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1. DNN Verification, Reachability, and the Exponential Function Problem

   Isac, Omri; Zohar, Yoni; Barrett, Clark; Katz, Guy (September 2023, Proceedings of the 34th International Conference on Concurrency Theory (CONCUR 2023))

   Pérez, Guillermo A.; Raskin, Jean-François (Ed.)

   Deep neural networks (DNNs) are increasingly being deployed to perform safety-critical tasks. The opacity of DNNs, which prevents humans from reasoning about them, presents new safety and security challenges. To address these challenges, the verification community has begun developing techniques for rigorously analyzing DNNs, with numerous verification algorithms proposed in recent years. While a significant amount of work has gone into developing these verification algorithms, little work has been devoted to rigorously studying the computability and complexity of the underlying theoretical problems. Here, we seek to contribute to the bridging of this gap. We focus on two kinds of DNNs: those that employ piecewise-linear activation functions (e.g., ReLU), and those that employ piecewise-smooth activation functions (e.g., Sigmoids). We prove the two following theorems: 1. The decidability of verifying DNNs with piecewise-smooth activation functions is equivalent to a well-known, open problem formulated by Tarski; and 2. The DNN verification problem for any quantifier-free linear arithmetic specification can be reduced to the DNN reachability problem, whose approximation is NP-complete. These results answer two fundamental questions about the computability and complexity of DNN verification, and the ways it is affected by the network’s activation functions and error tolerance; and could help guide future efforts in developing DNN verification tools. 

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2. Provable repair of deep neural networks

   https://doi.org/10.1145/3453483.3454064  

   Sotoudeh, Matthew; Thakur, Aditya V. (June 2021, Proceedings of the 42nd ACM SIGPLAN International Conference on Programming Language Design and Implementation (PLDI))

   Deep Neural Networks (DNNs) have grown in popularity over the past decade and are now being used in safety-critical domains such as aircraft collision avoidance. This has motivated a large number of techniques for finding unsafe behavior in DNNs. In contrast, this paper tackles the problem of correcting a DNN once unsafe behavior is found. We introduce the provable repair problem, which is the problem of repairing a network N to construct a new network N′ that satisfies a given specification. If the safety specification is over a finite set of points, our Provable Point Repair algorithm can find a provably minimal repair satisfying the specification, regardless of the activation functions used. For safety specifications addressing convex polytopes containing infinitely many points, our Provable Polytope Repair algorithm can find a provably minimal repair satisfying the specification for DNNs using piecewise-linear activation functions. The key insight behind both of these algorithms is the introduction of a Decoupled DNN architecture, which allows us to reduce provable repair to a linear programming problem. Our experimental results demonstrate the efficiency and effectiveness of our Provable Repair algorithms on a variety of challenging tasks. 

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3. Neural Network Verification with Proof Production

   Isac, Omri; Barrett, Clark; Zhang, Min; Katz, Guy (October 2022, Proceedings of the 22nd International Conference on Formal Methods In Computer-Aided Design (FMCAD))

   Griggio, Alberto; Rungta, Neha (Ed.)

   Deep neural networks (DNNs) are increasingly being employed in safety-critical systems, and there is an urgent need to guarantee their correctness. Consequently, the verification community has devised multiple techniques and tools for verifying DNNs. When DNN verifiers discover an input that triggers an error, that is easy to confirm; but when they report that no error exists, there is no way to ensure that the verification tool itself is not flawed. As multiple errors have already been observed in DNN verification tools, this calls the applicability of DNN verification into question. In this work, we present a novel mechanism for enhancing Simplex-based DNN verifiers with proof production capabilities: the generation of an easy-to-check witness of unsatisfiability, which attests to the absence of errors. Our proof production is based on an efficient adaptation of the well-known Farkas' lemma, combined with mechanisms for handling piecewise-linear functions and numerical precision errors. As a proof of concept, we implemented our technique on top of the Marabou DNN verifier. Our evaluation on a safety-critical system for airborne collision avoidance shows that proof production succeeds in almost all cases and requires only minimal overhead. 

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4. Input-Relational Verification of Deep Neural Networks

   https://doi.org/10.1145/3656377  

   Banerjee, Debangshu; Xu, Changming; Singh, Gagandeep (June 2024, Proceedings of the ACM on Programming Languages)

   We consider the verification of input-relational properties defined over deep neural networks (DNNs) such as robustness against universal adversarial perturbations, monotonicity, etc. Precise verification of these properties requires reasoning about multiple executions of the same DNN. We introduce a novel concept of difference tracking to compute the difference between the outputs of two executions of the same DNN at all layers. We design a new abstract domain, DiffPoly for efficient difference tracking that can scale large DNNs. DiffPoly is equipped with custom abstract transformers for common activation functions (ReLU, Tanh, Sigmoid, etc.) and affine layers and can create precise linear cross-execution constraints. We implement an input-relational verifier for DNNs called RaVeN which uses DiffPoly and linear program formulations to handle a wide range of input-relational properties. Our experimental results on challenging benchmarks show that by leveraging precise linear constraints defined over multiple executions of the DNN, RaVeN gains substantial precision over baselines on a wide range of datasets, networks, and input-relational properties. 

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5. Automated Verification of Soundness of DNN Certifiers

   https://doi.org/10.1145/3720509  

   Singh, Avaljot; Sarita, Yasmin Chandini; Mendis, Charith; Singh, Gagandeep (April 2025, Proceedings of the ACM on Programming Languages)

   The uninterpretability of Deep Neural Networks (DNNs) hinders their use in safety-critical applications. Abstract Interpretation-based DNN certifiers provide promising avenues for building trust in DNNs. Unsoundness in the mathematical logic of these certifiers can lead to incorrect results. However, current approaches to ensure their soundness rely on manual, expert-driven proofs that are tedious to develop, limiting the speed of developing new certifiers. Automating the verification process is challenging due to the complexity of verifying certifiers for arbitrary DNN architectures and handling diverse abstract analyses. We introduce ProveSound, a novel verification procedure that automates the soundness verification of DNN certifiers for arbitrary DNN architectures. Our core contribution is the novel concept of a symbolic DNN, using which, ProveSound reduces the soundness property, a universal quantification over arbitrary DNNs, to a tractable symbolic representation, enabling verification with standard SMT solvers. By formalizing the syntax and operational semantics of ConstraintFlow, a DSL for specifying certifiers, ProveSound efficiently verifies both existing and new certifiers, handling arbitrary DNN architectures. Our code is available at https://github.com/uiuc-focal-lab/constraintflow.git 

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