“Multivalued Analysis” is the theory of set-valued maps (called multifonctions) and has important applications in many different areas. Multivalued analysis is a remarkable mixture of many different parts of mathematics such as point-set topology, measure theory and nonlinear functional analysis. It is also closely related to “Nonsmooth Analysis” (Chapter 5) and in fact one of the main motivations behind the development of the theory, was in order to provide necessary analytical tools for the study of problems in nonsmooth analysis. It is not a coincidence that the development of the two fields coincide chronologically and follow parallel paths. Today multivalued analysis is a mature mathematical field with its own methods, techniques and applications that range from social and economic sciences to biological sciences and engineering. There is no doubt that a modern treatise on “Nonlinear Functional Analysis” can not afford the luxury of ignoring multivalued analysis. The omission of the theory of multifunctions will drastically limit the possible applications.

SET-Valued Analysis

We present AI2, the first sound and scalable analyzer for deep neural networks. Based on overapproximation, AI2 can automatically prove safety properties (e.g., robustness) of realistic neural networks (e.g., convolutional neural networks). The key insight behind AI2 is to phrase reasoning about safety and robustness of neural networks in terms of classic abstract interpretation, enabling us to leverage decades of advances in that area. Concretely, we introduce abstract transformers that capture the behavior of fully connected and convolutional neural network layers with rectified linear unit activations (ReLU), as well as max pooling layers. This allows us to handle real-world neural networks, which are often built out of those types of layers. We present a complete implementation of AI2 together with an extensive evaluation on 20 neural networks. Our results demonstrate that: (i) AI2 is precise enough to prove useful specifications (e.g., robustness), (ii) AI2 can be used to certify the effectiveness of state-of-the-art defenses for neural networks, (iii) AI2 is significantly faster than existing analyzers based on symbolic analysis, which often take hours to verify simple fully connected networks, and (iv) AI2 can handle deep convolutional networks, which are beyond the reach of existing methods.

/pdf/ai2-safety-and-robustness-certification-of-neural-networks-4junu05wnm.pdf

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

We present a novel method for scalable and precise certification of deep neural networks. The key technical insight behind our approach is a new abstract domain which combines floating point polyhedra with intervals and is equipped with abstract transformers specifically tailored to the setting of neural networks. Concretely, we introduce new transformers for affine transforms, the rectified linear unit (ReLU), sigmoid, tanh, and maxpool functions. We implemented our method in a system called DeepPoly and evaluated it extensively on a range of datasets, neural architectures (including defended networks), and specifications. Our experimental results indicate that DeepPoly is more precise than prior work while scaling to large networks. We also show how to combine DeepPoly with a form of abstraction refinement based on trace partitioning. This enables us to prove, for the first time, the robustness of the network when the input image is subjected to complex perturbations such as rotations that employ linear interpolation.

https://dl.acm.org/doi/pdf/10.1145/3290354

An abstract domain for certifying neural networks

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.

/pdf/differentiable-abstract-interpretation-for-provably-robust-3w0lfz9377.pdf

Differentiable Abstract Interpretation for Provably Robust Neural Networks

Finite element methods for incompressible viscous flow

Most modern safety-critical control programs, such as those embedded in fly-by-wire control systems, perform a lot of floating-point computations. The well-known pitfalls of IEEE 754 arithmetic make stability and accuracy analyses a requirement for this type of software. This need is traditionally addressed through a combination of testing and sophisticated intellectual analyses, but such a process is both costly and error-prone. FLUCTUAT is a static analyzer developed by CEA-LIST for studying the propagation of rounding errors in C programs. After a long time research collaboration with CEA-LIST on this tool, Airbus is now willing to use FLUCTUAT industrially, in order to automate part of the accuracy analyses of some control programs. In this paper, we present the IEEE 754 standard, the FLUCTUAT tool, the types of codes to be analyzed and the analysis methodology, together with code examples and analysis results.

/pdf/towards-an-industrial-use-of-fluctuat-on-safety-critical-uuwpwea34w.pdf

Towards an Industrial Use of FLUCTUAT on Safety-Critical Avionics Software

We present a new numerical abstract domain for static analysis of the errors introduced by the approximation by floating-point arithmetic of real numbers computation, by abstract interpretation [3]. This work extends a former domain [4,8], with an implicitly relational domain for the approximation of the floating-point values of variables, based on affine arithmetic [2]. It allows us to analyze non trivial numerical computations, that no other abstract domain we know of can analyze with such precise results, such as linear recursive filters of different orders, Newton methods for solving non-linear equations, polynomial iterations, conjugate gradient algorithms.

/pdf/static-analysis-of-numerical-algorithms-4o27f64csa.pdf

Static analysis of numerical algorithms

We define several abstract semantics for the static analysis of finite precision computations, that bound not only the ranges of values taken by numerical variables of a program, but also the difference with the result of the same sequence of operations in an idealized real number semantics. These domains point out with more or less detail (control point, block, function for instance) sources of numerical errors in the program and the way they were propagated by further computations, thus allowing to evaluate not only the rounding error, but also sensitivity to inputs or parameters of the program. 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 for sensitivity analysis and test generation. These abstract domains are implemented in the Fluctuat static analyzer, and we finally present some experiments.

/pdf/static-analysis-of-finite-precision-computations-3jko6xwsxb.pdf

Static analysis of finite precision computations

We present a new method for solving the fixed point equations that appear in the static analysis of programs by abstract interpretation. We introduce and analyze a policy iteration algorithm for monotone self-maps of complete lattices. We apply this algorithm to the particular case of lattices arising in the interval abstraction of values of variables. We demonstrate the improvements in terms of speed and precision over existing techniques based on Kleene iteration, including traditional widening/narrowing acceleration mecanisms.

/pdf/a-policy-iteration-algorithm-for-computing-fixed-points-in-3fkj9443xi.pdf

A policy iteration algorithm for computing fixed points in static analysis of programs

Static analysis by abstract interpretation [1] aims at automatically inferring properties on the behaviour of programs. We focus here on a specific kind of numerical invariants: the set of values taken by numerical variables, with a real numbers semantics, at each control point of a program.

/pdf/the-zonotope-abstract-domain-taylor1-2k3mifksix.pdf

Sylvie Putot

Papers

Towards an Industrial Use of FLUCTUAT on Safety-Critical Avionics Software

Static analysis of numerical algorithms

Static analysis of finite precision computations

A policy iteration algorithm for computing fixed points in static analysis of programs

The Zonotope Abstract Domain Taylor1