813 research outputs found
Efficient Symbolic Representation of Convex Polyhedra in High-Dimensional Spaces
peer reviewedThis work is aimed at developing an efficient data structure for representing symbolically convex polyhedra. We introduce an original data structure, the Decomposed Convex Polyhedron (DCP), that is closed under intersection and linear transformations, and allows to check inclusion, equality, and emptiness. The main feature of DCPs lies in their ability to represent concisely polyhedra that can be expressed as combinations of simpler sets, which can overcome combinatorial explosion in high dimensional spaces. DCPs also have the advantage of being reducible into a canonical form, which makes them efficient for representing simple sets constructed by long sequences of manipulations, such as those handled by state-space exploration tools. Their practical efficiency has been evaluated with the help of a prototype implementation, with promising results
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
Quadtrees as an Abstract Domain
Quadtrees have proved popular in computer graphics and spatial databases as a way of representing regions in two dimensional space. This hierarchical data-structure is flexible enough to support non-convex and even disconnected regions, therefore it is natural to ask whether this datastructure can form the basis of an abstract domain. This paper explores this question and suggests that quadtrees offer a new approach to weakly relational domains whilst their hierarchical structure naturally lends itself to representation with boolean functions
Software for Exact Integration of Polynomials over Polyhedra
We are interested in the fast computation of the exact value of integrals of
polynomial functions over convex polyhedra. We present speed ups and extensions
of the algorithms presented in previous work. We present the new software
implementation and provide benchmark computations. The computation of integrals
of polynomials over polyhedral regions has many applications; here we
demonstrate our algorithmic tools solving a challenge from combinatorial voting
theory.Comment: Major updat
LazySets.jl: Scalable symbolic-numeric set computations
LazySets.jl is a Julia library that provides ways to symbolically represent
sets of points as geometric shapes, with a special focus on convex sets and
polyhedral approximations. LazySets provides methods to apply common set
operations, convert between different set representations, and efficiently
compute with sets in high dimensions using specialized algorithms based on the
set types. LazySets is the core library of JuliaReach, a cutting-edge software
addressing the fundamental problem of reachability analysis: computing the set
of states that are reachable by a dynamical system from all initial states and
for all admissible inputs and parameters. While the library was originally
designed for reachability and formal verification, its scope goes beyond such
topics. LazySets is an easy-to-use, general-purpose and scalable library for
computations that mix symbolics and numerics. In this article we showcase the
basic functionality, highlighting some of the key design choices.Comment: published in the Proceedings of the JuliaCon Conferences 202
Polynomial-Time Amoeba Neighborhood Membership and Faster Localized Solving
We derive efficient algorithms for coarse approximation of algebraic
hypersurfaces, useful for estimating the distance between an input polynomial
zero set and a given query point. Our methods work best on sparse polynomials
of high degree (in any number of variables) but are nevertheless completely
general. The underlying ideas, which we take the time to describe in an
elementary way, come from tropical geometry. We thus reduce a hard algebraic
problem to high-precision linear optimization, proving new upper and lower
complexity estimates along the way.Comment: 15 pages, 9 figures. Submitted to a conference proceeding
Convergent Puiseux Series and Tropical Geometry of Higher Rank
We propose to study the tropical geometry specifically arising from
convergent Puiseux series in multiple indeterminates. One application is a new
view on stable intersections of tropical hypersurfaces. Another one is the
study of families of ordinary convex polytopes depending on more than one
parameter through tropical geometry. This includes cubes constructed by
Goldfarb and Sit (1979) as special cases.Comment: 32 pages, 3 figure
- …