# Robustness Verification

For ℓ0 perturbations that change only a few pixels, FocusBranch rewrites the “at most k perturbed pixels” constraint as a node made of a focus pixel set and a level, so verification can branch across levels. It builds super nodes through covering designs and concrete nodes through upper shadows, and picks a starting level from sampling estimates. Across 13 networks and 3,189 candidate instances it verified 2,799; in a 740-instance comparison it verified 641, against 375 for CoVerD+top-k-GP.

For ℓ0 perturbations that change only a few pixels, FocusBranch rewrites the “at most k perturbed pixels” constraint as a node made of a focus pixel set and a level, so verification can branch across levels. It builds super nodes through covering designs and concrete nodes through upper shadows, and picks a starting level from sampling estimates. Across 13 networks and 3,189 candidate instances it verified 2,799; in a 740-instance comparison it verified 641, against 375 for CoVerD+top-k-GP.

This work extends social laws from deterministic, goal-based settings to stochastic, reward-based multi-agent environments by introducing α-robustness, a measure of the fraction of guaranteed utility each agent retains while following its optimal single-agent policy under the assumption that all agents obey the social law, and by presenting a verification approach that reduces to solving a series of Markov decision processes; experiments on grid toy environments show that with action success probability 1, parallel lanes, opposite lanes, and switch corners with a clockwise social law reach α=1 robustness, while a slow-speed social law does not always improve guaranteed utility.

This work extends social laws from deterministic, goal-based settings to stochastic, reward-based multi-agent environments by introducing α-robustness, a measure of the fraction of guaranteed utility each agent retains while following its optimal single-agent policy under the assumption that all agents obey the social law, and by presenting a verification approach that reduces to solving a series of Markov decision processes; experiments on grid toy environments show that with action success probability 1, parallel lanes, opposite lanes, and switch corners with a clockwise social law reach α=1 robustness, while a slow-speed social law does not always improve guaranteed utility.