Signal Temporal Logic Formulas Research Articles

Estimating event probabilities via signal temporal logic and first occurrence distributions

Abstract Estimating the probability of events is a significant challenge in many fields, often requiring a probabilistic model or additional labels and tasks for accurate prediction. However, those methods have limited scalability or unnecessary computational resource consumption due to predicting unrelated values. To address these issues, we propose a novel approach that estimates event probabilities based on the distributions of their first occurrence in the time domain. By using Signal Temporal Logic formulas to describe events and applying an algorithm that estimates complex events’ probabilities through simple event occurrence distributions, this study presents an efficient approach that does not depend on high-precision prediction. We evaluate the performance of our method on simulated scenarios of unmanned aerial vehicle motion and autonomous driving.

BUSTLE: a Versatile Tool for the Evolutionary Learning of STL Specifications from Data.

Describing the properties of complex systems that evolve over time is a crucial requirement for monitoring and understanding them. Signal Temporal Logic (STL) is a framework that proved to be effective for this aim because it is expressive and allows state properties as human-readable formulae. Crafting STL formulae that fit a particular system is, however, a difficult task. For this reason, a few approaches have been proposed recently for the automatic learning of STL formulae starting from observations of the system. In this paper, we propose BUSTLE (Bi-level Universal STL Evolver), an approach based on evolutionary computation for learning STL formulae from data. BUSTLE advances the state-of-the-art because it (i) applies to a broader class of problems, in terms of what is known about the state of the system during its observation, and (ii) generates both the structure and the values of the parameters of the formulae employing a bi-level search mechanism (global for the structure, local for the parameters). We consider two cases where (a) observations of the system in both anomalous and regular state are available, or (b) only observations of regular state are available. We experimentally evaluate BUSTLE on problem instances corresponding to the two cases and compare it against previous approaches. We show that the evolved STL formulae are effective and human-readable: the versatility of BUSTLE does not come at the cost of lower effectiveness.

Online control synthesis for uncertain systems under signal temporal logic specifications

Signal temporal logic (STL) formulas have been widely used as a formal language to express complex robotic specifications, thanks to their rich expressiveness and explicit time semantics. Existing approaches for STL control synthesis suffer from limited scalability with respect to the task complexity and lack of robustness against the uncertainty, for example, external disturbances. In this paper, we study the online control synthesis problem for uncertain discrete-time systems subject to STL specifications. Different from existing techniques, we propose an approach based on STL, reachability analysis, and temporal logic trees. First, based on a real-time version of STL semantics, we develop the notion of tube-based temporal logic tree (tTLT) and its recursive (offline) construction algorithm. We show that the tTLT is an under-approximation of the STL formula, in the sense that a trajectory satisfying a tTLT also satisfies the corresponding STL formula. Then, an online control synthesis algorithm is designed using the constructed tTLT. It is shown that when the STL formula is robustly satisfiable and the initial state of the system belongs to the initial root node of the tTLT, it is guaranteed that the trajectory generated by the control synthesis algorithm satisfies the STL formula. We validate the effectiveness of the proposed approach by several simulation examples and further demonstrate its practical usability on a hardware experiment. These results show that our approach is able to handle complex STL formulas with long horizons and ensure the robustness against the disturbances, which is beyond the scope of the state-of-the-art STL control synthesis approaches.

Model predictive monitoring of dynamical systems for signal temporal logic specifications

Online monitoring aims to evaluate or to predict, at runtime, whether or not the behaviors of a system satisfy some desired specification. It plays a key role in safety-critical cyber–physical systems. In this work, we propose a new monitoring approach, called model predictive monitoring, for specifications described by Signal Temporal Logic (STL) formulae. Specifically, we assume that the observed state traces are generated by an underlying dynamical system whose model is known but the control law is unknown. The main idea is to use the dynamic of the system to predict future states when evaluating the satisfaction of the STL formulae. To this end, effective approaches for the computation of feasible sets of STL formulae are provided. We show that, by explicitly utilizing the model information of the dynamical system, the proposed online monitoring algorithm can falsify or certify of the specification in advance compared with existing algorithms, where no model information is used. We also demonstrate the proposed monitoring algorithm by several real world case studies.

Signal temporal logic synthesis under Model Predictive Control: A low complexity approach

In this paper, we focus on the challenging problem of model predictive control (MPC) for dynamics systems with high-level tasks formulated as signal temporal logic (STL). The state-of-art for STL synthesis mainly suffers from limited scalability with respect to the complexity of the task and the planning horizon, hindering the real-time implementation of MPC. This work tackles this issue by STL formula reformulation and input blocking. Specifically, simplifications are applied on disjunctive STL (sub)formulae recursively in the framework of MPC to limit formula size. We show that the simplified STL can be reformulated into mixed integer linear programming (MILP) constraints with a modifiable number of binary variables being required. The move blocking scheme is then employed to further reduce problem complexity by fixing input variables to be constant over several time intervals. In order to trade off the control performance and computational load, a blocking structure design with on-line correction is proposed. The extension of the proposed STL-MPC algorithm to uncertain systems is achieved through STL constraint tightening. Simulations and experiments show the effectiveness of the proposed algorithm.

Dynamic Quantization based Control Synthesis for Signal Temporal Logics Specifications

For dynamical systems under temporal logic tasks, the control objective is mainly to propose control methods for temporal logic tasks. Existing methods are based on the assumption where temporal logic tasks are realizable, which is not always valid and results in the separation between the realizability verification and the controller design. In this paper, we apply dynamic quantization techniques to propose a framework to combine the verification of signal temporal logic (STL) specifications and the controller design. In particular, a sequence of quantization regions is generated to show the realizability of the STL formula, and results in the decomposition of the global STL formula into finite local ones. All local STL formulas are formulated into optimization problems, which are solved iteratively and result in local controllers, whose combination gives the global controller for the satisfaction of the STL formula. Finally, a numerical example is presented to illustrate the proposed approach.

Distributed Control of Coupled Leader-follower Multi-agent Systems under Spatiotemporal Logic Tasks

This paper addresses the problem of cooperative control of leader-follower multi-agent systems under local signal temporal logic (STL) tasks in a distributed fashion, where the overall system is composed of several leader-follower subsystems with coupled dynamics. In this work, only the leaders know the related STL specifications and are designed to drive the followers in a way such that the STL specifications are globally satisfied. Under the local feasibility assumption, we propose a funnel-based control approach for each leader-follower subsystem such that the local STL specifications are achieved, which further implies the global satisfaction of all STL specifications. In order to enforce the satisfaction of the STL formulas, the funnel parameters are appropriately designed to prescribe certain transient behavior that constrains the closed-loop trajectories. The proposed approach is illustrated by a simulation example.

Exploiting Invariance Properties to Certify Always and Eventually Signal Temporal Logic Operators for Hybrid Dynamical Systems

In this paper, semantics and characterizations of signal temporal logic formulas for hybrid dynamical systems are presented. Hybrid dynamical systems are given in terms of constrained differential and difference inclusions, which, respectively, capture the continuous evolution and the instantaneous events exhibited by solutions. For such systems, the always and eventually operator of signal temporal logic are studied and characterizations in terms of dynamical properties of hybrid systems are presented – in particular, using invariance and finite-time attractivity properties. Sufficient conditions that guarantee the satisfaction of a signal temporal logic formula for a given system through the satisfaction of an untimed formula for an appropriately defined new system are introduced. Specifically, it is shown that satisfying an (untimed) temporal logic formula involving until operators suffices to certify always and eventually signal temporal logic formulas for hybrid systems.

Differentiable Inference of Temporal Logic Formulas

We demonstrate the first Recurrent Neural Network architecture for learning Signal Temporal Logic formulas, and present the first systematic comparison of formula inference methods. Legacy systems embed much expert knowledge which is not explicitly formalized. There is great interest in learning formal specifications that characterize the ideal behavior of such systems -- that is, formulas in temporal logic that are satisfied by the system's output signals. Such specifications can be used to better understand the system's behavior and improve design of its next iteration. Previous inference methods either assumed certain formula templates, or did a heuristic enumeration of all possible templates. This work proposes a neural network architecture that infers the formula structure via gradient descent, eliminating the need for imposing any specific templates. It combines learning of formula structure and parameters in one optimization. Through systematic comparison, we demonstrate that this method achieves similar or better mis-classification rates (MCR) than enumerative and lattice methods. We also observe that different formulas can achieve similar MCR, empirically demonstrating the under-determinism of the problem of temporal logic inference.

Cooperative Object Manipulation Under Signal Temporal Logic Tasks and Uncertain Dynamics

We address the problem of cooperative manipulation of an object whose tasks are specified by a Signal Temporal Logic (STL) formula. We employ the Prescribed Performance Control (PPC) methodology to guarantee predefined transient and steady-state performance on the object trajectory in order to satisfy the STL formula. More specifically, we first provide a way that translates the problem of satisfaction of an STL task to the problem of state evolution within a user-defined time-varying funnel. We then design a control strategy for the robotic agents that guarantees compliance with this funnel. The control strategy is decentralized, in the sense that each agent calculates its own control signal, and does not use any information on the agents' and object's dynamic terms, which are assumed to be unknown. We experimentally verify the results on two manipulator arms, cooperatively working to manipulate an object based on a STL formula.

Parameter synthesis of polynomial dynamical systems

Parametric dynamical systems emerge as a formalism for modeling natural and engineered systems ranging from biology, epidemiology, and medicine to cyber-physics. Parameter tuning is a complex task usually performed exploiting heavy simulations having high computational complexity and not ensuring the correctness of the synthesized systems. We consider the problem of parameter synthesis for discrete-time polynomial systems. We propose a formal method based on Bernstein coefficients that allows refining the set of parameters according to a temporal specification defined as a Signal Temporal Logic formula. The synthesized system is correct with respect to the specification and we demonstrate the scalability of the approach by implementing it in the C++ library Sapo, also available within a stand-alone application and as a web application. Finally, we illustrate the tool usage and the interface through a simple yet realistic epidemiological model and consider an intriguing application enhancing accuracy of the verification of neural networks.

Distributed model predictive control for multi‐robot systems with conflicting signal temporal logic tasks

In this paper, the control problem of multi-robot systems under temporal logic tasks with limited time and logic constraints is studied, where each robot is required to reach the specified region in a given time and avoid collision all the time. Since the cooperative collision avoidance task of one robot depends on other robots' behaviors, the satisfaction of all the tasks may be conflicting. In this work, a distributed model predictive control (DMPC) strategy is proposed for conflicting temporal logic tasks. First, signal temporal logic (STL) is adopted to formally describe the temporal logic tasks. Based on robust semantics of STL formulas, a reference trajectory for the satisfaction degree of the task is designed to determine the short-term task in the optimization horizon. In the DMPC optimization problem, the compatibility constraints are introduced to redesign the collision avoidance constraints, such that the collision avoidance tasks can be fulfilled using neighbouring robots' predicted information of the last time instant. Then, the terminal constraint is designed by the short-term motion task which enforces each robot to move towards the goal region within the specified time interval. For conflicting tasks, a slack parameter is introduced in the terminal set to relax the motion task of each robot. The recursive feasibility of the DMPC algorithm is guaranteed, and the relaxed temporal logic tasks are fulfilled. In the proposed method, the discrete events including limited time and logic requirements are incorporated into the DMPC optimization problem, such that the motion task can be satisfied as much as possible and safety requirements are fulfilled. The effectiveness of the algorithm is illustrated by simulation and experiment results.

Recurrent Neural Network Controllers for Signal Temporal Logic Specifications Subject to Safety Constraints

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.

Signal Temporal Logic Task Decomposition via Convex Optimization

In this letter we focus on the problem of decomposing a global Signal Temporal Logic formula (STL) assigned to a multi-agent system to local STL tasks when the team of agents is a-priori decomposed to disjoint sub-teams. The predicate functions associated to the local tasks are parameterized as hypercubes depending on the states of the agents in a given sub-team. The parameters of the functions are, then, found as part of the solution of a convex program that aims implicitly at maximizing the volume of the zero superlevel set of the corresponding predicate function. Two alternative definitions of the local STL tasks are proposed and the satisfaction of the global STL formula is proven when the conjunction of the local STL tasks is satisfied.

Data-driven verification of stochastic linear systems with signal temporal logic constraints

Cyber–physical systems usually have complex dynamics and are required to fulfill complex tasks. In recent years, formal methods from Computer Science have been used by control theorists for both describing the required tasks and ensuring that they are fulfilled by the systems. The crucial drawback of formal methods is that a complete model of the system often needs to be available. The main goal of this paper is to study formal verification of linear time-invariant systems with respect to a fragment of temporal logic specifications when only a partial knowledge of the model is available, i.e., a parameterized model of the system is known but the exact values of the parameters are unknown. We provide a probabilistic measure for the satisfaction of the specification by trajectories of the system under the influence of uncertainty. We assume these specifications are expressed as signal temporal logic formulae and provide an approach that relies on gathering input–output data from the system and employs Bayesian inference on the collected data to associate a notion of confidence to the satisfaction of the specification. The main novelty of our approach is to combine both data-driven and model-based techniques in order to have a two-layer probabilistic reasoning over the behavior of the system. The inner layer is with respect to the uncertainties in dynamics and observed data while the outer layer is with respect to the distribution over the parameter space. The latter is updated using Bayesian inference on the collected data. The proposed approach is demonstrated in two case studies.

Formel Yaklaşımlar ile Sürüm Değerlendirme

In this paper, a formal method-based release evaluation method was developed. During the release evaluation process, two versions of a server are run under similar (or the same) configurations and the system logs are compared. This comparison can be based on graphical analysis, applying fixed rules over logs or regression analysis. This paper presents a novel release evaluation approach based on formal methods. The proposed method consists of three main steps. The first step is the collection of data from both versions of the server. The second step is the synthesis of Signal Temporal Logic (STL) formulas for each dataset. The final step is the generation of the release evaluation result by comparing the formulas that have the same structure and that can represent both datasets with high accuracy. Thus, the proposed approach represents the release evaluation rules as STL formulas and generates such formulas from system logs in an automated way. Due to the resemblance of temporal logics to natural language, the resulting formulas explain the evaluation result. The proposed method automates the release evaluation process. The findings of the paper are shown over sample datasets.

Temporal Logic Inference for Fault Detection of Switched Systems With Gaussian Process Dynamics

In this article, we present a method for constructing the fault detector in the form of signal temporal logic (STL) formulas, which can be understood by human users and formally proven to detect faults with probabilistic satisfaction guarantees, for a class of switched nonlinear systems with partially unknown dynamics. First, the partially unknown internal dynamics are approximated by the Gaussian process with stability guarantees. Second, a novel temporal logic inference algorithm is proposed to find the fault detector, which takes advantage of the internal properties of temporal logic and searches for the optimal formula along a partially ordered direction. Moreover, the algorithm is not allowed for missing faults but allowed for false alarms during the temporal logic inference process. In addition, we simulate finitely many trajectories with Chua’s circuit and infer the temporal logic formulas with the Gaussian optimization. The results show that the proposed method can find a temporal logic formula to detect the faulty trajectory with a probability guarantee. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Note to Practitioners</i> —The method proposed in this article can be used to detect faults for switched systems with partially unknown dynamics. STL is used to describe the behaviors of the system, which acts as a classifier and detector, such that all normal behaviors of the system will satisfy the description, while the faulty behaviors will violate the description. Moreover, STL formulas can be understood by human operators, which is important for the timely response to faulty events. For example, the normal behavior of a smart grid can be described as follows: “if the smart grid is safe, it should reach 9 kV within 15 min when the voltage to region A is above 12 kV,” which can be expressed with STL. Due to the unknown dynamics, the Gaussian process regression is applied to estimate the model and the region that is robust to noises.

Towards a framework of enforcing resilient operation of cyber‐physical systems with unknown dynamics

Ensuring that safety-critical cyber-physical systems (CPSs) continue to satisfy correctness and safety specifications even under faults or adversarial attacks is very challenging, especially in the presence of legacy components for which accurate models are unknown to the designer. Current techniques for secure-by-design systems engineering do not provide an end-to-end methodology for a designer to provide real-time assurance for safety-critical CPSs by identifying system dynamics and updating control strategies in response to newly discovered faults, attacks or other changes such as system upgrades. We propose a new methodology, along with an integrated framework implemented in MATLAB to guarantee the resilient operation of safety-critical CPSs with unknown dynamics. The proposed framework consists of three main components. The runtime monitor evaluates the system behaviour on-the-fly against its correctness specifications expressed as signal temporal logic formulas. The model synthesiser incorporates a sparse identification approach that is used to continually update the plant model and control policies to adapt to any changes in the system or the environment. The decision and control module designs a controller to ensure that the correctness specifications are satisfied at runtime. For evaluation, we apply our proposed framework to ensure the resilient operations of two CPS case studies.

Offline and Online Learning of Signal Temporal Logic Formulae Using Decision Trees

In this article, we focus on inferring high-level descriptions of a system from its execution traces. Specifically, we consider a classification problem where system behaviors are described using formulae of Signal Temporal Logic (STL). Given a finite set of pairs of system traces and labels, where each label indicates whether the corresponding trace exhibits some system property, we devised a decision-tree-based framework that outputs an STL formula that can distinguish the traces. We also extend this approach to the online learning scenario. In this setting, it is assumed that new signals may arrive over time and the previously inferred formula should be updated to accommodate the new data. The proposed approach presents some advantages over traditional machine learning classifiers. In particular, the produced formulae are interpretable and can be used in other phases of the system’s operation, such as monitoring and control. We present two case studies to illustrate the effectiveness of the proposed algorithms: (1) a fault detection problem in an automotive system and (2) an anomaly detection problem in a maritime environment.

Analysis of E-commerce Ranking Signals via Signal Temporal Logic

The timed position of documents retrieved by learning to rank models can be seen as signals. Signals carry useful information such as drop or rise of documents over time or user behaviors. In this work, we propose to use the logic formalism called Signal Temporal Logic (STL) to characterize document behaviors in ranking accordingly to the specified formulas. Our analysis shows that interesting document behaviors can be easily formalized and detected thanks to STL formulas. We validate our idea on a dataset of 100K product signals. Through the presented framework, we uncover interesting patterns, such as cold start, warm start, spikes, and inspect how they affect our learning to ranks models.