Specifications play a crucial role in neural network verification. They define the precise input regions we aim to verify, typically represented as L-infinity norm balls. While recent research suggests using neural activation patterns (NAPs) as specifications for verifying unseen test set data, it focuses on computing the most refined NAPs, often limited to very small regions in the input space. In this paper, we study the following problem: Given a neural network, find a minimal (coarsest) NAP that is sufficient for formal verification of the network's robustness. Finding the minimal NAP specification not only expands verifiable bounds but also provides insights into which neurons contribute to the model's robustness. To address this problem, we propose several exact and approximate approaches. Our exact approaches leverage the verification tool to find minimal NAP specifications in either a deterministic or statistical manner. Whereas the approximate methods efficiently estimate minimal NAPs using adversarial examples and local gradients, without making calls to the verification tool. This allows us to inspect potential causal links between neurons and the robustness of state-of-the-art neural networks, a task for which existing verification frameworks fail to scale. Our experimental results suggest that minimal NAP specifications require much smaller fractions of neurons compared to the most refined NAP specifications, yet they can significantly expand the verifiable boundaries to several orders of magnitude larger.

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