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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. 

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: (i) the decidability of verifying DNNs with a particular set of piecewise-smooth activation functions, including Sigmoid and tanh, is equivalent to a well-known, open problem formulated by Tarski; and (ii) 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. 

Abstract This paper serves as a comprehensive system description of version 2.0 of the Marabou framework for formal analysis of neural networks. We discuss the tool’s architectural design and highlight the major features and components introduced since its initial release. 

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.