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Directed Specifications and Assumption Mining for Monotone Dynamical Systems

Given a dynamical system and a specification, assumption mining is the problem of identifying the set of admissible disturbance signals and initial states that generate trajectories satisfying the specification. We first introduce the notion of a directed specification, which describes either upper or lower sets in a partially ordered signal space, and show that this notion encompasses an expressive temporal logic fragment. We next show that the order preserving nature of monotone dynamical systems makes them amenable to a systematic form of assumption mining that checks numerical simulations of system trajectories against directed specifications. The assumption set is then located with a multidimensional bisection method that converges to the boundary from above and below. Typical objectives in vehicular traffic control, such as avoiding or clearing congestion, are directed specifications. In an application to a freeway flow model with monotone dynamics, we identify the set of vehicular demand profiles that satisfy a specification that congestion be intermittent.