  Safety-critical systems, such as autonomous vehicles, often incorporate
perception modules that can anticipate upcoming disturbances to system
dynamics, expecting that such preview information can improve the performance
and safety of the system in complex and uncertain environments. However, there
is a lack of formal analysis of the impact of preview information on safety. In
this work, we introduce a notion of safety regret, a properly defined
difference between the maximal invariant set of a system with finite preview
and that of a system with infinite preview, and show that this quantity
corresponding to finite-step preview decays exponentially with preview horizon.
Furthermore, algorithms are developed to numerically evaluate the safety regret
of the system for different preview horizons. Finally, we demonstrate the
established theory and algorithms via multiple examples from different
application domains.
