  This paper studies the satisfaction of a class of temporal properties for
cyber-physical systems (CPSs) over a finite-time horizon in the presence of an
adversary, in an environment described by discrete-time dynamics. The temporal
logic specification is given in safe-LTL_F, a fragment of linear temporal logic
over traces of finite length. The interaction of the CPS with the adversary is
modeled as a two-player zero-sum discrete-time dynamic stochastic game with the
CPS as defender. We formulate a dynamic programming based approach to determine
a stationary defender policy that maximized the probability of satisfaction of
a safe-LTL_F formula over a finite time-horizon under any stationary adversary
policy. We introduce secure control barrier certificates (S-CBCs), a
generalization of barrier certificates and control barrier certificates that
accounts for the presence of an adversary, and use S-CBCs to provide a lower
bound on the above satisfaction probability. When the dynamics of the evolution
of the system state has a specific underlying structure, we present a way to
determine an S-CBC as a polynomial in the state variables using sum-of-squares
optimization. An illustrative example demonstrates our approach.
