In this paper, we introduce a notion of input-to-state safety (ISSf) and input-to-state attractivity (ISA) controllers for fi- nite state dynamical systems. Using such controllers, the deviation of the closed-loop trajectories from a safe or from a target set can be bounded by a gain function that is increasing with respect to the amplitude of the disturbances. We show the existence of controllers that are least violating (LV) in the sense that their gain function is minimal (with respect to a certain order on the set of gain functions) over all possible ISSf and ISA controllers. Then, we consider the problem of synthesizing these LV-ISSf and LV-ISA controllers for the colexicographic order on gain functions. We present an approach that is based on successive refinements of controllers: starting from a controller synthesized against worst- case disturbances, the controller is iteratively refined in order to improve the closed-loop behavior under disturbances of lower am- plitude. We prove that our method makes it possible to synthesize a LV-ISSf controller, and an ISA controller that is shown to be a LV-ISA controller when an easily checkable condition is satisfied. We discuss how these results can be used to synthesize robust controllers for nonlinear continuous-time systems via symbolic control techniques. Finally, we show an application to adaptive cruise control to demonstrate the effectiveness of our approach