiments on some benchmark systems show that our approach is effective
Verifying Hybrid Systems Involving Transcendental Functions
Abstract. We explore uses of a link we have constructed between the KeYmaera hybrid systems theorem prover and the MetiTarski proof en-gine for problems involving special functions such as sin, cos, exp, etc. Transcendental functions arise in the specification of hybrid systems and often occur in the solutions of the differential equations that govern how the states of hybrid systems evolve over time. To date, formulas ex-changed between KeYmaera and external tools have involved polynomi-als over the reals, but not transcendental functions, chiefly because of the lack of tools capable of proving such goals.
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