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The Zonotope Abstract Domain Taylor1+ [chapter]

Khalil Ghorbal, Eric Goubault, Sylvie Putot
2009 Lecture Notes in Computer Science  
doi:10.1007/978-3-642-02658-4_47 fatcat:4ucmn7na2nd3dkxzfe3wkwtpji

Characterizing Algebraic Invariants by Differential Radical Invariants [chapter]

Khalil Ghorbal, André Platzer
2014 Lecture Notes in Computer Science  
We prove that any invariant algebraic set of a given polynomial vector field can be algebraically represented by one polynomial and a finite set of its successive Lie derivatives. This so-called differential radical characterization relies on a sound abstraction of the reachable set of solutions by the smallest variety that contains it. The characterization leads to a differential radical invariant proof rule that is sound and complete, which implies that invariance of algebraic equations over
more » ... aic equations over real-closed fields is decidable. Furthermore, the problem of generating invariant varieties is shown to be as hard as minimizing the rank of a symbolic matrix, and is therefore NP-hard. We investigate symbolic linear algebra tools based on Gaussian elimination to efficiently automate the generation. The approach can, e.g., generate nontrivial algebraic invariant equations capturing the airplane behavior during take-off or landing in longitudinal motion.
doi:10.1007/978-3-642-54862-8_19 fatcat:5fbanhbgzvbk3meu6m2luoqhg4

Characterizing Positively Invariant Sets: Inductive and Topological Methods [article]

Khalil Ghorbal, Andrew Sogokon
2021 arXiv   pre-print
Our implementation is available from[Ghorbal, 2020] 9 The set of escape points is fundamental to the Ważewski principle. See Conley [1978] where it is denoted as W • for a set W .  ...  condition to ideals generated by successive Lie derivatives of polynomials in order to prove invariance of algebraic varieties in polynomial vector fields was employed by Novikov and Yakovenko [1999] , Ghorbal  ... 
arXiv:2009.09797v2 fatcat:pm47nhta4fhs5o3yq74z2zcgfi

Vector Barrier Certificates and Comparison Systems [chapter]

Andrew Sogokon, Khalil Ghorbal, Yong Kiam Tan, André Platzer
2018 Lecture Notes in Computer Science  
Vector Lyapunov functions are a multi-dimensional extension of the more familiar (scalar) Lyapunov functions, commonly used to prove stability properties in systems of non-linear ordinary differential equations (ODEs). This paper explores an analogous vector extension for so-called barrier certificates used in safety verification. As with vector Lyapunov functions, the approach hinges on constructing appropriate comparison systems, i.e., related differential equation systems from which
more » ... from which properties of the original system may be inferred. The paper presents an accessible development of the approach, demonstrates that most previous notions of barrier certificate are special cases of comparison systems, and discusses the potential applications of vector barrier certificates in safety verification and invariant synthesis.
doi:10.1007/978-3-319-95582-7_25 fatcat:hrgch7sazvfmdcn2fhjrcbygna

Decoupling Abstractions of Non-linear Ordinary Differential Equations [chapter]

Andrew Sogokon, Khalil Ghorbal, Taylor T. Johnson
2016 Lecture Notes in Computer Science  
HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L'archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou
more » ... he français ou étrangers, des laboratoires publics ou privés.
doi:10.1007/978-3-319-48989-6_38 fatcat:6nsv4hajczfoxgatzt7j63x4be

A Method for Invariant Generation for Polynomial Continuous Systems [chapter]

Andrew Sogokon, Khalil Ghorbal, Paul B. Jackson, André Platzer
2015 Lecture Notes in Computer Science  
This paper presents a method for generating semi-algebraic invariants for systems governed by non-linear polynomial ordinary differential equations under semi-algebraic evolution constraints. Based on the notion of discrete abstraction, our method eliminates unsoundness and unnecessary coarseness found in existing approaches for computing abstractions for non-linear continuous systems and is able to construct invariants with intricate boolean structure, in contrast to invariants typically
more » ... nts typically generated using template-based methods. In order to tackle the state explosion problem associated with discrete abstraction, we present invariant generation algorithms that exploit sound proof rules for safety verification, such as differential cut (DC), and a new proof rule that we call differential divide-and-conquer (DDC), which splits the verification problem into smaller sub-problems. The resulting invariant generation method is observed to be much more scalable and efficient than the naïve approach, exhibiting orders of magnitude performance improvement on many of the problems.
doi:10.1007/978-3-662-49122-5_13 fatcat:wed2znuprjb6fb2boyynodfbg4

Operational Models for Piecewise-Smooth Systems

Andrew Sogokon, Khalil Ghorbal, Taylor T. Johnson
2017 ACM Transactions on Embedded Computing Systems  
[Ghorbal and Platzer 2014; Novikov and Yakovenko 1999] .  ...  ., Ghorbal, K., Johnson, T.T. increasingly used in modelling and analyzing the behaviour of modern control systems employing embedded devices.  ... 
doi:10.1145/3126506 fatcat:qtr5cirzlvg75essiv26takv5e

Multi-Mode DAE Models - Challenges, Theory and Implementation [chapter]

Albert Benveniste, Benoît Caillaud, Hilding Elmqvist, Khalil Ghorbal, Martin Otter, Marc Pouzet
2019 Lecture Notes in Computer Science  
Our objective is to model and simulate Cyber-Physical Systems (CPS) such as robots, vehicles, and power plants. The structure of CPS models may change during simulation due to the desired operation, due to failure situations or due to changes in physical conditions. Corresponding models are called multi-mode. We are interested in multidomain, component-oriented modeling as performed, for example, with the modeling language Modelica that leads naturally to Differential Algebraic Equations
more » ... ic Equations (DAEs). This paper is thus about multi-mode DAE systems. In particular, new methods are discussed to overcome one key problem that was only solved for specific subclasses of systems before: How to switch from one mode to another one when the number of equations may change and variables may exhibit impulsive behavior? An evaluation is performed both with the experimental modeling and simulation system Modia, a domain specific language extension of the programming language Julia, and with SunDAE, a novel structural analysis library for multi-mode DAE systems.
doi:10.1007/978-3-319-91908-9_16 fatcat:bpjyurkbtbbibd3m4fx27v44qe

Invariance of Conjunctions of Polynomial Equalities for Algebraic Differential Equations [chapter]

Khalil Ghorbal, Andrew Sogokon, André Platzer
2014 Lecture Notes in Computer Science  
Public reporting burden for the collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this collection of information, including suggestions for reducing this burden, to Washington Headquarters Services, Directorate for Information
more » ... te for Information Operations and Reports, 1215 Jefferson Davis Highway, Suite 1204, Arlington VA 22202-4302. Respondents should be aware that notwithstanding any other provision of law, no person shall be subject to a penalty for failing to comply with a collection of information if it does not display a currently valid OMB control number. 1. REPORT DATE continuous and hybrid dynamical systems. We present an efficient procedure to check invariance of conjunctions of polynomial equalities under the flow of polynomial ordinary differential equations. The procedure is based on a necessary and sufficient condition that characterizes invariant conjunctions of polynomial equalities. We contrast this approach to an alternative one which combines fast and sufficient (but not necessary) conditions using differential cuts for soundly restricting the system evolution domain. 15 . SUBJECT TERMS 16. SECURITY CLASSIFICATION OF: 17. LIMITATION OF ABSTRACT Same as Report (SAR) 18. NUMBER OF PAGES 37 19a. NAME OF RESPONSIBLE PERSON a. REPORT unclassified b. ABSTRACT unclassified c. THIS PAGE unclassified Standard Form 298 (Rev. 8-98) Prescribed by ANSI Std Z39-18 Abstract In this paper we seek to provide greater automation for formal deductive verification tools working with continuous and hybrid dynamical systems. We present an efficient procedure to check invariance of conjunctions of polynomial equalities under the flow of polynomial ordinary differential equations. The procedure is based on a necessary and sufficient condition that characterizes invariant conjunctions of polynomial equalities. We contrast this approach to an alternative one which combines fast and sufficient (but not necessary) conditions using differential cuts for soundly restricting the system evolution domain.
doi:10.1007/978-3-319-10936-7_10 fatcat:ytgh7vnw2zhslhkgawp3chonbi

On Provably Safe Obstacle Avoidance for Autonomous Robotic Ground Vehicles

Stefan Mitsch, Khalil Ghorbal, Andre Platzer
2013 Robotics: Science and Systems IX  
Nowadays, robots interact more frequently with a dynamic environment outside limited manufacturing sites and in close proximity with humans. Thus, safety of motion and obstacle avoidance are vital safety features of such robots. We formally study two safety properties of avoiding both stationary and moving obstacles: (i) passive safety, which ensures that no collisions can happen while the robot moves, and (ii) the stronger passive friendly safety in which the robot further maintains sufficient
more » ... aintains sufficient maneuvering distance for obstacles to avoid collision as well. We use hybrid system models and theorem proving techniques that describe and formally verify the robot's discrete control decisions along with its continuous, physical motion. Moreover, we formally prove that safety can still be guaranteed despite location and actuator uncertainty.
doi:10.15607/rss.2013.ix.014 dblp:conf/rss/MitschGP13 fatcat:6suwybwk7rclrdwqr7z5w2ys4a

Donut Domains: Efficient Non-convex Domains for Abstract Interpretation [chapter]

Khalil Ghorbal, Franjo Ivančić, Gogul Balakrishnan, Naoto Maeda, Aarti Gupta
2012 Lecture Notes in Computer Science  
Program analysis using abstract interpretation has been successfully applied in practice to find runtime bugs or prove software correct. Most abstract domains that are used widely rely on convexity for their scalability. However, the ability to express non-convex properties is sometimes required in order to achieve a precise analysis of some numerical properties. This work combines already known abstract domains in a novel way in order to design new abstract domains that tackle some non-convex
more » ... le some non-convex invariants. The abstract objects of interest are encoded as a pair of two convex abstract objects: the first abstract object defines an over-approximation of the possible reached values, as is done customarily. The second abstract object under-approximates the set of impossible values within the state-space of the first abstract object. Therefore, the geometrical concretization of our objects is defined by a convex set minus another convex set (or hole). We thus call these domains donut domains.
doi:10.1007/978-3-642-27940-9_16 fatcat:xdnwvbxrmzejhox5arjoknwxsa

Efficient Probabilistic Model Checking of Systems with Ranged Probabilities [chapter]

Khalil Ghorbal, Parasara Sridhar Duggirala, Vineet Kahlon, Franjo Ivančić, Aarti Gupta
2012 Lecture Notes in Computer Science  
We introduce a new technique to model check reachability properties on Interval Discrete-Time Markov Chains (IDTMC). We compute a sound overapproximation of the probabilities of satisfying a given property where the accuracy is characterized in terms of error bounds. We leverage affine arithmetic to propagate the first-order error terms. Higher-order error terms are bounded using interval arithmetic.
doi:10.1007/978-3-642-33512-9_10 fatcat:acfaol4nqbewbeszg6yky7snjy

A Hierarchy of Proof Rules for Checking Differential Invariance of Algebraic Sets [chapter]

Khalil Ghorbal, Andrew Sogokon, André Platzer
2015 Lecture Notes in Computer Science  
This paper presents a theoretical and experimental comparison of sound proof rules for proving invariance of algebraic sets, that is, sets satisfying polynomial equalities, under the flow of polynomial ordinary differential equations. Problems of this nature arise in formal verification of continuous and hybrid dynamical systems, where there is an increasing need for methods to expedite formal proofs. We study the trade-off between proof rule generality and practical performance and evaluate
more » ... nce and evaluate our theoretical observations on a set of heterogeneous benchmarks. The relationship between increased deductive power and running time performance of the proof rules is far from obvious; we discuss and illustrate certain classes of problems where this relationship is interesting.
doi:10.1007/978-3-662-46081-8_24 fatcat:wa3q2u5n3bdarj7cazaiksyo6a

Ordered Functional Decision Diagrams: A Functional Semantics For Binary Decision Diagrams [article]

Joan Thibault, Khalil Ghorbal
2020 arXiv   pre-print
We introduce a novel framework, termed λDD, that revisits Binary Decision Diagrams from a purely functional point of view. The framework allows to classify the already existing variants, including the most recent ones like Chain-DD and ESRBDD, as implementations of a special class of ordered models. We enumerate, in a principled way, all the models of this class and isolate its most expressive model. This new model, termed λDD-O-NUCX, is suitable for both dense and sparse Boolean functions, and
more » ... lean functions, and is moreover invariant by negation. The canonicity of λDD-O-NUCX is formally verified using the Coq proof assistant. We furthermore give bounds on the size of the different diagrams: the potential gain achieved by more expressive models can be at most linear in the number of variables n.
arXiv:2003.09340v4 fatcat:fkafwperdfgejdnk6uys6tvz64

A hierarchy of proof rules for checking positive invariance of algebraic and semi-algebraic sets

Khalil Ghorbal, Andrew Sogokon, André Platzer
2017 Computer languages, systems & structures  
We used the notation DI = for the same proof rule in (Ghorbal et al., 2015) . See(Dumortier et al., 2006, Proposition 8.4) for a similar proposition over the complex numbers.  ...  We extend our earlier analysis presented in (Ghorbal et al., 2015) to include proof rules that are concerned with checking positive invariance 1 That is explicitly given in terms of elementary functions  ... 
doi:10.1016/ fatcat:f3ujzbyofvfsfguwhtg6g7vhwm
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