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A Logical Product Approach to Zonotope Intersection [chapter]

Khalil Ghorbal, Eric Goubault, Sylvie Putot
2010 Lecture Notes in Computer Science  
always zonotopes, and there is not even a best zonotopic over-approximation of the intersection.  ...  This fixes a known drawback of zonotopic methods, as used for reachability analysis for hybrid sys- tems as well as for invariant generation in abstract interpretation: intersection of zonotopes are not  ...  We thus choose in this paper a direct approach to the logical product of zonotopes with other classical abstract domains.  ... 
doi:10.1007/978-3-642-14295-6_22 fatcat:opdbpzxj35ai7aif3chrbdwbea

Enhancing Data-Driven Reachability Analysis using Temporal Logic Side Information [article]

Amr Alanwar, Frank J. Jiang, Maryam Sharifi, Dimos V. Dimarogonas, Karl H. Johansson
2021 arXiv   pre-print
In certain scenarios, the data-driven reachable sets of a robot can be prohibitively conservative due to the inherent noise in the robot's historical measurement data.  ...  In this work, we show that if we can model this side information using a signal temporal logic (STL) fragment, we can constrain the data-driven reachability analysis and safely limit the conservatism of  ...  We prove the guaranteed intersection in the nonlinear case as follows: We aim to find the zonotope that over-approximates the intersection.  ... 
arXiv:2109.07121v1 fatcat:bynwp72w2na7thow2ozbbjd66q

A Verified Algorithm for Geometric Zonotope/Hyperplane Intersection

Fabian Immler
2015 Proceedings of the 2015 Conference on Certified Programs and Proofs - CPP '15  
We have implemented a functional algorithm to compute the zonotope/hyperplane intersection and verified it in Isabelle/HOL.  ...  The interesting fact is that we combine a mixture of different fields: a discrete geometrical algorithm to perform operations on the continuous sets represented by zonotopes.  ...  I would like to thank Lars Hupel, Johannes Hölzl, and the anonymous reviewers for providing helpful feedback on earlier versions of this paper.  ... 
doi:10.1145/2676724.2693164 dblp:conf/cpp/Immler15 fatcat:vw54zxfvejhfpmmoq75w445thy

Ellipsotopes: Combining Ellipsoids and Zonotopes for Reachability Analysis and Fault Detection [article]

Shreyas Kousik, Adam Dai, Grace Gao
2021 arXiv   pre-print
Recently, constrained zonotopes and constrained polynomial zonotopes have been shown to overcome some of these conservatism challenges, and are closed under intersection.  ...  Zonotopes, a type of symmetric, convex polytope, are similarly frequently used due to efficient numerical implementation of affine maps and exact Minkowski sums.  ...  To explain our approach, we consider a simple case. Consider E = E 2 (c, G, A, b, I) ⊂ R n with m > n generators and with k linear constraints.  ... 
arXiv:2108.01750v2 fatcat:l5hb4texsfb5bgt2ziivkywxay

Model-Based Fault Diagnosis System Verification Using Reachability Analysis

Jinya Su, Wen-Hua Chen
2017 IEEE Transactions on Systems, Man & Cybernetics. Systems  
Besides, other functions (such as the convex hull, Cartesian product) of two zonotopes are referred to [20] . 1) Computation Tool: Consider an uncertain continuoustime linear system described by a differential  ...  This approach has recently been applied to FDI problem of wind turbine system in [12] and [17] , where uncertain set is represented by zonotopes due to its computational efficacy.  ... 
doi:10.1109/tsmc.2017.2710132 fatcat:q2tavzr4tje5dhzsec726ajkya

6 Lattices and positive quadratic forms [chapter]

2020 Introduction to Louis Michel's lattice geometry through group action  
This approach requires us to introduce initially a lattice basis and to represent the translation lattice Λ n in this basis (6.1) From the associated scalar products one can form the Gram matrix Q: Using  ...  The term A is linear in the q ij and hence, it determines a flat wall W ⊂ Φ. • The case det(Q) = 0, or det( f 1 , . . . , f n ) = 0 means that Q ∈ C 0 and the lattice L n degenerates to L k , k < n.  ...  A very interesting and new situation (as compared with the threedimensional case) appears for 4-dimensional zonotopes. Let us extend our visualization approach to the 4-dimensional case.  ... 
doi:10.1051/978-2-7598-1952-2-007 fatcat:fz47ozs3trhmnkmbgdv7353aym

Combining Set Propagation with Finite Element Methods for Time Integration in Transient Solid Mechanics Problems [article]

Marcelo Forets, Daniel Freire Caporale, Jorge M. Pérez Zerpa
2021 arXiv   pre-print
We present the numerical results obtained in five examples to illustrate the capabilities of our approach, and compare its performance against reference numerical integration methods.  ...  In this article we propose a novel approach, based in set propagation methods and motivated by recent advances in the field of Reachability Analysis.  ...  analysis, and to Ander Gray for comments on non-probabilistic approaches.  ... 
arXiv:2105.05841v2 fatcat:dbhrj6bpjbe3noh2jzcmdppn44

Refining Abstract Interpretation Based Value Analysis with Constraint Programming Techniques [chapter]

Olivier Ponsini, Claude Michel, Michel Rueher
2012 Lecture Notes in Computer Science  
interpretation based value analysis is a classical approach for verifying programs with floating-point computations.  ...  ., 0.1 has no exact representation), arithmetic operators are not associative and may be subject to phenomena such as absorption (e.g., a + b is rounded to a when a is far greater than b) or cancellation  ...  An approximation of a solution is described by a box, i.e., the Cartesian product of the domains of the variables.  ... 
doi:10.1007/978-3-642-33558-7_43 fatcat:e2l2xakg4vf2xj422fzz426lpi

A Verified ODE Solver and the Lorenz Attractor

Fabian Immler
2018 Journal of automated reasoning  
A rigorous numerical algorithm, formally verified with Isabelle/HOL, is used to certify the computations that Tucker used to prove chaos for the Lorenz attractor.  ...  The verification is based on a formalization of a diverse variety of mathematics and algorithms. Formalized mathematics include ordinary differential equations and Poincaré maps.  ...  the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.  ... 
doi:10.1007/s10817-017-9448-y pmid:30069071 pmcid:PMC6044317 fatcat:bfnqyafba5ah3njzjprip2tgz4

Using Redundant Constraints for Refinement [chapter]

Eugene Asarin, Thao Dang, Oded Maler, Romain Testylier
2010 Lecture Notes in Computer Science  
We introduce the notion of directional distance which is appropriate for measuring approximation effectiveness with respect to verifying a safety property.  ...  This paper is concerned with a method for computing reachable sets of linear continuous systems with uncertain input.  ...  However, a major difficulty that comes with this set representation is that intersection of zonotopes is hard, which limits the extension of this approach to hybrid systems.  ... 
doi:10.1007/978-3-642-15643-4_5 fatcat:o6eezrxxybhbln472gf7gcmcr4

Verifying floating-point programs with constraint programming and abstract interpretation techniques

Olivier Ponsini, Claude Michel, Michel Rueher
2014 Automated Software Engineering : An International Journal  
Static value analysis is a classical approach for verifying programs with floating-point computations.  ...  More precisely, we introduce a hybrid approach combining abstract interpretation and constraint programming techniques in a single static and automatic analysis.  ...  An approximation of a solution is described by a box, that is the Cartesian product of the domains of the variables.  ... 
doi:10.1007/s10515-014-0154-2 fatcat:5jp5yqpswjd3xipkxnfyev33be

Online Verification of Cognitive Car Decisions

Matthias Althoff, Olaf Stursberg, Martin Buss
2007 IEEE Intelligent Vehicles Symposium  
An efficient online algorithm is presented that leads to the result whether the reachable sets of different traffic participants can intersect, meaning that the control strategy of the cognitive car is  ...  This paper suggests an approach for verifying safety by computing reachable sets for the position of relevant traffic participants, i.e. a cognitive car as well as moving objects in its environment.  ...  This allows to calculate the probability of a crash and due to the conservative computation of reachable sets, safety can be guaranteed if the set intersection is empty.  ... 
doi:10.1109/ivs.2007.4290203 fatcat:4h34bqxv3bc4phhhqy3kso2qae

Verification of temporal properties on hybrid automata by simulation relations

A. D'Innocenzo, A.A. Julius, G.J. Pappas, M.D. Di Benedetto, S. Di Gennaro
2007 2007 46th IEEE Conference on Decision and Control  
This approach allows checking properties such as safety, but also timing properties of hybrid automata. We apply our framework to a relevant case study in the context of air traffic management (ATM).  ...  The contribution of this paper is to show that, given a hybrid automaton and the durational graph abstraction, there exists a simulation relation between the two systems.  ...  A/C 1 Approach A/C 2 Approach Ω is the set of logic propositions on the set of variables {φ 1 , φ 2 }, where φ 1 is true if A/C 1 is in landing mode, and φ 2 is true if A/C 2 is in landing mode.  ... 
doi:10.1109/cdc.2007.4434716 dblp:conf/cdc/DInnocenzoJPBG07 fatcat:armkmiyqzzfvpfjqwso7al34ce

Static Analysis of Finite Precision Computations [chapter]

Eric Goubault, Sylvie Putot
2011 Lecture Notes in Computer Science  
We describe two classes of abstractions, a non relational one based on intervals, and a weakly relational one based on parametrized zonotopic abstract domains called affine sets, especially well suited  ...  evaluate not only the rounding error, but also sensitivity to inputs or parameters of the program.  ...  From this, the finite precision value of variables can be bounded by the reduced product of r x − e x with the intersection of a direct interval computation of f x .  ... 
doi:10.1007/978-3-642-18275-4_17 fatcat:mxus7c4yxrgvvnyyhg3capj34m

COMs: Complexes of Oriented Matroids [article]

Hans-Juergen Bandelt, Victor Chepoi, Kolja Knauer
2018 arXiv   pre-print
Relaxing realizability to local realizability, we capture a wider class of combinatorial objects: we show that non-positively curved Coxeter zonotopal complexes give rise to locally realizable COMs.  ...  A realizable COM is represented by a hyperplane arrangement restricted to an open convex set.  ...  ) for careful reading of previous versions and suggesting a simpler argument in the proof of Theorem 1.  ... 
arXiv:1507.06111v3 fatcat:c74oh7ljvfdt7aeky7qkeu2bw4
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