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The following article showcases the basic functionality, highlighting some of the key design choices:
Forets, Marcelo, and Christian Schilling. LazySets.jl: Scalable Symbolic-Numeric Set Computations. Proceedings of the JuliaCon Conferences (2021).
See below for how to cite it.
- Manual
- Release notes of the development version
- Release notes of previous versions
- How to contribute
- Developers team
LazySets.jl is a registered Julia package and as such you can install it by activating the pkg mode (type ], and to leave it, type <backspace>),
followed by
pkg> add LazySetsSee the Getting started section of the manual for other options.
This library has been applied in a number of scientific works.
Click to see the full list of publications that use LazySets.
The articles appear in reverse chronological order.
[33] Verified propagation of imprecise probabilities in non-linear ODEs. Ander Gray, Marcelo Forets, Christian Schilling, Scott Ferson, and Luis Benet (2024). International Journal of Approximate Reasoning, vol. 164. doi: 10.1016/j.ijar.2023.109044.
[32] Safety verification of decision-tree policies in continuous time. Christian Schilling, Anna Lukina, Emir Demirović, and Kim G. Larsen (2023). 37th Conference on Neural Information Processing Systems (NeurIPS). pdf.
[31] Shielded reinforcement learning for hybrid systems. Asger H. Brorholt, Peter G. Jensen, Kim G. Larsen, Florian Lorber, and Christian Schilling (2023). 1st International Conference on Bridging the Gap between AI and Reality (AISoLA), LNCS, vol. 14380, pp. 33-54. doi: 10.1007/978-3-031-46002-9_3, arXiv: 2308.14424.
[30] The inverse problem for neural networks. Marcelo Forets and Christian Schilling (2023). 1st International Conference on Bridging the Gap between AI and Reality (AISoLA), LNCS, vol. 14380, pp. 241-255. doi: 10.1007/978-3-031-46002-9_14, arXiv: 2308.14093.
[29] ARCH-COMP23 category report: Continuous and hybrid systems with linear continuous dynamics. Matthias Althoff, Marcelo Forets, Yangge Li, Sayan Mitra, Christian Schilling, Mark Wetzlinger, and Daniel Zhuang (2023). 10th International Workshop on Applied Verification of Continuous and Hybrid Systems (ARCH23), vol 96, pp. 34-60. doi: 10.29007/nl86.
[28] ARCH-COMP23 category report: Continuous and hybrid systems with nonlinear dynamics. Luca Geretti, Julien Alexandre Dit Sandretto, Matthias Althoff, Luis Benet, Pieter Collins, Marcelo Forets, Elena Ivanova, Yangge Li, Sayan Mitra, Stefan Mitsch, Christian Schilling, Mark Wetzlinger, and Daniel Zhuang (2023). 10th International Workshop on Applied Verification of Continuous and Hybrid Systems (ARCH23), vol 96, pp. 61-88. doi: 10.29007/93f2.
[27] ARCH-COMP23 category report: Artificial intelligence and neural network control systems for continuous and hybrid systems plants. Diego Manzanas Lopez, Matthias Althoff, Marcelo Forets, Taylor T. Johnson, Tobias Ladner, and Christian Schilling (2023). 10th International Workshop on Applied Verification of Continuous and Hybrid Systems (ARCH23), vol 96, pp. 89-125. doi: 10.29007/x38n.
[26] ARCH-COMP22 category report: Continuous and hybrid systems with linear continuous dynamics. Matthias Althoff, Marcelo Forets, Christian Schilling, and Mark Wetzlinger (2022). 9th International Workshop on Applied Verification of Continuous and Hybrid Systems (ARCH22), vol 90, pp. 58-85. doi: 10.29007/mmzc.
[25] ARCH-COMP22 category report: Continuous and hybrid systems with nonlinear dynamics. Luca Geretti, Julien Alexandre Dit Sandretto, Matthias Althoff, Luis Benet, Pieter Collins, Parasara Sridhar Duggirala, Marcelo Forets, Edward Kim, Stefan Mitsch, Christian Schilling, and Mark Wetzlinger (2022). 9th International Workshop on Applied Verification of Continuous and Hybrid Systems (ARCH22), vol 90, pp. 86-112. doi: 10.29007/fnzc.
[24] ARCH-COMP22 category report: Artificial intelligence and neural network control systems for continuous and hybrid systems plants. Diego Manzanas Lopez, Matthias Althoff, Luis Benet, Xin Chen, Jiameng Fan, Marcelo Forets, Chao Huang, Taylor T. Johnson, Tobias Ladner, Wenchao Li, Christian Schilling, and Qi Zhu (2022). 9th International Workshop on Applied Verification of Continuous and Hybrid Systems (ARCH22), vol 90, pp. 142-184. doi: 10.29007/wfgr.
[23] Synthesis of parametric hybrid automata from time series. Miriam García Soto, Thomas A. Henzinger, and Christian Schilling (2022). Proceedings of the 20th International Symposium on Automated Technology for Verification and Analysis, LNCS, vol. 13505, pp. 337-353. doi: 10.1007/978-3-031-19992-9_22, arXiv: 2208.06383.
[22] Decomposing reach set computations with low-dimensional sets and high-dimensional matrices (extended version). Sergiy Bogomolov, Marcelo Forets, Goran Frehse, Andreas Podelski, and Christian Schilling (2022). Information and Computation, vol. 289. doi: 10.1016/j.ic.2022.104937.
[21] Conservative Time Discretization: A Comparative Study. Marcelo Forets and Christian Schilling (2022). Proceedings of the 17th International Conference on integrated Formal Methods (iFM), LNCS, vol. 13274, pp. 149-167. doi: 10.1007/978-3-031-07727-2_9, arXiv: 2111.01454.
[20] Verification of Neural-Network Control Systems by Integrating Taylor Models and Zonotopes. Christian Schilling, Marcelo Forets, and Sebastián Guadalupe (2022). Proceedings of the 36th Conference on Artificial Intelligence (AAAI). doi: 10.1609/aaai.v36i7.20790.
[19] Combining Set Propagation with Finite Element Methods for Time Integration in Transient Solid Mechanics Problems. Marcelo Forets, Daniel Freire Caporale, and Jorge M. Pérez Zerpa (2022). Computers & Structures, vol 259. doi: 10.1016/j.compstruc.2021.106699, arXiv: 2105.05841.
[18] LazySets.jl: Scalable Symbolic-Numeric Set Computations. Marcelo Forets and Christian Schilling (2021). Proceedings of the JuliaCon Conferences. doi: 10.21105/jcon.00097.
[17] Reachability of weakly nonlinear systems using Carleman linearization. Marcelo Forets and Christian Schilling (2021). Proceedings of the 15th International Conference on Reachability Problems (RP), LNCS, vol. 13035, pp. 85-99. doi: 10.1007/978-3-030-89716-1_6, arXiv: 2108.10390.
[16] Combined Exact and Heuristics Based Approach to Hamiltonian Path Problem Optimization for Route Planning. Fernando Hernandez, Rafael Sotelo, and Marcelo Forets (2021). Technical Proceedings of the 2021 Amazon Last Mile Routing Research Challenge.
[15] ARCH-COMP21 Category Report: Continuous and Hybrid Systems with Linear Continuous Dynamics. Matthias Althoff, Erika Abraham, Marcelo Forets, Goran Frehse, Daniel Freire, Christian Schilling, Stefan Schupp, and Mark Wetzlinger (2021). 8th International Workshop on Applied Verification of Continuous and Hybrid Systems (ARCH21), vol 80, pp. 1-31. doi: 10.29007/lhbw.
[14] ARCH-COMP21 Category Report: Continuous and Hybrid Systems with Nonlinear Dynamics. Luca Geretti, Julien Alexandre dit Sandretto, Matthias Althoff, Luis Benet, Alexandre Chapoutot, Pieter Collins, Parasara Sridhar Duggirala, Marcelo Forets, Edward Kim, Uziel Linares, David P. Sanders, Christian Schilling, and Mark Wetzlinger (2021). 8th International Workshop on Applied Verification of Continuous and Hybrid Systems (ARCH21), vol 80, pp. 32-54. doi: 10.29007/2jw8.
[13] ARCH-COMP21 Category Report: Artificial Intelligence and Neural Network Controlled Systems for Continuous and Hybrid Systems Plants. Taylor T. Johnson, Diego Manzanas Lopez, Luis Benet, Marcelo Forets, Christian Schilling, Radoslav Ivanov, Taylor Carpenter, James Weimer, and Insup Lee (2021). 8th International Workshop on Applied Verification of Continuous and Hybrid Systems (ARCH21), vol 80, pp. 90-119. doi: 10.29007/kfk9.
[12] Synthesis of hybrid automata with affine dynamics from time-series data. Miriam García Soto, Thomas A. Henzinger, and Christian Schilling (2021). 24th International Conference on Hybrid Systems: Computation and Control (HSCC). doi: 10.1145/3447928.3456704, arXiv: 2102.12734.
[11] Algorithms for verifying deep neural networks. Changliu Liu, Tomer Arnon, Christopher Lazarus, Christopher A. Strong, Clark W. Barrett, and Mykel J. Kochenderfer (2021). Foundations and Trends in Optimization, vol 4, pp. 244-404. doi: 10.1561/2400000035, arXiv: 1903.06758.
[10] Efficient reachability analysis of parametric linear hybrid systems with time-triggered transitions. Marcelo Forets, Daniel Freire, and Christian Schilling (2020). Proceedings of the 18th International Conference on Formal Methods and Models for System Design (MEMOCODE), pp. 137-142. doi: 10.1109/MEMOCODE51338.2020.9314994, arXiv: 2006.12325.
[9] ARCH-COMP20 Category Report: Continuous and Hybrid Systems with Linear Continuous Dynamics. Matthias Althoff, Stanley Bak, Zongnan Bao, Marcelo Forets, Daniel Freire, Goran Frehse, Niklas Kochdumper, Yangge Li, Sayan Mitra, Rajarshi Ray, Christian Schilling, Stefan Schupp, and Mark Wetzlinger (2020). 7th International Workshop on Applied Verification of Continuous and Hybrid Systems (ARCH20), vol 74, pp. 16-48. doi: 10.29007/7dt2.
[8] ARCH-COMP20 Category Report: Continuous and Hybrid Systems with Nonlinear Dynamics. Luca Geretti, Julien Alexandre dit Sandretto, Matthias Althoff, Luis Benet, Alexandre Chapoutot, Xin Chen, Pieter Collins, Marcelo Forets, Daniel Freire, Fabian Immler, Niklas Kochdumper, David P. Sanders, and Christian Schilling (2020). 7th International Workshop on Applied Verification of Continuous and Hybrid Systems (ARCH20), vol 74, pp. 49-75. doi: 10.29007/zkf6.
[7] Case Study: Reachability Analysis of a unified Combat-Command-and-Control Model. Sergiy Bogomolov, Marcelo Forets, and Kostiantyn Potomkin (2020). Proceedings of the 14th International Conference on Reachability Problems (RP), LNCS, vol 12448, pp. 52-66. doi: 10.1007/978-3-030-61739-4_4.
[6] Reachability analysis of linear hybrid systems via block decomposition. Sergiy Bogomolov, Marcelo Forets, Goran Frehse, Kostiantyn Potomkin, and Christian Schilling (2020). IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, vol. 39, pp. 4018-4029. Presented at Embedded Systems Week 2020. doi: 10.1109/TCAD.2020.3012859, arXiv: 1905.02458.
[5] ARCH-COMP19 Category Report: Continuous and Hybrid Systems with Linear Continuous Dynamics. Matthias Althoff, Stanley Bak, Marcelo Forets, Goran Frehse, Niklas Kochdumper, Rajarshi Ray, Christian Schilling, and Stefan Schupp (2019). 6th International Workshop on Applied Verification of Continuous and Hybrid Systems (ARCH19), vol 61, pp. 14-40. doi: 10.29007/bj1w.
[4] ARCH-COMP19 Category Report: Continuous and Hybrid Systems with Nonlinear Dynamics. Fabian Immler, Matthias Althoff, Luis Benet, Alexandre Chapoutot, Xin Chen, Marcelo Forets, Luca Geretti, Niklas Kochdumper, David P. Sanders, and Christian Schilling (2019). 6th International Workshop on Applied Verification of Continuous and Hybrid Systems (ARCH19), vol 61, pp. 41-61. doi: 10.29007/m75b.
[3] JuliaReach: a Toolbox for Set-Based Reachability. Sergiy Bogomolov, Marcelo Forets, Goran Frehse, Kostiantyn Potomkin, Christian Schilling (2019). Proceedings of the 22nd International Conference on Hybrid Systems: Computation and Control (HSCC), pp. 39-44. doi: 10.1145/3302504.3311804, arXiv: 1901.10736.
[2] ARCH-COMP18 Category Report: Continuous and Hybrid Systems with Linear Continuous Dynamics. Matthias Althoff, Stanley Bak, Xin Chen, Chuchu Fan, Marcelo Forets, Goran Frehse, Niklas Kochdumper, Yangge Li, Sayan Mitra, Rajarshi Ray, Christian Schilling, and Stefan Schupp (2018). 5th International Workshop on Applied Verification of Continuous and Hybrid Systems (ARCH18), vol 54, pp. 23–52. doi: 10.29007/73mb.
[1] Reach Set Approximation through Decomposition with Low-dimensional Sets and High-dimensional Matrices. Sergiy Bogomolov, Marcelo Forets, Goran Frehse, Frédéric Viry, Andreas Podelski, and Christian Schilling (2018). Proceedings of the 21st International Conference on Hybrid Systems: Computation and Control (HSCC), pp. 41–50. doi: 10.1145/3178126.3178128, arXiv: 1801.09526.
Several projects in the Julia technical computing stack use this library.
Click to see the full list of Julia packages that use LazySets.
- ConvexBodyProximityQueries.jl -- Proximity computation between convex bodies in 2D/3D.
- ClosedLoopReachability.jl -- Reachability analysis for closed-loop control systems.
- HySynthParametric -- Synthesis of parametric linear hybrid automata.
- IntervalLinearAlgebra.jl -- Routines to perform numerical linear algebra using interval arithmetic.
- InvariantSets.jl -- Compute, approximate and display invariant sets.
- InvariantSetApproximation.jl -- Invariant sets of general discrete-time dynamical systems with controls and uncertainties using graph-based algorithms.
- NeuralVerification.jl -- Methods to verify deep neural networks.
- OpticSim.jl -- Ray tracing for procedurally generated systems.
- Photometry.jl -- Utilities for characterizing sources in astronomical images.
- ReachabilityAnalysis.jl -- Methods to compute the sets of states reachable in dynamical systems.
- Swalbe.jl -- Simple Julia Lattice Boltzmann Solver for Thin Liquid Films and Droplets.
- TrajectoryGamesBase.jl -- Interface to define trajectory games.
Abstract
We present JuliaReach, a Julia ecosystem to perform reachability analysis of dynamical systems. JuliaReach builds on sound scientific approaches and was, in two occasions (2018 and 2020) the winner of the annual friendly competition on Applied Verification for Continuous and Hybrid Systems (ARCH-COMP).
The workshop consists of three parts (respectively packages) in JuliaReach: our core package for set representations, our main package for reachability analysis, and a new package applying reachability analysis with potential use in domain of control, robotics and autonomous systems.
In the first part we present LazySets.jl, which provides ways to symbolically represent sets of points as geometric shapes, with a special focus on convex sets and polyhedral approximations. LazySets.jl provides methods to apply common set operations, convert between different set representations, and efficiently compute with sets in high dimensions.
In the second part we present ReachabilityAnalysis.jl, which provides tools to approximate the set of reachable states of systems with both continuous and mixed discrete-continuous dynamics, also known as hybrid systems. It implements conservative discretization and set-propagation techniques at the state-of-the-art.
In the third part we present NeuralNetworkAnalysis.jl, which is an application of ReachabilityAnalysis.jl to analyze dynamical systems that are controlled by neural networks. This package can be used to validate or invalidate specifications, for instance about the safety of such systems.
Workshop materials are available here: https://github.com/JuliaReach/JuliaCon-2021-Workshop-Its-All-Set
If you use this package in your work, please cite it using the metadata here or below.
Click to see BibTeX entry.
@article{lazysets21,
title = {{LazySets.jl: Scalable Symbolic-Numeric Set Computations}},
author = {Forets, Marcelo and Schilling, Christian},
journal = {Proceedings of the JuliaCon Conferences},
year = {2021},
publisher = {The Open Journal},
volume = {1},
number = {1},
pages = {11},
doi = {10.21105/jcon.00097}
}