In this paper, we consider the problem of coordinating a large collection of homogeneous agents subject to a novel class of constraints: counting temporal logic constraints. Counting constraints arise naturally in many multi-agent planning problems where the identity of the agents is not important for the task to be completed. We introduce a formal language to capture such tasks and present an optimization-based technique to synthesize plans for large collections of agents in a way to guarantee the satisfaction of tasks speci ed in this formalism. CCS CONCEPTS • eory of computation →Modal and temporal logics; Abstraction; •Computing methodologies →Planning for deterministic actions; Multi-agent planning; Motion path planning; •Computer systems organization →Robotic autonomy;

This paper employs correct-by-construction control synthesis, in particular controlled invariant set computations, for falsification. Our hypothesis is that if it is possible to compute a "large enough" controlled invariant set either for the actual system model or some simplification of the system model, interesting corner cases for other control designs can be generated by sampling initial conditions from the boundary of this controlled invariant set. Moreover, if falsifying trajectories for a given control design can be found through such sampling, then the controlled invariant set can be used as a supervisor to ensure safe operation of the control design under consideration. In addition to interesting initial conditions, which are mostly related to safety violations in transients, we use solutions from a dual game, a reachability game for the safety specification, to find falsifying inputs. We also propose optimization-based heuristics for input generation for cases when the state is outside the winning set of the dual game. To demonstrate the proposed ideas, we consider case studies from basic autonomous driving functionality, in particular, adaptive cruise control and lane keeping. We show how the proposed technique can be used to find interesting falsifying trajectories for classical control designs like proportional controllers, proportional integral controllers and model predictive controllers, as well as an open source real-world autonomous driving package. MAIN INGREDIENTSOur goal in this paper is to search for interesting corner cases for falsification of closed-loop control systems. By interesting corner case, we mean a pair of initial condition and external (disturbance) input signal that leads to a trajectory violating a given safety specification, together with a certificate that it is possible to satisfy the specification for this initial condition and external input; therefore arXiv:1807.09537v2 [cs.SY]

In many multirobot applications, planning trajectories in a way to guarantee that the collective behavior of the robots satisfies a certain high-level specification is crucial. Motivated by this problem, we introduce counting temporal logics-formal languages that enable concise expression of multirobot task specifications over possibly infinite horizons. We first introduce a general logic called counting linear temporal logic plus (cLTL+), and propose an optimization-based method that generates individual trajectories such that satisfaction of a given cLTL+ formula is guaranteed when these trajectories are synchronously executed. We then introduce a fragment of cLTL+, called counting linear temporal logic (cLTL), and show that a solution to planning problem with cLTL constraints can be obtained more efficiently if all robots have identical dynamics. In the second part of the paper, we relax the synchrony assumption and discuss how to generate trajectories that can be asynchronously executed, while preserving the satisfaction of the desired cLTL+ specification. In particular, we show that when the asynchrony between robots is bounded, the method presented in this paper can be modified to generate robust trajectories. We demonstrate these ideas with an experiment and provide numerical results that showcase the scalability of the method.

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