State set representations and their usage in the reachability analysis of hybrid systems

Hybrid systems in computer science are systems with combined discretecontinuous behavior. This work presents results obtained in the field of safety verification for linear hybrid systems whose continuous behavior can be described by linear differential equations. We focus on a special technique named flowpipe-construction-based reachability analysis, which over-approximates the reachable states of a given hybrid system as a finite union of state sets. In these computations we can use different geometric and symbolic representations for state sets as datatypes. The choice of the state set representation has a strong impact on the precision of the approximation and on the running time of the analysis method. Additionally, numerous further parameters and heuristics influence the analysis outcome. In this work we investigate on the influence and optimal usage of these parameters. Our results are collected in a publicly available open-source C++ programming library named HyPro. The major contributions of this work are threefold: • We present our HyPro library offering implementations for several state set representations that are commonly used in flowpipe-construction-based reachability analysis. A unified interface in combination with reduction and conversion methods supports the fast implementation of versatile analysis methods for linear hybrid systems. • We put our library to practice and show its applicability by embedding a flowpipe-construction-based reachability analysis method in a CEGAR-based abstraction refinement framework. The parallelization of this approach further increases its performance. • We introduce methods to decompose the search space and replace high-dimensional computations by computations in lowerdimensional subspaces. This method is applicable under certain conditions. An automated check of these conditions, an automated decomposition, and the integration of dedicated analysis methods for subspace computations extend our approach.

[1]  T. Dang.  Vérification et synthèse des systèmes hybrides , 2000 .

[2]  André Platzer,et al.  KeYmaera: A Hybrid Theorem Prover for Hybrid Systems (System Description) , 2008, IJCAR.

[3]  Rajeev Alur,et al.  Counterexample-guided predicate abstraction of hybrid systems , 2003, Theor. Comput. Sci..

[4]  Amir Pnueli,et al.  Orthogonal Polyhedra: Representation and Computation , 1999, HSCC.

[5]  Stanley Bak,et al.  Direct Verification of Linear Systems with over 10000 Dimensions , 2017, ARCH@CPSWeek.

[6]  Eduardo F. Camacho,et al.  Guaranteed state estimation by zonotopes , 2005, Autom..

[7]  Jacob K. White,et al.  A trajectory piecewise-linear approach to model order reduction and fast simulation of nonlinear circuits and micromachined devices , 2001, IEEE/ACM International Conference on Computer Aided Design. ICCAD 2001. IEEE/ACM Digest of Technical Papers (Cat. No.01CH37281).

[8]  Klaus Wehrle,et al.  Perfect difference sets for neighbor discovery: energy efficient and fair , 2011, ExtremeCom.

[9]  David A. McAllester,et al.  Solving Polynomial Systems Using a Branch and Prune Approach , 1997 .

[10]  Stefan Kowalewski,et al.  Safety Assessment of Networked Vehicle Platoon Controllers - Practical Experiences With Available Tools , 2009, ADHS.

[11]  Pravin Varaiya,et al.  Ellipsoidal Techniques for Reachability Analysis , 2000, HSCC.

[12]  Jeremy Sproston.  Decidable Model Checking of Probabilistic Hybrid Automata , 2000, FTRTFT.

[13]  Michael Joswig,et al.  polymake: a Framework for Analyzing Convex Polytopes , 2000 .

[14]  David P. Dobkin,et al.  The quickhull algorithm for convex hulls , 1996, TOMS.

[15]  Edmund M. Clarke,et al.  Counterexample-guided abstraction refinement , 2003, 10th International Symposium on Temporal Representation and Reasoning, 2003 and Fourth International Conference on Temporal Logic. Proceedings..

[16]  Leslie Lamport,et al.  A fast mutual exclusion algorithm , 1987, TOCS.

[17]  Christopher Kugler,et al.  A Polytope Library for the Reachability Analysis of Hybrid Systems , 2014 .

[18]  Erika Ábrahám,et al.  Efficient Dynamic Error Reduction for Hybrid Systems Reachability Analysis , 2018, TACAS.

[19]  Sergiy Bogomolov,et al.  Reach Set Approximation through Decomposition with Low-dimensional Sets and High-dimensional Matrices , 2018, HSCC.

[20]  Erika Ábrahám,et al.  Analysis and synthesis of hybrid systems in engineering applications , 2017 .

[21]  Jean-Louis Imbert.  About Redundant Inequalities Generated by Fourier's Algorithm , 1990, AIMSA.

[22]  Lijun Zhang,et al.  Safety Verification for Probabilistic Hybrid Systems , 2010, Eur. J. Control.

[23]  R. Alur,et al.  Modelchecking for real-time systems , 1990 .

[24]  Erika Ábrahám,et al.  Spread the Work: Multi-threaded Safety Analysis for Hybrid Systems , 2018, SEFM.

[25]  Erika Ábrahám,et al.  Analyzing Hybrid Petri nets with multiple stochastic firings using HyPro , 2017, VALUETOOLS.

[26]  A.D. Ames,et al.  Characterization of Zeno behavior in hybrid systems using homological methods , 2005, Proceedings of the 2005, American Control Conference, 2005..

[27]  Eugene M. Izhikevich,et al.  Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting , 2006 .

[28]  Lei Bu,et al.  ARCH-COMP17 Category Report: Bounded Model Checking of Hybrid Systems with Piecewise Constant Dynamics , 2017, ARCH@CPSWeek.

[29]  Sergiy Bogomolov,et al.  Abstraction-Based Guided Search for Hybrid Systems , 2013, SPIN.

[30]  Roberto Bagnara,et al.  The Parma Polyhedra Library: Toward a complete set of numerical abstractions for the analysis and verification of hardware and software systems , 2006, Sci. Comput. Program..

[31]  Xin Chen,et al.  Flow*: An Analyzer for Non-linear Hybrid Systems , 2013, CAV.

[32]  David Avis,et al.  A pivoting algorithm for convex hulls and vertex enumeration of arrangements and polyhedra , 1992, Discret. Comput. Geom..

[33]  Matthias Althoff,et al.  ARCH-COMP17 Category Report: Continuous and Hybrid Systems with Linear Continuous Dynamics , 2017, ARCH@CPSWeek.

[34]  M. Hirsch,et al.  Differential Equations, Dynamical Systems, and Linear Algebra , 1974 .

[35]  O. Stursberg,et al.  Computing Reachable Sets of Hybrid Systems Using a Combination of Zonotopes and Polytopes , 2010 .

[36]  Sergiy Bogomolov,et al.  A Box-Based Distance between Regions for Guiding the Reachability Analysis of SpaceEx , 2012, CAV.

[37]  Sergiy Bogomolov,et al.  Guided search for hybrid systems based on coarse-grained space abstractions , 2015, International Journal on Software Tools for Technology Transfer.

[38]  A. T. Hofkamp,et al.  CIF 3: Model-Based Engineering of Supervisory Controllers , 2014, TACAS.

[39]  A. Girard,et al.  Reachability analysis of linear systems using support functions , 2010 .

[40]  Colas Le Guernic.  Reachability Analysis of Hybrid Systems with Linear Continuous Dynamics. (Calcul d'Atteignabilité des Systèmes Hybrides à Partie Continue Linéaire) , 2009 .

[41]  Erika Ábrahám,et al.  Interval Constraint Propagation in SMT Compliant Decision Procedures , 2013 .

[42]  Stanley Bak,et al.  HyLAA: A Tool for Computing Simulation-Equivalent Reachability for Linear Systems , 2017, HSCC.

[43]  Sriram Sankaranarayanan,et al.  Symbolic Model Checking of Hybrid Systems Using Template Polyhedra , 2008, TACAS.

[44]  Thomas A. Henzinger,et al.  Counterexample-Guided Refinement of Template Polyhedra , 2017, TACAS.

[45]  Xin Chen,et al.  Efficient Bounded Reachability Computation for Rectangular Automata , 2011, RP.

[46]  Xin Chen,et al.  Current Challenges in the Verification of Hybrid Systems , 2015, CyPhy.

[47]  Luan Viet Nguyen,et al.  International Workshop on Applied Verification for Continuous and Hybrid Systems Large-Scale Linear Systems from Order-Reduction ( Benchmark Proposal ) , 2017 .

[48]  Matthias Althoff,et al.  Online Verification of Automated Road Vehicles Using Reachability Analysis , 2014, IEEE Transactions on Robotics.

[49]  David Monniaux,et al.  On using floating-point computations to help an exact linear arithmetic decision procedure , 2009, CAV.

[50]  Davide Bresolin,et al.  Computing the Evolution of Hybrid Systems using Rigorous Function Calculus , 2012, ADHS.

[51]  Erika Ábrahám,et al.  Context-Dependent Reachability Analysis for Hybrid Systems , 2018, 2018 IEEE International Conference on Information Reuse and Integration (IRI).

[52]  Wang Yi,et al.  Timed Automata: Semantics, Algorithms and Tools , 2003, Lectures on Concurrency and Petri Nets.

[53]  Ezio Bartocci,et al.  Parallel reachability analysis of hybrid systems in XSpeed , 2019, International Journal on Software Tools for Technology Transfer.

[54]  Ray A. Jarvis,et al.  On the Identification of the Convex Hull of a Finite Set of Points in the Plane , 1973, Inf. Process. Lett..

[55]  E. Ábrahám,et al.  DEVELOPMENT OF A MODULAR APPROACH FOR HYBRID SYSTEMS REACHABILITY ANALYSIS , 2016 .

[56]  Matthias Althoff,et al.  ARCH-COMP19 Category Report: Continuous and Hybrid Systems with Linear Continuous Dynamics , 2019, ARCH@CPSIoTWeek.

[57]  Nicolas Halbwachs,et al.  Automatic discovery of linear restraints among variables of a program , 1978, POPL.

[58]  Thomas A. Henzinger,et al.  The Algorithmic Analysis of Hybrid Systems , 1995, Theor. Comput. Sci..

[59]  Goran Frehse,et al.  Modular, hierarchical models of control systems in SpaceEx , 2013, 2013 European Control Conference (ECC).

[60]  M. Fisher,et al.  A semiclosed-loop algorithm for the control of blood glucose levels in diabetics , 1991, IEEE Transactions on Biomedical Engineering.

[61]  Stefan Kowalewski,et al.  Safety Verification of a Controlled Cooperative Platoon Under Loss of Communication Using Zonotopes , 2012, ADHS.

[62]  Xin Chen,et al.  Decomposed Reachability Analysis for Nonlinear Systems , 2016, 2016 IEEE Real-Time Systems Symposium (RTSS).

[63]  Andreas Eggers.  Direct handling of ordinary differential equations in constraint-solving-based analysis of hybrid systems , 2014 .

[64]  Xin Chen,et al.  A Benchmark Suite for Hybrid Systems Reachability Analysis , 2015, NFM.

[65]  Antoine Girard,et al.  Zonotope/Hyperplane Intersection for Hybrid Systems Reachability Analysis , 2008, HSCC.

[66]  Alexandre Chapoutot,et al.  Enclosing Temporal Evolution of Dynamical Systems Using Numerical Methods , 2013, NASA Formal Methods.

[67]  Ronald L. Graham,et al.  An Efficient Algorithm for Determining the Convex Hull of a Finite Planar Set , 1972, Inf. Process. Lett..

[68]  Pravin Varaiya,et al.  Ellipsoidal Techniques for Reachability Analysis of Discrete-Time Linear Systems , 2007, IEEE Transactions on Automatic Control.

[69]  Nacim Meslem,et al.  A Hybrid Bounding Method for Computing an Over-Approximation for the Reachable Set of Uncertain Nonlinear Systems , 2009, IEEE Transactions on Automatic Control.

[70]  Thomas A. Henzinger,et al.  The theory of hybrid automata , 1996, Proceedings 11th Annual IEEE Symposium on Logic in Computer Science.

[71]  Jing Liu,et al.  Spatio-temporal Hybrid Automata for Cyber-Physical Systems , 2013, ICTAC.

[72]  Komei Fukuda,et al.  From the zonotope construction to the Minkowski addition of convex polytopes , 2004, J. Symb. Comput..

[73]  René David,et al.  On Hybrid Petri Nets , 2001, Discret. Event Dyn. Syst..

[74]  Xin Chen,et al.  Reachability analysis of non-linear hybrid systems using Taylor Models , 2015 .

[75]  Edmund M. Clarke,et al.  dReal: An SMT Solver for Nonlinear Theories over the Reals , 2013, CADE.

[76]  David L. Dill,et al.  Timing Assumptions and Verification of Finite-State Concurrent Systems , 1989, Automatic Verification Methods for Finite State Systems.

[77]  Antoine Girard,et al.  SpaceEx: Scalable Verification of Hybrid Systems , 2011, CAV.

[78]  Joseph Sifakis,et al.  An Approach to the Description and Analysis of Hybrid Systems , 1992, Hybrid Systems.

[79]  Erika Ábrahám,et al.  Divide and Conquer: Variable Set Separation in Hybrid Systems Reachability Analysis , 2017, QAPL@ETAPS.

[80]  Stefan Kowalewski,et al.  HyReach: A Reachability Tool for Linear Hybrid Systems Based on Support Functions , 2016, ARCH@CPSWeek.

[81]  Stefan Ratschan,et al.  Safety Verification of Hybrid Systems by Constraint Propagation Based Abstraction Refinement , 2005, HSCC.

[82]  Goran Frehse.  PHAVer: Algorithmic Verification of Hybrid Systems Past HyTech , 2005, HSCC.

[83]  Armando Tacchella,et al.  Robot Swarms as Hybrid Systems: Modelling and Verification , 2022, SNR.

[84]  Stefan Kowalewski,et al.  HyPro: A C++ Library of State Set Representations for Hybrid Systems Reachability Analysis , 2017, NFM.

[85]  Martin Fränzle,et al.  SAT Modulo ODE: A Direct SAT Approach to Hybrid Systems , 2008, ATVA.

[86]  Komei Fukuda,et al.  Computing faces up to k dimensions of a Minkowski Sum of Polytopes , 2005, CCCG.

[87]  Joël Ouaknine,et al.  Abstraction and Counterexample-Guided Refinement in Model Checking of Hybrid Systems , 2003, Int. J. Found. Comput. Sci..

[88]  Antoine Girard,et al.  Reachability of Uncertain Linear Systems Using Zonotopes , 2005, HSCC.

[89]  Kazunori Ueda,et al.  An interval-based SAT modulo ODE solver for model checking nonlinear hybrid systems , 2011, International Journal on Software Tools for Technology Transfer.

[90]  Xin Chen,et al.  Taylor Model Flowpipe Construction for Non-linear Hybrid Systems , 2012, 2012 IEEE 33rd Real-Time Systems Symposium.

[91]  Rajeev Alur,et al.  A Theory of Timed Automata , 1994, Theor. Comput. Sci..

[92]  John Lygeros,et al.  Toward a General Theory of Stochastic Hybrid Systems , 2006 .

[93]  Toby Walsh,et al.  Handbook of Satisfiability: Volume 185 Frontiers in Artificial Intelligence and Applications , 2009 .

[94]  Olaf Stursberg,et al.  Efficient Representation and Computation of Reachable Sets for Hybrid Systems , 2003, HSCC.

[95]  Matthias Althoff,et al.  An Introduction to CORA 2015 , 2015, ARCH@CPSWeek.

[96]  René David,et al.  Continuous and Hybrid Petri Nets , 1998, J. Circuits Syst. Comput..

[97]  Goran Frehse,et al.  Flowpipe-Guard Intersection for Reachability Computations with Support Functions , 2012, ADHS.

[98]  Sebastian Junges,et al.  SMT-RAT: An Open Source C++ Toolbox for Strategic and Parallel SMT Solving , 2015, SAT.

[99]  Martin Berz,et al.  Rigorous integration of flows and ODEs using taylor models , 2009, SNC '09.

[100]  George B. Dantzig,et al.  Linear programming and extensions , 1965 .

[101]  Erika Ábrahám,et al.  OPTIMIZING REACHABILITY ANALYSIS FOR NON-AUTONOMOUS HYBRID SYSTEMS USING ELLIPSOIDS , 2016 .

[102]  Karl Pearson F.R.S..  LIII. On lines and planes of closest fit to systems of points in space , 1901 .

[103]  Goran Frehse,et al.  Non-convex Invariants and Urgency Conditions on Linear Hybrid Automata , 2014, FORMATS.

[104]  Pravin Varaiya,et al.  What's decidable about hybrid automata? , 1995, STOC '95.

[105]  André Platzer,et al.  Differential Dynamic Logic for Hybrid Systems , 2008, Journal of Automated Reasoning.

[106]  Xin Chen,et al.  Counterexample Generation for Hybrid Automata , 2013, FTSCS.

[107]  Goran Frehse,et al.  Flowpipe approximation and clustering in space-time , 2013, HSCC '13.

[108]  Nikolaj Bjørner,et al.  Z3: An Efficient SMT Solver , 2008, TACAS.

[109]  Goran Frehse,et al.  Design Principles for an Extendable Verification Tool for Hybrid Systems , 2009, ADHS.

[110]  Christel Baier,et al.  Principles of model checking , 2008 .

[111]  Armando Tacchella,et al.  Engineering Controllers For Swarm Robotics Via Reachability Analysis In Hybrid Systems , 2019, ECMS.

[112]  Lei Bu,et al.  ARCH-COMP18 Category Report: Bounded Model Checking of Hybrid Systems with Piecewise Constant Dynamics , 2018, ARCH@ADHS.

[113]  Roland Wunderling.  Paralleler und Objektorientierter Simplex , 1996 .

[114]  Ansgar Fehnker,et al.  Benchmarks for Hybrid Systems Verification , 2004, HSCC.