227 research outputs found
Analysis of parametric biological models with non-linear dynamics
In this paper we present recent results on parametric analysis of biological
models. The underlying method is based on the algorithms for computing
trajectory sets of hybrid systems with polynomial dynamics. The method is then
applied to two case studies of biological systems: one is a cardiac cell model
for studying the conditions for cardiac abnormalities, and the second is a
model of insect nest-site choice.Comment: In Proceedings HSB 2012, arXiv:1208.315
Sapo: Reachability Computation and Parameter Synthesis of Polynomial Dynamical Systems
Sapo is a C++ tool for the formal analysis of polynomial dynamical systems.
Its main features are: 1) Reachability computation, i.e., the calculation of
the set of states reachable from a set of initial conditions, and 2) Parameter
synthesis, i.e., the refinement of a set of parameters so that the system
satisfies a given specification. Sapo can represent reachable sets as unions of
boxes, parallelotopes, or parallelotope bundles (symbolic representation of
polytopes). Sets of parameters are represented with polytopes while
specifications are formalized as Signal Temporal Logic (STL) formulas
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
Automatic Dynamic Parallelotope Bundles for Reachability of Nonlinear Dynamical Systems
Reachable set computation is an important technique for the verification of safety properties of dynamical systems. In this thesis, we investigate reachable set computation for discrete non-linear systems based on parallelotope bundles. The crux of the reachability algorithm relies on computing an upper and lower bound on the supremum and infimum respectively of a non-linear function over a rectangular domain. Bernstein Expansion of a polynomial function has been explored as a traditional method for computing these bounds efficiently. In light of this, we aim to improve the traditional parallelotope-based reachability method by removing the manual step of parallelotope template selection in order to make the procedure fully automatic. Furthermore, we show that adding templates dynamically during computations can improve accuracy. To this end, we investigate two techniques for generating template directions. The first technique approximates the dynamics as a linear transformation and generates templates using this transformation. The second technique uses Principal Component Analysis (PCA) of sample trajectories for generating templates. We have implemented our approach in a Python-based tool called Kaa, which uses two types of global optimization solvers, the first using Bernstein polynomials and the second usingthe Kodiak library. We demonstrate the improved accuracy of our approach on several standard nonlinear benchmark systems, including a high-dimensional COVID19 model. Finally, we explore a potential application of the Bernstein expansion technique to real-time reachability. We present evidence of several hurdles and barriers against effectively utilizing our Bernstein coefficient pruning method.Master of Scienc
Reachability computation for polynomial dynamical systems
This paper is concerned with the problem of computing the bounded time reachable set of a polynomial discrete-time dynamical system. The problem is well-known for being difficult when nonlinear systems are considered. In this regard, we propose three reachability methods that differ in the set representation. The proposed algorithms adopt boxes, parallelotopes, and parallelotope bundles to construct flowpipes that contain the actual reachable sets. The latter is a new data structure for the symbolic representation of polytopes. Our methods exploit the Bernstein expansion of polynomials to bound the images of sets. The scalability and precision of the presented methods are analyzed on a number of dynamical systems, in comparison with other existing approaches
PolyARBerNN: A Neural Network Guided Solver and Optimizer for Bounded Polynomial Inequalities
Constraints solvers play a significant role in the analysis, synthesis, and
formal verification of complex embedded and cyber-physical systems. In this
paper, we study the problem of designing a scalable constraints solver for an
important class of constraints named polynomial constraint inequalities (also
known as non-linear real arithmetic theory). In this paper, we introduce a
solver named PolyARBerNN that uses convex polynomials as abstractions for
highly nonlinear polynomials. Such abstractions were previously shown to be
powerful to prune the search space and restrict the usage of sound and complete
solvers to small search spaces. Compared with the previous efforts on using
convex abstractions, PolyARBerNN provides three main contributions namely (i) a
neural network guided abstraction refinement procedure that helps selecting the
right abstraction out of a set of pre-defined abstractions, (ii) a Bernstein
polynomial-based search space pruning mechanism that can be used to compute
tight estimates of the polynomial maximum and minimum values which can be used
as an additional abstraction of the polynomials, and (iii) an optimizer that
transforms polynomial objective functions into polynomial constraints (on the
gradient of the objective function) whose solutions are guaranteed to be close
to the global optima. These enhancements together allowed the PolyARBerNN
solver to solve complex instances and scales more favorably compared to the
state-of-art non-linear real arithmetic solvers while maintaining the soundness
and completeness of the resulting solver. In particular, our test benches show
that PolyARBerNN achieved 100X speedup compared with Z3 8.9, Yices 2.6, and
NASALib (a solver that uses Bernstein expansion to solve multivariate
polynomial constraints) on a variety of standard test benches
- …