Cyber-physical system applications such as autonomous vehicles, wearable devices, and avionic systems generate a large volume of time-series data. Designers often look for tools to help classify and categorize the data. Traditional machine learning techniques for time-series data offer several solutions to solve these problems; however, the artifacts trained by these algorithms often lack interpretability. On the other hand, temporal logics, such as Signal Temporal Logic (STL) have been successfully used in the formal methods community as specifications of time-series behaviors. In this work, we propose a new technique to automatically learn temporal logic formulae that are able to cluster and classify real-valued time-series data. Previous work on learning STL formulas from data either assumes a formula-template to be given by the user, or assumes some special fragment of STL that enables exploring the formula structure in a systematic fashion. In our technique, we relax these assumptions, and provide a way to systematically explore the space of all STL formulas. As the space of all STL formulas is very large, and contains many semantically equivalent formulas, we suggest a technique to heuristically prune the space of formulas considered. Finally, we illustrate our technique on various case studies from the automotive, transportation and healthcare domain.
Cyber-physical systems of today are generating large volumes of time-series data. As manual inspection of such data is not tractable, the need for learning methods to help discover logical structure in the data has increased. We propose a logic-based framework that allows domain-specific knowledge to be embedded into formulas in a parametric logical specification over time-series data. The key idea is to then map a time series to a surface in the parameter space of the formula. Given this mapping, we identify the Hausdorff distance between surfaces as a natural distance metric between two time-series data under the lens of the parametric specification. This enables embedding non-trivial domain-specific knowledge into the distance metric and then using off-the-shelf machine learning tools to label the data. After labeling the data, we demonstrate how to extract a logical specification for each label. Finally, we showcase our technique on real world traffic data to learn classifiers monitors for slow-downs and traffic jams.  This paper introduces a framework for inference of timed temporal logic properties from data. The dataset is given as a finite set of pairs of finite-time system traces and labels, where the labels indicate whether the traces exhibit some desired behavior (e.g., a ship traveling along a safe route). We propose a decision-tree based approach for learning signal temporal logic classifiers. The method produces binary decision trees that represent the inferred formulae. Each node of the tree contains a test associated with the satisfaction of a simple formula, optimally tuned from a predefined finite set of primitives. Optimality is assessed using heuristic impurity measures, which capture how well the current primitive splits the data with respect to the traces' labels. We propose extensions of the usual impurity measures from machine learning literature to handle classification of system traces by leveraging upon the robustness degree concept. The proposed incremental construction procedure greatly improves the execution time and the accuracy compared to existing algorithms. We present two case studies that illustrate the usefulness and the computational advantages of the algorithms. The first is an anomaly detection problem in a maritime environment. The second is a fault detection problem in an automotive powertrain system. This paper presents an inference algorithm that can discover temporal logic properties of a system from data. Our algorithm operates on finite time system trajectories that are labeled according to whether or not they demonstrate some desirable system properties (e.g. "the car successfully stops before hitting an obstruction"). A temporal logic formula that can discriminate between the desirable behaviors and the undesirable ones is constructed. The formulae also indicate possible causes for each set of behaviors (e.g. "If the speed of the car is greater than 15 m s within 0.5s of brake application, the obstruction will be struck") which can be used to tune designs or to perform on-line monitoring to ensure the desired behavior. We introduce reactive parameter signal temporal logic (rPSTL), a fragment of parameter signal temporal logic (PSTL) that is expressive enough to capture causal, spatial, and temporal relationships in data. We define a partial order over the set of rPSTL formulae that is based on language inclusion. This order enables a directed search over this set, i.e. given a candidate rPSTL formula that does not adequately match the observed data, we can automatically construct a formula that will fit the data at least as well. Two case studies, one involving a cattle herding scenario and one involving a stochastic hybrid gene circuit model, are presented to illustrate our approach. As the complexity of cyber-physical systems increases, so does the number of ways an adversary can disrupt them. This necessitates automated anomaly detection methods to detect possible threats. In this paper, we extend our recent results in the field of inference via formal methods to develop an unsupervised learning algorithm. Our procedure constructs from data a signal temporal logic (STL) formula that describes normal system behavior. Trajectories that do not satisfy the learned formula are flagged as anomalous. STL can be used to formulate properties such as “If the train brakes within 500 m of the platform at a speed of 50 km hr, then it will stop in at least 30 s and at most 50 s.” STL gives a more human-readable representation of behavior than classifiers represented as surfaces in high-dimensional feature spaces. STL formulae can also be used for early detection via online monitoring and for anomaly mitigation via formal synthesis. We demonstrate the power of our method with a physical model of a train's brake system. To our knowledge, this paper is the first instance of formal methods being applied to anomaly detection.