Metric Interval Temporal Logic Research Articles

Robust Satisfaction of Metric Interval Temporal Logic Objectives in Adversarial Environments

This paper studies the synthesis of controllers for cyber-physical systems (CPSs) that are required to carry out complex time-sensitive tasks in the presence of an adversary. The time-sensitive task is specified as a formula in the metric interval temporal logic (MITL). CPSs that operate in adversarial environments have typically been abstracted as stochastic games (SGs); however, because traditional SG models do not incorporate a notion of time, they cannot be used in a setting where the objective is time-sensitive. To address this, we introduce durational stochastic games (DSGs). DSGs generalize SGs to incorporate a notion of time and model the adversary’s abilities to tamper with the control input (actuator attack) and manipulate the timing information that is perceived by the CPS (timing attack). We define notions of spatial, temporal, and spatio-temporal robustness to quantify the amounts by which system trajectories under the synthesized policy can be perturbed in space and time without affecting satisfaction of the MITL objective. In the case of an actuator attack, we design computational procedures to synthesize controllers that will satisfy the MITL task along with a guarantee of its robustness. In the presence of a timing attack, we relax the robustness constraint to develop a value iteration-based procedure to compute the CPS policy as a finite-state controller to maximize the probability of satisfying the MITL task. A numerical evaluation of our approach is presented on a signalized traffic network to illustrate our results.

Online Motion Planning With Soft Metric Interval Temporal Logic in Unknown Dynamic Environment

Motion planning of an autonomous system with high-level specifications has wide applications. However, research of formal languages involving timed temporal logic is still under investigation. Furthermore, many existing results rely on a key assumption that user-specified tasks are feasible in the given environment. Challenges arise when the operating environment is dynamic and unknown since the environment can be found prohibitive, leading to potentially conflicting tasks where pre-specified timed missions cannot be fully satisfied. Such issues become even more challenging when considering time-bound requirements. To address these challenges, this work proposes a control framework that considers hard constraints to enforce safety requirements and soft constraints to enable task relaxation. The metric interval temporal logic (MITL) specifications are employed to deal with time-bound constraints. By constructing a relaxed timed product automaton, an online motion planning strategy is synthesized with a receding horizon controller to generate policies, achieving multiple objectives in decreasing order of priority 1) formally guarantee the satisfaction of hard safety constraints; 2) mostly fulfill soft timed tasks; and 3) collect time-varying rewards as much as possible. Another novelty of the relaxed structure is to consider violations of both time and tasks for infeasible cases. Simulation results are provided to validate the proposed approach.

Scalable time-constrained planning of multi-robot systems

This paper presents a scalable procedure for time-constrained planning of a class of uncertain nonlinear multi-robot systems. In particular, we consider N robotic agents operating in a workspace which contains regions of interest (RoI), in which atomic propositions for each robot are assigned. The main goal is to design decentralized and robust control laws so that each robot meets an individual high-level specification given as a metric interval temporal logic (MITL), while using only local information based on a limited sensing radius. Furthermore, the robots need to fulfill certain desired transient constraints such as collision avoidance between them. The controllers, which guarantee the transition between regions, consist of two terms: a nominal control input, which is computed online and is the solution of a decentralized finite-horizon optimal control problem (DFHOCP); and an additive state feedback law which is computed offline and guarantees that the real trajectories of the system will belong to a hyper-tube centered along the nominal trajectory. The controllers serve as actions for the individual weighted transition system (WTS) of each robot, and the time duration required for the transition between regions is modeled by a weight. The DFHOCP is solved at every sampling time by each robot and then necessary information is exchanged between neighboring robots. The proposed approach is scalable since it does not require a product computation among the WTS of the robots. The proposed framework is experimentally tested and the results show that the proposed framework is promising for solving real-life robotic as well as industrial applications.

Model Checking MITL Formulae on Timed Automata

Timed Automata (TA) is de facto a standard modelling formalism to represent systems when the interest is the analysis of their behaviour as time progresses. This modelling formalism is mostly used for checking whether the behaviours of a system satisfy a set of properties of interest. Even if efficient model-checkers for Timed Automata exist, these tools are not easily configurable. First, they are not designed to easily allow adding new Timed Automata constructs, such as new synchronization mechanisms or communication procedures, but they assume a fixed set of Timed Automata constructs. Second, they usually do not support the Metric Interval Temporal Logic (MITL) and rely on a precise semantics for the logic in which the property of interest is specified, which cannot be easily modified and customized. Finally, they do not easily allow using different solvers that may speed up verification in different contexts. This article presents a novel technique to perform model checking of Metric Interval Temporal Logic (MITL) properties on TA. The technique relies on the translation of both the TA and the MITL formula into an intermediate Constraint LTL over clocks (CLTLoc) formula, which is verified through an available decision procedure. The technique is flexible, since the intermediate logic allows the encoding of new semantics as well as new TA constructs, by just adding new CLTLoc formulae. Furthermore, our technique is not bound to a specific solver as the intermediate CLTLoc formula can be verified using different procedures.

Robust Design and Validation of Cyber-physical Systems

Co-simulation--based validation of hardware controllers adjoined with plant models, with continuous dynamics, is an important step in model-based design of controllers for Cyber-physical Systems (CPS). Co-simulation suffers from many problems, such as timing delays, skew, race conditions, and so on, making it unsuitable for checking timing properties of CPS. In our approach to validation of controllers, synthesised from their models, the synthesised controller is adjoined with a synthesised hardware plant unit. The synthesised plant and controller are then executed synchronously and Metric Interval Temporal Logic (MITL) properties are validated on the closed-loop system. The clock period is chosen, using robustness estimates, such that all timing properties that hold on the controller guiding the discretised plant model also hold on the original case of the continuous-time plant model guided by the controller. Benchmark results show that real-time MITL properties that are vacuously satisfied or violated due to co-simulation artefacts hold correctly in the proposed closed-loop validation framework.

Transfer of Temporal Logic Formulas in Reinforcement Learning.

Transferring high-level knowledge from a source task to a target task is an effective way to expedite reinforcement learning (RL). For example, propositional logic and first-order logic have been used as representations of such knowledge. We study the transfer of knowledge between tasks in which the timing of the events matters. We call such tasks temporal tasks. We concretize similarity between temporal tasks through a notion of logical transferability, and develop a transfer learning approach between different yet similar temporal tasks. We first propose an inference technique to extract metric interval temporal logic (MITL) formulas in sequential disjunctive normal form from labeled trajectories collected in RL of the two tasks. If logical transferability is identified through this inference, we construct a timed automaton for each sequential conjunctive subformula of the inferred MITL formulas from both tasks. We perform RL on the extended state which includes the locations and clock valuations of the timed automata for the source task. We then establish mappings between the corresponding components (clocks, locations, etc.) of the timed automata from the two tasks, and transfer the extended Q-functions based on the established mappings. Finally, we perform RL on the extended state for the target task, starting with the transferred extended Q-functions. Our implementation results show, depending on how similar the source task and the target task are, that the sampling efficiency for the target task can be improved by up to one order of magnitude by performing RL in the extended state space, and further improved by up to another order of magnitude using the transferred extended Q-functions.

On the timed temporal logic planning of coupled multi-agent systems

This paper presents a fully automated procedure for controller synthesis for multi-agent systems under coupling constraints. Each agent is modeled with dynamics consisting of two terms: the first one models the coupling constraints and the other one is an additional bounded control input. We aim to design these inputs so that each agent meets an individual high-level specification given as a Metric Interval Temporal Logic (MITL). First, a decentralized abstraction that provides a space and time discretization of the multi-agent system is designed. Second, by utilizing this abstraction and techniques from formal verification, we propose an algorithm that computes the individual runs which provably satisfy the high-level tasks. The overall approach is demonstrated in a simulation example conducted in MATLAB environment.

Control Synthesis for Multi-Agent Systems under Metric Interval Temporal Logic Specifications

This paper presents a framework for automatic synthesis of a control sequence for multi-agent systems governed by continuous linear dynamics under timed constraints. First, the motion of the agents in the workspace is abstracted into individual Transition Systems (TS). Second, each agent is assigned with an individual formula given in Metric Interval Temporal Logic (MITL) and in parallel, the team of agents is assigned with a collaborative team formula. The proposed method is based on a correct-by-construction control synthesis method, and hence guarantees that the resulting closed-loop system will satisfy the desired specifications. The specifications considers boolean-valued properties under real-time bounds. Extended simulations has been performed in order to demonstrate the efficiency of the proposed methodology.

A logical characterization of timed regular languages

CLTLoc (Constraint LTL over clocks) is a quantifier-free extension of LTL allowing variables behaving like clocks over real numbers. CLTLoc is in PSPACE [1] and its satisfiability can polynomially be reduced to a Satisfiability Modulo Theories (SMT) problem, allowing a feasible implementation of a decision procedure. We used CLTLoc to capture the semantics of metric temporal logics over continuous time, such as Metric Interval Temporal Logic (MITL), resulting in the first successful implementation of a tool for checking MITL satisfiability [2,3]. In this paper, we assess the expressive power of CLTLoc, by comparing it with various temporal formalisms over dense time. We restrict the analysis to well initialized models of formulae where the value of all clocks in the origin is either 0 or equal to a positive real constant. Under this assumption, when interpreted over timed words, the class of timed languages defined by CLTLoc formulae coincides with the class defined by Timed Automata. We also define a timed monadic first order logic of order, extending the one introduced by Kamp, which is expressively equivalent to CLTLoc for the class of timed languages that are defined by well initialized models.

A Mori-Zwanzig and MITL Based Approach to Statistical Verification of Continuous-time Dynamical Systems

In this work, we introduce a framework for the statistical verification of Metric Interval Temporal Logic (MITL) formulas on continuous-time dynamical systems. By considering the continuous-time Markov process associated with the dynamical system, we apply the Mori-Zwanzig method to reduce the original system to a Continuous-Time Markov Chain (CTMC). Accordingly, the MITL formulas on the original system can be reduced to MITL formulas on the CTMC. Furthermore, we propose a statistical verification algorithm for checking the MITL formulas on the CTMCand show that the original MITL formulas on the original system can be checked by this procedure.

Incremental verification of component-based timed systems

We are interested in the incremental development, by integration of components, of component-based timed systems, and in particular, in the preservation of their properties during such a development process. We model timed components with timed automata. Their composition is achieved with the classic parallel composition operator for timed automata. The specifications of these timed systems are expressed with the timed linear logic Metric Interval Temporal Logic (MITL). To guarantee the preservation of properties during an incremental development process, we propose to use τ-simulation relations, adapted for timed systems. First, we extend the classic notion of τ-simulation with timed aspects. As in the untimed case, this relation, called timed τ-simulation, preserves safety properties. To preserve more properties, in particular liveness ones, we present another relation, called divergence-sensitive and stability-respecting (DS) timed τ-simulation. This last relation preserves all MITL properties (and thus liveness ones), but also strong non-zenoness and deadlock-freedom. Moreover, as we put ourselves in a component-based framework, we study if the relations are appropriate to the use of the composition operator that we consider. For this purpose, we study if the relations are compatible with this operator, and if composability and compositionality hold. These three properties are a way to reduce the cost of the verification of the preservation, or even to get it for free. It results that the timed τ-simulation is appropriate with the classic operator since the properties hold without any assumption. However, this is not the case for the DS timed τ-simulation. We implemented the algorithmic verification of the simulations in a tool called Verification of Simulation for Timed Automata (VeSTA). The structure of the tool was inspired from the one of the OPEN-KRONOS tool. This allows, as additional feature, to connect the models considered in VeSTA to the modules of the verification platform OPEN-CAESAR. We show the interest of our method by applying it on a case study, concerning a production cell example.

Robustness of temporal logic specifications for continuous-time signals

In this paper, we consider the robust interpretation of Metric Temporal Logic (MTL) formulas over signals that take values in metric spaces. For such signals, which are generated by systems whose states are equipped with non-trivial metrics, for example continuous or hybrid, robustness is not only natural, but also a critical measure of system performance. Thus, we propose multi-valued semantics for MTL formulas, which capture not only the usual Boolean satisfiability of the formula, but also topological information regarding the distance, ε , from unsatisfiability. We prove that any other signal that remains ε -close to the initial one also satisfies the same MTL specification under the usual Boolean semantics. Finally, our framework is applied to the problem of testing formulas of two fragments of MTL, namely Metric Interval Temporal Logic (MITL) and closed Metric Temporal Logic (clMTL), over continuous-time signals using only discrete-time analysis. The motivating idea behind our approach is that if the continuous-time signal fulfills certain conditions and the discrete-time signal robustly satisfies the temporal logic specification, then the corresponding continuous-time signal should also satisfy the same temporal logic specification.

Timed automata for metric interval temporal logic formulae in prototype verification system

Based on analysis of the syntax structure and semantics model of the metric interval temporal logic (MITL) formulas, it is shown how to transform a formula written in the real-time temporal logic MITL formula into a fair timed automaton (TA) that recognizes its satisfying models with prototype verification system (PVS) in this paper. Both the tabular construction’s principles and the PVS implementation details are given for the different type of MITL formula according to the corresponding semantics interpretations. After this transformation procedure, specifications expressed with MITL formula can be verified formally in the timed automata framework developed previously.

On the Contribution of a τ-simulation in the Incremental Modeling of Timed Systems

We are interested in the preservation of linear-time properties during incremental modeling of timed systems. We consider timed systems modeled by timed automata in a compositional framework. Their requirements are expressed by the logical formalism MITL (Metric Interval Temporal Logic).We propose to use τ-simulations as a way to preserve such properties during an incremental modeling, i.e., either integration of components or refinement. We define τ-simulation relations on the semantics of timed automata in order to handle the preservation of liveness properties. Moreover, we implemented a tool to verify such τ-simulations, based on Open-Kronos libraries and using the tool Profounder.