Interpretable classification of time-series data using efficient enumerative techniques

We propose a framework based on Recurrent Neural Networks (RNNs) to determine an optimal control strategy for a discrete-time system that is required to satisfy specifications given as Signal Temporal Logic (STL) formulae. RNNs can store information of a system over time, thus, enable us to determine satisfaction of the dynamic temporal requirements specified in STL formulae. Given a STL formula, a dataset of satisfying system executions and corresponding control policies, we can use RNNs to predict a control policy at each time based on the current and previous states of system. We use Control Barrier Functions (CBFs) to guarantee the safety of the predicted control policy. We validate our theoretical formulation and demonstrate its performance in an optimal control problem subject to partially unknown safety constraints through simulations.

We present a general framework for risk semantics on Signal Temporal Logic (STL) specifications for stochastic dynamical systems using axiomatic risk theory. We show that under our recursive risk semantics, risk constraints on STL formulas can be expressed in terms of risk constraints on atomic predicates. We then show how this allows a (stochastic) STL risk constraint to be transformed into a risk-tightened deterministic STL constraint on a related deterministic nominal system, enabling the application of existing STL methods. For affine predicate functions and a (coherent) Distributionally Robust Value at Risk measure, we show how risk constraints on atomic predicates can be reformulated as tightened deterministic affine constraints. We demonstrate the framework using a Model Predictive Control (MPC) design with an STL risk constraint.

As the complexity of Cyber-Physical Systems (CPSs) increases, designers can understand it hardly and manual analysis is not feasible. In recent years, specification mining has emerged as an effective approach to describe the behaviors of complex systems. We propose a novel learning approach, LRx, to synthesize signal temporal logic formulas that characterize the behaviors of the CPS using observed signal traces of the system. The approach requires two inputs: a set of observed traces and a parametric Signal Temporal Logic (PSTL) formula. We define a new Metric Discriminant Model for identifying all traces and focusing on two situations: positive label traces and negative label traces; only positive label traces. However, most studies can only deal with one of the situations. Finally, we demonstrate the effectiveness of our technique on two case studies: Autonomous-Car Driving, Automatic Transmission Controller.

We consider the problem of controlling a system with unknown, stochastic dynamics to achieve a complex, time-sensitive task. An example of this problem is controlling a noisy aerial vehicle with partially known dynamics to visit a pre-specified set of regions in any order while avoiding hazardous areas. In particular, we are interested in tasks which can be described by signal temporal logic (STL) specifications. STL is a rich logic that can be used to describe tasks involving bounds on physical parameters, continuous time bounds, and logical relationships over time and states. STL is equipped with a continuous measure called the robustness degree that measures how strongly a given sample path exhibits an STL property [4, 3]. This measure enables the use of continuous optimization problems to solve learning [7, 6] or formal synthesis problems [9] involving STL.

We propose a novel passive learning approach, TeLEx, to infer signal temporal logic formulas that characterize the behavior of a dynamical system using only observed signal traces of the system. The approach requires two inputs: a set of observed traces and a template Signal Temporal Logic (STL) formula. The unknown parameters in the template can include time-bounds of the temporal operators, as well as the thresholds in the inequality predicates. TeLEx finds the value of the unknown parameters such that the synthesized STL property is satisfied by all the provided traces and it is tight. This requirement of tightness is essential to generating interesting properties when only positive examples are provided and there is no option to actively query the dynamical system to discover the boundaries of legal behavior. We propose a novel quantitative semantics for satisfaction of STL properties which enables TeLEx to learn tight STL properties without multidimensional optimization. The proposed new metric is also smooth. This is critical to enable use of gradient-based numerical optimization engines and it produces a 30\(\times \)–100\(\times \) speed-up with respect to the state-of-art gradient-free optimization. The approach is implemented in a publicly available tool.

Deep Reinforcement Learning (DRL) has the potential to be used for synthesizing feedback controllers (agents) for various complex systems with unknown dynamics. These systems are expected to satisfy diverse safety and liveness properties best captured using temporal logic. In RL, the reward function plays a crucial role in specifying the desired behaviour of these agents. However, the problem of designing the reward function for an RL agent to satisfy complex temporal logic specifications has received limited attention in the literature. To address this, we provide a systematic way of generating rewards in real-time by using the quantitative semantics of Signal Temporal Logic (STL), a widely used temporal logic to specify the behaviour of cyber-physical systems. We propose a new quantitative semantics for STL having several desirable properties, making it suitable for reward generation. We evaluate our STL-based reinforcement learning mechanism on several complex continuous control benchmarks and compare our STL semantics with those available in the literature in terms of their efficacy in synthesizing the controller agent. Experimental results establish our new semantics to be the most suitable for synthesizing feedback controllers for complex continuous dynamical systems through reinforcement learning.

Sampling-based Motion Planning with Temporal Logic Missions and Spatial Preferences

This paper reports on a new approach to Signal Temporal Logic (STL) control synthesis, that 1) utilizes a navigation function as the basis to construct a Control Barrier Function (CBF), and 2) composes navigation function-based barrier functions using nonsmooth mappings to encode Boolean operations between the predicates that those barrier functions encode. Because of these two key features, the reported approach 1) covers a larger fragment of STL compared to existing approaches, 2) alleviates the computational cost associated with evaluation of the control law for the system in existing STL control barrier function methodologies, and 3) simultaneously relaxes some of the conservativeness of smooth combinations of barrier functions as a means of implementing Boolean operators. The paper demonstrates the efficacy of this new approach with three simulation case studies, one aiming at illustrating how complex STL motion planning specification can be realized, the second highlights the less-conservativeness of the approach in comparison to the existing methods, and another that shows how this technology can be brought to bear to push the envelope in the context of human-robot social interaction.

What is the frequency content of temporal logic formulas? That is, when we monitor a signal against a formula, which frequency bands of the signal are relevant to the logic and should be preserved, and which can be safely discarded? This question is relevant whenever signals are filtered or compressed before being monitored, which is almost always the case for analog signals. To answer this question, we focus on monitors that measure the robustness of a signal relative to a specification in Signal Temporal Logic. We prove that robustness monitors can be modeled using Volterra series. We then study the Fourier transforms of these Volterra representations, and provide a method to derive the Fourier transforms of entire formulas. We also make explicit the measurement process in temporal logic and re-define it on the basis of distributions to make it compatible with measurements in signal processing. Experiments illustrate these results. Beyond compression, this work enables the integration of temporal logic monitoring into common signal processing tool chains as just another signal processing operation, and enables a common formalism to study both logical and non-logical operations in the frequency domain, which we refer to as Logical Signal Processing.

We address the problem of learning temporal properties from the branching-time behavior of systems. Existing research in this field has mostly focused on learning linear temporal properties specified using popular logics, such as Linear Temporal Logic (LTL) and Signal Temporal Logic (STL). Branching-time logics such as Computation Tree Logic (CTL) and Alternating-time Temporal Logic (ATL), despite being extensively used in specifying and verifying distributed and multi-agent systems, have not received adequate attention. Thus, in this paper, we investigate the problem of learning CTL and ATL formulas from examples of system behavior. As input to the learning problems, we rely on the typical representations of branching behavior as Kripke structures and concurrent game structures, respectively. Given a sample of structures, we learn concise formulas by encoding the learning problem into a satisfiability problem, most notably by symbolically encoding both the search for prospective formulas and their fixed-point based model checking algorithms. We also study the decision problem of checking the existence of prospective ATL formulas for a given sample. We implement our algorithms in a Python prototype and have evaluated them to extract several common CTL and ATL formulas used in practical applications.

We present a bounded-memory receding horizon approach to robot control for complex specifications in dynamic environments. We use Signal Temporal Logic, a logic that quantifies how robustly trajectories satisfy the specification, to specify robot behavior. To handle unbounded specifications, we consider a short planning horizon, only searching for nonviolating trajectories. We identify the subset of Signal Temporal Logic for which this approach needs only a bounded memory of the past, and leverage syntactic separation to summarize the robust satisfaction of the trajectory as it evolves. We implement our approach using receding horizon control in dynamic environments. We demonstrate the effectiveness and scalability of our approach compared to the state-of-the-art approach in several case studies.

Recently, rule-based time series classification models are widely used in safety-critical scenarios demanding strong interpretability, through providing explicit and rigorous rationales. However, due to the discrepancy between discrete rules and continuous neural networks, the generated rules are not completely consistent with the actual decision-making process, rendering users hesitant to trust the model. Additionally, existing methods learn the role (weight) of each time point independently, which is not in line with human understanding and the consecutive nature of temporal properties. In this paper, we propose a novel neuro-symbolic model named TemporalRule, aiming to automatically learn Signal Temporal Logic (STL) rules for interpretable time series classification. Our model directly optimizes the neural networks representing discrete rules via gradient grafting, producing reliable rules that can exactly determine the classification results. Notably, a temporal logical layer is designed to simulate expressive temporal operators, including one 2-predicate operator (Until) and two 2-level composite operators (EA and AE), through adaptively learning the time bounds of involved intervals. That creates consistent weights for consecutive time points, enhancing the intuitiveness of derived rules. Extensive experiments on diverse real-world datasets in safety-critical domains show that TemporalRule achieves superior and stable classification accuracy compared to state-of-the-art rule-based approaches, and the learned rules can precisely and concisely explicate the classification results, leading to a trustable model.

We develop a distributed motion planner for multi-robot systems with Signal Temporal Logic (STL) objectives. Existing approaches to STL-based motion planning are limited to fragments of STL that might not capture all desired behaviors or safety requirements. Focusing on the case of a fleet of multi-rotor Unmanned Aerial Vehicles (UAVs) jointly tasked with satisfying a given STL mission, we develop a distributed method, Fly-by-Distributed-Logic (FBDL), for trajectory planning. The proposed method generates trajectories that maximize the smooth robustness of STL specifications, where the associated optimization is solved in a distributed manner. This, to the best of our knowledge, is the first distributed method that can handle the full grammar of STL (bounded time). Simulation studies show the applicability of our approach to a variety of STL missions, including those with nested STL operators and the Until operator. We also compare its performance to a centralized approach and show that our method is computationally faster, but generally computes trajectories with lower robustness.

We study control synthesis under Signal Temporal Logic (STL) specifications for driving scenarios where strict rule satisfaction is not always feasible and human experts exhibit context-dependent flexibility. We represent such behavior using robustness slackness—learned rule-wise lower bounds on STL robustness—and introduce sub-goals that encode intermediate intent in the state/output space (e.g., lane-level waypoints). Prior learning-based MPC–STL methods typically infer slackness with VAE priors and plug it into MPC, but these priors can underrepresent multimodal and rare yet valid expert behaviors and do not explicitly model intermediate intent. We propose a diffusion-guided MPC–STL framework that jointly learns slackness and sub-goals from demonstrations and integrates both into STL-constrained MPC. A conditional diffusion model generates pairs of (rule-wise slackness, sub-goal) conditioned on features from the ego vehicle, surrounding traffic, and road context. At run time, a few denoising steps produce samples for the current situation; slackness values define soft STL margins, while sub-goals shape the MPC objective via a terminal (optionally stage) cost, enabling context-dependent trade-offs between rule relaxation and task completion. In closed-loop simulations on held-out highD track-driving scenarios, our method improves task success and yields more realistic lane-changing behavior compared to imitation-learning baselines and MPC–STL variants using CVAE slackness or strict rule enforcement, while remaining computationally tractable for receding-horizon MPC in our experimental setting.

Vehicle trajectory prediction is essential to autonomous driving tasks. Accurate trajectory prediction of other traffic agents can significantly improve the ability of scene understanding and eventually improve the driving safety level of autonomous vehicles. Popular trajectory prediction methods leverage traffic rules by considering maps or lane graphs as a part of the input. However, this implicit representation is insufficient and can not guarantee an accurate understanding of traffic rules through the model. In this paper, we introduce a novel framework called forecasting using continuous unlikelihood and signal temporal logic (FCUS), which incorporates traffic rules in vehicle motion forecasting by explicitly modeling them as continuous signal temporal logic (STL) features into a GAN-style neural network. By doing so, a strong rule prior can be introduced. Meanwhile, an auxiliary unlikelihood loss is adopted to make the model more rule-aware. Extensive experiments evaluated on the real-world driving dataset show that our framework enables state-of-the-art prediction models to improve 8.92% performance on MinADE <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">5</inf> and 19.73% on avoiding rule violation. Demo videos and code are available at https://chantsss.github.io/FCUS/.