Probabilistic model checking is a useful technique for specifying and
verifying properties of stochastic systems including randomized protocols and
the theoretical underpinnings of reinforcement learning models. However, these
methods rely on the assumed structure and probabilities of certain system
transitions. These assumptions may be incorrect, and may even be violated in
the event that an adversary gains control of some or all components in the

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In this paper, motivated by research in adversarial machine learning on
adversarial examples, we develop a formal framework for adversarial robustness
in systems defined as discrete time Markov chains (DTMCs), and extend to
include deterministic, memoryless policies acting in Markov decision processes
(MDPs). Our framework includes a flexible approach for specifying several
adversarial models with different capabilities to manipulate the system. We
outline a class of threat models under which adversaries can perturb system
transitions, constrained by an $varepsilon$ ball around the original
transition probabilities and define four specific instances of this threat

We define three main DTMC adversarial robustness problems and present two
optimization-based solutions, leveraging traditional and parametric
probabilistic model checking techniques. We then evaluate our solutions on two
stochastic protocols and a collection of GridWorld case studies, which model an
agent acting in an environment described as an MDP. We find that the parametric
solution results in fast computation for small parameter spaces. In the case of
less restrictive (stronger) adversaries, the number of parameters increases,
and directly computing property satisfaction probabilities is more scalable. We
demonstrate the usefulness of our definitions and solutions by comparing system
outcomes over various properties, threat models, and case studies.

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