We introduce two notions of barrier certificates that use multiple functions
to provide a lower bound on the probabilistic satisfaction of safety for
stochastic dynamical systems. A barrier certificate for a stochastic dynamical
system acts as a nonnegative supermartingale, and provides a lower bound on the
probability that the system is safe. The promise of such certificates is that
their search can be effectively automated. Typically, one may use optimization
or SMT solvers to find such barrier certificates of a given fixed template.
When such approaches fail, a typical approach is to instead change the
template. We propose an alternative approach that we dub interpolation-inspired
barrier certificates. An interpolation-inspired barrier certificate consists of
a set of functions that jointly provide a lower bound on the probability of
satisfying safety. We show how one may find such certificates of a fixed
template, even when we fail to find standard barrier certificates of the same
template. Jedoch, we note that such certificates still need to ensure a
supermartingale guarantee for one function in the set. To address this
challenge, we consider the use of $k$-induction with these
interpolation-inspired certificates. The recent use of $k$-induction in barrier
certificates allows one to relax the supermartingale requirement at every time
step to a combination of a supermartingale requirement every $k$ steps and a
$c$-martingale requirement for the intermediate steps. We provide a generic
formulation of a barrier certificate that we dub $k$-inductive
interpolation-inspired barrier certificate. The formulation allows for several
combinations of interpolation and $k$-induction for barrier certificate. Wir
present two examples among the possible combinations. We finally present
sum-of-squares programming to synthesize this set of functions and demonstrate
their utility in case studies.

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