We highlight the contributions made in the field of Statistical Model Checking (SMC) since its inception in 2002. As the formal setting, we use a very general model of Stochastic Systems (an SS is simply a family of time-indexed random variables), and Bounded LTL (BLTL) as the temporal logic. Let S be an SS and ϕ a BLTL formula. Our survey of the area is centered around the following five main contributions. Qualitative approach to SMC: Is the probability that S satisfies ϕ greater or equal to a certain threshold? Quantitative approach to SMC: What is the probability that S satisfies ϕ? Typically this results in a confidence interval being computed for this probability. Rare Events: What happens when the probability that S satisfies ϕ is extremely small, i.e. it is a rare event? To make the SMC approach viable in this setting, rare-event estimation techniques Importance Sampling and Importance Splitting are deployed to great advantage. Optimal Planning: Motivated by the success of Importance Sampling and Importance Splitting in rare-event SMC, we explore the use of these techniques in the context of optimal planning. In particular, we consider ARES, an optimal-planning approach based on a notion of adaptive receding-horizon planning. We illustrate the utility of ARES on the planning problem of bringing a flock of birds (autonomous agents) from a random initial configuration to a Vformation, an energy-conservation formation deployed by migrating geese. Somewhat ironically, the performance of ARES can be evaluated using (quantitative) SMC, as the problem to be solved is of the form F (J ≤ θ); i.e. does an ARES-generated plan eventually bring the flock to a configuration where the flock-wide cost function J is below a given threshold θ? Optimal Control: We show that the techniques we presented for optimal planning in the form of ARES carry over to the control setting in the form of Adaptive-Horizon Model-Predictive Control (AMPC). We again use the V-formation problem for evaluation purposes. We also introduce the concept of V-formation games, and show how the power of AMPC can be used to ward off cyber-physical attacks.

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Decision tree learning is a widely used approach in machine learning, favoured in applications that require concise and interpretable models. Heuristic methods are traditionally used to quickly produce models with reasonably high accuracy. A commonly criticised point, however, is that the resulting trees may not necessarily be the best representation of the data in terms of accuracy, size, and other considerations such as fairness. In recent years, this motivated the development of optimal classification tree algorithms that globally optimise the decision tree in contrast to heuristic methods that perform a sequence of locally optimal decisions. We follow this line of work and provide a novel algorithm for learning optimal classification trees based on dynamic programming and search. Our algorithm supports constraints on the depth of the tree and number of nodes and we argue it can be extended with other requirements. The success of our approach is attributed to a series of specialised techniques that exploit properties unique to classification trees. Whereas algorithms for optimal classification trees have traditionally been plagued by high runtimes and limited scalability, we show in a detailed experimental study that our approach uses only a fraction of the time required by the state-of-the-art and can handle datasets with tens of thousands of instances, providing several orders of magnitude improvements and notably contributing towards the practical realisation of optimal decision trees.

Abstract. We introduce ARES, an efficient approximation algorithm for generating optimal plans (action sequences) that take an initial state of a Markov Decision Process (MDP) to a state whose cost is below a specified (convergence) threshold. ARES uses Particle Swarm Optimization, with adaptive sizing for both the receding horizon and the particle swarm. Inspired by Importance Splitting, the length of the horizon and the number of particles are chosen such that at least one particle reaches a next-level state, that is, a state where the cost decreases by a required delta from the previous-level state. The level relation on states and the plans constructed by ARES implicitly define a Lyapunov function and an optimal policy, respectively, both of which could be explicitly generated by applying ARES to all states of the MDP, up to some topological equivalence relation. We also assess the effectiveness of ARES by statistically evaluating its rate of success in generating optimal plans. The ARES algorithm resulted from our desire to clarify if flying in V-formation is a flocking policy that optimizes energy conservation, clear view, and velocity alignment. That is, we were interested to see if one could find optimal plans that bring a flock from an arbitrary initial state to a state exhibiting a single connected V-formation. For flocks with 7 birds, ARES is able to generate a plan that leads to a V-formation in 95% of the 8,000 random initial configurations within 63 seconds, on average. ARES can also be easily customized into a model-predictive controller (MPC) with an adaptive receding horizon and statistical guarantees of convergence. To the best of our knowledge, our adaptive-sizing approach is the first to provide convergence guarantees in receding-horizon techniques.

Abstract. We introduce the concept of a V-formation game between a controller and an attacker, where controller's goal is to maneuver the plant (a simple model of flocking dynamics) into a V-formation, and the goal of the attacker is to prevent the controller from doing so. Controllers in V-formation games utilize a new formulation of model-predictive control we call Adaptive-Horizon MPC (AMPC), giving them extraordinary power: we prove that under certain controllability assumptions, an AMPC controller is able to attain V-formation with probability 1. We define several classes of attackers, including those that in one move can remove R birds from the flock, or introduce random displacement into flock dynamics. We consider both naive attackers, whose strategies are purely probabilistic, and AMPC-enabled attackers, putting them on par strategically with the controllers. While an AMPC-enabled controller is expected to win every game with probability 1, in practice, it is resourceconstrained : its maximum prediction horizon and the maximum number of game execution steps are fixed. Under these conditions, an attacker has a much better chance of winning a V-formation game. Our extensive performance evaluation of V-formation games uses statistical model checking to estimate the probability an attacker can thwart the controller. Our results show that for the bird-removal game with R = 1, the controller almost always wins (restores the flock to a V-formation). For R = 2, the game outcome critically depends on which two birds are removed. For the displacement game, our results again demonstrate that an intelligent attacker, i.e. one that uses AMPC in this case, significantly outperforms its naive counterpart that randomly executes its attack.