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A zonotopic framework for functional abstractions [article]

Eric Goubault, Sylvie Putot
2009 arXiv   pre-print
This article formalizes an abstraction of input/output relations, based on parameterized zonotopes, which we call affine sets.  ...  We describe the abstract transfer functions and prove their correctness, which allows the generation of accurate numerical invariants.  ...  We plan to develop them for this domain in a longer version. Conclusion and future work We set up a formal framework for a fast and accurate abstract analysis based on zonotopes.  ... 
arXiv:0910.1763v1 fatcat:pinbqzuasrdlrjmnpj2lh4i4fq

AI2: Safety and Robustness Certification of Neural Networks with Abstract Interpretation

Timon Gehr, Matthew Mirman, Dana Drachsler-Cohen, Petar Tsankov, Swarat Chaudhuri, Martin Vechev
2018 2018 IEEE Symposium on Security and Privacy (SP)  
We present a complete implementation of AI 2 together with an extensive evaluation on 20 neural networks.  ...  We present AI 2 , the first sound and scalable analyzer for deep neural networks.  ...  Then, we specified a robustness property for each image and each robustness bound in Δ, resulting in 60 properties per dataset.  ... 
doi:10.1109/sp.2018.00058 dblp:conf/sp/GehrMDTCV18 fatcat:6o35qb3wurhadnbaylclnv7hsq

Differentiable Abstract Interpretation for Provably Robust Neural Networks

Matthew Mirman, Timon Gehr, Martin T. Vechev
2018 International Conference on Machine Learning  
We introduce a scalable method for training robust neural networks based on abstract interpretation.  ...  We present several abstract transformers which balance efficiency with precision and show these can be used to train large neural networks that are certifiably robust to adversarial perturbations.  ...  An abstract domain D is a set equipped with an abstraction function α : P(R p ) → D and a concretization function γ : D → P(R p ) for some p ∈ N.  ... 
dblp:conf/icml/MirmanGV18 fatcat:3zwfqkkj4jayzkm53tk7otblty

Optimization and Abstraction: A Synergistic Approach for Analyzing Neural Network Robustness [article]

Greg Anderson, Shankara Pailoor, Isil Dillig, Swarat Chaudhuri
2019 arXiv   pre-print
Our method synergistically combines gradient-based optimization methods for counterexample search with abstraction-based proof search to obtain a sound and (δ-)complete decision procedure.  ...  Our method also employs a data-driven approach to learn a verification policy that guides abstract interpretation during proof search.  ...  Acknowledgments We thank our shepherd Michael Pradel as well as our anonymous reviewers and members of the UToPiA group for their helpful feedback.  ... 
arXiv:1904.09959v1 fatcat:7khytmrwprfvlppxugkrf3drae

Analyzing Deep Neural Networks with Symbolic Propagation: Towards Higher Precision and Faster Verification [article]

Pengfei Yang, Jiangchao Liu, Jianlin Li, Liqian Chen, Xiaowei Huang
2019 arXiv   pre-print
This paper improves on a recent proposal of analyzing DNNs through the classic abstract interpretation technique, by a novel symbolic propagation technique.  ...  Deep neural networks (DNNs) have been shown lack of robustness for the vulnerability of their classification to small perturbations on the inputs.  ...  abstraction for f (X). (3) There exists a CNN g : R m → R n and a box region X ⊆ R m s.t. the Zonotope abstract domain with symbolic propagation give a strictly more precise abstraction for g(X) than  ... 
arXiv:1902.09866v1 fatcat:asa27grlwzcchgtrorl35fsybu

Sum of Abstract Domains [chapter]

Gianluca Amato, Simone Di Nardo Di Maio, Francesca Scozzari
2015 Lecture Notes in Computer Science  
We provide a general framework equipped with all the necessary abstract operators for static analysis of imperative languages.  ...  We propose a new method to combine numerical abstract domains based on the Minkowski sum.  ...  Abstract interpretation In this paper we adopt a framework for abstract interpretation which is weaker than the common one based on Galois' connections/insertions (see [10, Section 7] ).  ... 
doi:10.1007/978-3-319-17524-9_4 fatcat:x7zrpecxmrgzrghbkt3lzvr4t4

Introduction to Neural Network Verification

Aws Albarghouthi
2021 Foundations and Trends® in Programming Languages  
A number of insightful people sent me comments that radically improved the presentation: Frantisek Plasil, Georg Weissenbacher, Sayan Mitra, Benedikt Böing, Vivek Garg, Guy Van den Broeck, Matt Fredrikson  ...  Abstract interpretation is a general framework, based on lattice theory, for defining and reasoning about program analyses.  ...  A Zonotope Transformer for ReLU Let's slowly build the ReLU abstract transformer for zonotopes. We're given a 1-dimensional zonotope ⟨c i ⟩ i as input.  ... 
doi:10.1561/2500000051 fatcat:2wbm374jcrc3pd5mhn5pfukoae

PaRoT: A Practical Framework for Robust Deep Neural Network Training [article]

Edward Ayers, Francisco Eiras, Majd Hawasly, Iain Whiteside
2020 arXiv   pre-print
Raising unique challenges for assurance due to their black-box nature, DNNs pose a fundamental problem for regulatory acceptance of these types of systems.  ...  In this paper we introduce a novel framework, PaRoT, developed on the popular TensorFlow platform, that greatly reduces the barrier to entry.  ...  Here we generalize the work in [34] to find optimal hybrid zonotopes for a given activation function.  ... 
arXiv:2001.02152v3 fatcat:74vra6lsu5ecjngsbbxczyejpy

Quadratic Zonotopes:An extension of Zonotopes to Quadratic Arithmetics [article]

Assalé Adjé and Pierre-Loïc Garoche and Alexis Werey
2015 arXiv   pre-print
, providing relational abstraction of functions with a cost linear in the number of errors terms.  ...  In static analysis, the zonotope domain, a relational abstract domain based on affine forms has been used in a wide set of settings, e.g. set-based simulation for hybrid systems, or floating point analysis  ...  Quadratic zonotopes shows here to be a good alternative to interval or affine zonotopes abstractions. About the Householder function, it converges towards 1{ ?  ... 
arXiv:1411.5847v2 fatcat:cgg2yyzlozeutlzcprel4phb24

Quadratic Zonotopes [chapter]

Assalé Adjé, Pierre-Loïc Garoche, Alexis Werey
2015 Lecture Notes in Computer Science  
, providing relational abstraction of functions with a cost linear in the number of errors terms.  ...  In static analysis, the zonotope domain, a relational abstract domain based on affine forms has been used in a wide set of settings, e.g. setbased simulation for hybrid systems, or floating point analysis  ...  Quadratic zonotopes seems more suited than linear abstractions when analyzing non linear functions such as multiplications.  ... 
doi:10.1007/978-3-319-26529-2_8 fatcat:y4ti5sdkobgsbotd4xd4toph4q

NNV: The Neural Network Verification Tool for Deep Neural Networks and Learning-Enabled Cyber-Physical Systems [chapter]

Hoang-Dung Tran, Xiaodong Yang, Diego Manzanas Lopez, Patrick Musau, Luan Viet Nguyen, Weiming Xiang, Stanley Bak, Taylor T. Johnson
2020 Lecture Notes in Computer Science  
The crux of NNV is a collection of reachability algorithms that make use of a variety of set representations, such as polyhedra, star sets, zonotopes, and abstract-domain representations.  ...  This paper presents the Neural Network Verification (NNV) software tool, a set-based verification framework for deep neural networks (DNNs) and learning-enabled cyber-physical systems (CPS).  ...  54] [55] [56] , star sets [29, 38, 39, 41] , zonotopes [32] , and abstract domain representations [33] .  ... 
doi:10.1007/978-3-030-53288-8_1 fatcat:7ls6jm5w4rh2veqfzlytl3eciq

Interval observer fault detection ensuring detectability and isolability by using a set-invariance approach

Masoud Pourasghar, Vicenҫ Puig, Carlos Ocampo-Martinez
2018 IFAC-PapersOnLine  
The effect of the uncertainty is taken into account using zonotopic-set representations.  ...  The effect of the uncertainty is taken into account using zonotopic-set representations.  ...  In this case, the residual can be obtained as a zonotope since using the zonotopic definition of a set for propagating the uncertainty.  ... 
doi:10.1016/j.ifacol.2018.09.727 fatcat:wwgx7vzqfjdrdmgt2uuzswzu7i

Reachset Conformance Testing of Hybrid Automata

Hendrik Roehm, Jens Oehlerking, Matthias Woehrle, Matthias Althoff
2016 Proceedings of the 19th International Conference on Hybrid Systems: Computation and Control - HSCC '16  
For this reason, a common approach is to formally verify abstractions of (parts of) the original system.  ...  Additionally, we present a test selection algorithm that uses a coverage measure to reduce the number of test cases for conformance testing.  ...  ACKNOWLEDGMENT The authors would like to thank the anonymous reviewers as well as Christoph Gladisch, Thomas Heinz and Christian Heinzemann for their valuable comments and suggestions to improve the quality  ... 
doi:10.1145/2883817.2883828 dblp:conf/hybrid/RoehmOWA16 fatcat:pxhgj63h3rah7b3i73wbuno7sm

NNV: The Neural Network Verification Tool for Deep Neural Networks and Learning-Enabled Cyber-Physical Systems [article]

Hoang-Dung Tran, Xiaodong Yang, Diego Manzanas Lopez, Patrick Musau, Luan Viet Nguyen, Weiming Xiang, Stanley Bak, Taylor T. Johnson
2020 arXiv   pre-print
The crux of NNV is a collection of reachability algorithms that make use of a variety of set representations, such as polyhedra, star sets, zonotopes, and abstract-domain representations.  ...  This paper presents the Neural Network Verification (NNV) software tool, a set-based verification framework for deep neural networks (DNNs) and learning-enabled cyber-physical systems (CPS).  ...  The approximate star method is much less conservative than the zonotope and abstract domain methods.  ... 
arXiv:2004.05519v1 fatcat:siuyyneoprbffosliqbmyewy3a

A Logical Product Approach to Zonotope Intersection [chapter]

Khalil Ghorbal, Eric Goubault, Sylvie Putot
2010 Lecture Notes in Computer Science  
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  ...  While abstract transfer functions are still rather inexpensive and accurate even for interpreting non-linear computations, we are able to also interpret tests (i.e. intersections) efficiently.  ...  We first introduce in Section 2 affine sets, a zonotopic abstract domain for abstract interpretation, that abstracts input/output relations in a program.  ... 
doi:10.1007/978-3-642-14295-6_22 fatcat:opdbpzxj35ai7aif3chrbdwbea
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