Linear Temporal Logic Formulas Research Articles

The Complexity of LTL Rational Synthesis

In rational synthesis , we automatically construct a reactive system that satisfies its specification in all rational environments, namely environments that have objectives and act to fulfill them. We complete the study of the complexity of LTL rational synthesis, when the objectives are given by formulas in Linear Temporal Logic. Our contribution is threefold. First, we tighten the known upper bounds for settings that were left open in earlier work. Second, our complexity analysis is parametric, and we describe tight upper and lower bounds in each of the problem parameters: the game graph, the objectives of the system components, and the objectives of the environment components. Third, we generalize the definition of rational synthesis by adding hostile players to the setting and by combining the cooperative and non-cooperative approaches studied in earlier work.

On reward distribution in reinforcement learning of multi-agent surveillance systems with temporal logic specifications

In multi-agent systems, it is important to design a reward based on the contribution of each agent for efficient learning. In this paper, we propose a reward distribution method for a surveillance system based on our previously proposed multi-agent reinforcement learning method with an aggregator, in which a control specification is described by a linear temporal logic formula. In this method, the aggregator computes and distributes rewards according to the actions that agents take on the surveillance system. Finally, the effectiveness of the proposed method is presented through a numerical simulation of a surveillance problem addressing a specific type of linear temporal logic specification.

Fast and Adaptive Multi-Agent Planning under Collaborative Temporal Logic Tasks via Poset Products

Efficient coordination and planning is essential for large-scale multi-agent systems that collaborate in a shared dynamic environment. Heuristic search methods or learning-based approaches often lack the guarantee on correctness and performance. Moreover, when the collaborative tasks contain both spatial and temporal requirements, e.g., as linear temporal logic (LTL) formulas, formal methods provide a verifiable framework for task planning. However, since the planning complexity grows exponentially with the number of agents and the length of the task formula, existing studies are mostly limited to small artificial cases. To address this issue, a new planning paradigm is proposed in this work for system-wide temporal task formulas that are released online and continually. It avoids two common bottlenecks in the traditional methods, i.e., (a) the direct translation of the complete task formula to the associated Büchi automaton and (b) the synchronized product between the Büchi automaton and the transition models of all agents. Instead, an adaptive planning algorithm is proposed, which computes the product of relaxed partially ordered sets (R-posets) on-the-fly and assigns these subtasks to the agents subject to the ordering constraints. It is shown that the first valid plan can be derived with a polynomial time and memory complexity with respect to the system size and the formula length. Our method can take into account task formulas with a length of more than 400 and a fleet with more than 400 agents, while most existing methods fail at the formula length of 25 within a reasonable duration. The proposed method is validated on large fleets of service robots in both simulation and hardware experiments.

Generating LTL Formulas for Process Mining by Example of Trace

Process mining enables efficient and exhaustive analysis of business processes based on event data. Process mining tools such as LTL Checker allow users to verify temporal properties of business processes for traces by providing a description based on linear temporal logic (LTL). However, it is difficult for many users to understand and use LTL-based mathematical notation. Therefore, there is a need for a method to describe temporal properties even for those who are not familiar with mathematical notation. In this study, we proposed a method based on the satisfiability problem (SAT) for event logs in XES format used in process mining, and it is verified how well it can automatically generate temporal properties in business processes. We conducted experiments using two types of event logs to demonstrate the effectiveness of the proposed method. 

A Uniform Framework for Diagnosis of Discrete-Event Systems With Unreliable Sensors Using Linear Temporal Logic

In this paper, we investigate the diagnosability verification problem of partially-observed discrete-event systems (DES) subject to unreliable sensors. In this setting, upon the occurrence of each event, the sensor reading may be non-deterministic due to measurement noises or possible sensor failures. Existing works on this topic mainly consider specific types of unreliable sensors such as the cases of intermittent sensors failures, permanent sensor failures or their combinations. In this work, we propose a novel <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">uniform framework</i> for diagnosability of DES subject to, not only sensor failures, but also a very general class of unreliable sensors. Our approach is to use linear temporal logic (LTL) with semantics on infinite traces to describe the possible behaviors of the sensors. A new notion of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\varphi$</tex-math></inline-formula> -diagnosability is proposed as the necessary and sufficient condition for the existence of a diagnoser when the behaviors of sensors satisfy the LTL formula <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\varphi$</tex-math></inline-formula> . Effective approach is provided to verify this notion. We show that, our new notion of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\varphi$</tex-math></inline-formula> -diagnosability subsumes all existing notions of robust diagnosability of DES subject to sensor failures. Furthermore, the proposed framework is user-friendly and flexible since it supports an arbitrary user-defined unreliable sensor type based on the specific scenario of the application. As examples, we provide two new notions of diagnosability, which have never been investigated in the literature, using our uniform framework.

Robust stutter bisimulation for abstraction and controller synthesis with disturbance

This paper proposes a method to synthesise controllers for cyber-physical systems subjected to disturbances, such that the controlled system satisfies specifications given as linear temporal logic formulas. To solve this problem, a finite-state abstraction of the original system is first constructed, and then a controller is synthesised for the abstraction. Due to the disturbances and uncertainty in the environment, future states cannot be predicted exactly, and the abstraction must take this into account. For this purpose, the robust stutter bisimulation relation is introduced, which preserves the existence of controllers for any given linear temporal logic formula that excludes the next operator. States are related by the robust stutter bisimulation relation if the same target sets can be guaranteed to be reached or avoided under control of some controller, thus ensuring that disturbances have similar effect on paths that start in related states. It is shown that there exists a controller enforcing a linear temporal logic formula for the original system if and only if a controller exists for the abstracted system. The approach is illustrated by a robot navigation example.

Hierarchical Motion Planning Under Probabilistic Temporal Tasks and Safe-Return Constraints

Safety is crucial for robotic missions within an uncertain environment. Common safety requirements such as collision avoidance are only state-dependent, which can be restrictive for complex missions. In this work, we address a more general formulation as safe-return constraints, which require the existence of a return-policy to drive the system back to a set of safe states with high probability. The robot motion is modeled as a Markov Decision Process (MDP) with probabilistic labels, which can be highly non-ergodic. The robotic task is specified as Linear Temporal Logic (LTL) formulas over these labels, such as surveillance and transportation. We first provide theoretical guarantees on the re-formulation of such safe-return constraints, and a baseline solution based on computing two complete product automata. Furthermore, to tackle the computational complexity, we propose a hierarchical planning algorithm that combines the feature-based symbolic and temporal abstraction with constrained optimization. It synthesizes simultaneously two dependent motion policies: the outbound policy minimizes the overall cost of satisfying the task with a high probability, while the return policy ensures the safe-return constraints. The problem formulation is versatile regarding the robot model, task specifications and safety constraints. The proposed hierarchical algorithm is more efficient and can solve much larger problems than the baseline solution, with only a slight loss of optimality. Numerical validations include simulations and hardware experiments of a search-and-rescue mission and a planetary exploration mission over various system sizes.

Time minimization and online synchronization for multi-agent systems under collaborative temporal logic tasks

Multi-agent systems can be efficient when solving a team-wide task in a concurrent manner. However, without proper synchronization, the correctness of the combined behavior is hard to guarantee, such as to follow a specific ordering of subtasks or to perform a simultaneous collaboration. This work addresses the minimum-time task planning problem for multi-agent systems under complex global tasks stated as syntactically co-safe Linear Temporal Logic (sc-LTL) formulas. These tasks include the temporal and spatial requirements on both independent local actions and direct sub-team collaborations. The proposed solution is an anytime algorithm that combines the partial-ordering analysis of the underlying task automaton for task decomposition, and the branch and bound (BnB) search method for task assignment. We analyze the soundness, completeness and optimality regarding the minimal completion time. We show that a feasible and near-optimal solution is quickly reached while the search continues within the time budget. Furthermore, to handle fluctuations in task duration and agent failures during online execution, we propose an adaptation algorithm to synchronize execution status and re-assign unfinished subtasks dynamically to maintain correctness and optimality. Both algorithms are validated over systems of more than 10 agents via numerical simulations and hardware experiments, against several established baselines.

Outcome-Oriented Prescriptive Process Monitoring based on Temporal Logic Patterns

Background:Prescriptive Process Monitoring systems aim at recommending, during the execution of a business process, interventions that, if followed, prevent poor performance of the process. Such interventions have to be (i) reliable: they have to guarantee the achievement of the desired outcome or performance and (ii) flexible: they cannot overturn the normal process execution. Problem:Most of the Prescriptive Process Monitoring solutions perform well in terms of recommendation reliability but provide the users with recommendations expressed in terms of specific activities that have to be executed without caring about their feasibility. Method:We propose a new Outcome-Oriented Prescriptive Process Monitoring system recommending temporal relations among activities that have to be guaranteed during the process execution. The proposed system is based on a Machine Learning model that learns the correlations between temporal relations among activities and the (positive) outcome of the process. Then, given the prefix of an ongoing process, the model is queried to return the most promising recommendations. Contribution:The main contribution is that the proposed system softens the mandatory execution of an activity at a given point in time, thus leaving more freedom to the user in deciding the interventions to put in place. This is achieved by providing recommendations that are expressed as Linear Temporal Logic formulas over activities. Results:The proposed system has been widely assessed using a pool of 22 real-life datasets. The results demonstrate the reliability of the provided recommendations by achieving an F1 score higher than 90% on 18 datasets out of 22.

Computably Continuous Reinforcement-Learning Objectives Are PAC-Learnable

In reinforcement learning, the classic objectives of maximizing discounted and finite-horizon cumulative rewards are PAC-learnable: There are algorithms that learn a near-optimal policy with high probability using a finite amount of samples and computation. In recent years, researchers have introduced objectives and corresponding reinforcement-learning algorithms beyond the classic cumulative rewards, such as objectives specified as linear temporal logic formulas. However, questions about the PAC-learnability of these new objectives have remained open. This work demonstrates the PAC-learnability of general reinforcement-learning objectives through sufficient conditions for PAC-learnability in two analysis settings. In particular, for the analysis that considers only sample complexity, we prove that if an objective given as an oracle is uniformly continuous, then it is PAC-learnable. Further, for the analysis that considers computational complexity, we prove that if an objective is computable, then it is PAC-learnable. In other words, if a procedure computes successive approximations of the objective's value, then the objective is PAC-learnable. We give three applications of our condition on objectives from the literature with previously unknown PAC-learnability and prove that these objectives are PAC-learnable. Overall, our result helps verify existing objectives' PAC-learnability. Also, as some studied objectives that are not uniformly continuous have been shown to be not PAC-learnable, our results could guide the design of new PAC-learnable objectives.

Learning Interpretable Temporal Properties from Positive Examples Only

We consider the problem of explaining the temporal behavior of black-box systems using human-interpretable models. Following recent research trends, we rely on the fundamental yet interpretable models of deterministic finite automata (DFAs) and linear temporal logic (LTL_f) formulas. In contrast to most existing works for learning DFAs and LTL_f formulas, we consider learning from only positive examples. Our motivation is that negative examples are generally difficult to observe, in particular, from black-box systems. To learn meaningful models from positive examples only, we design algorithms that rely on conciseness and language minimality of models as regularizers. Our learning algorithms are based on two approaches: a symbolic and a counterexample-guided one. The symbolic approach exploits an efficient encoding of language minimality as a constraint satisfaction problem, whereas the counterexample-guided one relies on generating suitable negative examples to guide the learning. Both approaches provide us with effective algorithms with minimality guarantees on the learned models. To assess the effectiveness of our algorithms, we evaluate them on a few practical case studies.

NL2LTL – a Python Package for Converting Natural Language (NL) Instructions to Linear Temporal Logic (LTL) Formulas

This is a demonstration of our newly released Python package NL2LTL which leverages the latest in natural language understanding (NLU) and large language models (LLMs) to translate natural language instructions to linear temporal logic (LTL) formulas. This allows direct translation to formal languages that a reasoning system can use, while at the same time, allowing the end-user to provide inputs in natural language without having to understand any details of an underlying formal language. The package comes with support for a set of default LTL patterns, corresponding to popular DECLARE templates, but is also fully extensible to new formulas and user inputs. The package is open-source and is free to use for the AI community under the MIT license. Open Source: https://github.com/IBM/nl2ltl. Video Link: https://bit.ly/3dHW5b1

Optimal multi-robot path planning for cyclic tasks using Petri nets

In this paper, we investigate the problem of optimal multi-robot path planning for cyclic tasks represented by a particular type of linear-temporal logic (LTL) formulae. Specifically, the team of robot needs to fulfill a given LTL formula and at the same time, accomplishes some particular tasks infinitely often. To avoid the state-space explosion when the number of robot increases, we use Petri nets to model the team of multi-robot. Our goal is to find an optimal infinite sequence in the prefix-suffix form for each robot such that the average cost per task is minimized. We propose an efficient planning method based on the notion of basis reachability graph, which is a compact representation of the reachability space of the PN. We demonstrate the computational efficiency and scalability of our method through illustrative examples. The proposed methods have also been implemented in real-world experiments.

Mapping singly-linked rules to linear temporal logic formulas

Business services are provided by enacting interrelated business processes. Service providers must ensure enactments comply with policies, regulations, and business rules, including rules with quantitative time constraints. Enforcing such rules at design-time may be too restrictive, so effective service provisioning includes expressing rules in a formal specification language and detecting violations of these rules at runtime. Many specification languages do not include quantitative time constraints; for languages with such constraints, it is often unknown if they have runtime monitors whose auxiliary data storage is of bounded size. In this paper, we formulate a technical model of services, a logic language with quantitative time constraints for specifying rules, and develop techniques for automatically generating monitors to detect rule violations. This approach involves two steps, translating: (1) rules to formulas in linear temporal logic (LTL) on finite traces, and (2) LTL formulas to finite state machines. Since algorithms exist for step (2), we focus on step (1), i.e., mapping rules to equivalent LTL formulas. We present and establish the correctness of two translation techniques for “singly-linked” rules. We also compare the size of formulas produced by these techniques with a method of translation derived from Kamp’s Theorem, showing an improvement from hyper-exponential to exponential size.

Learning Linear Temporal Properties for Autonomous Robotic Systems

The problem of passive learning of linear temporal logic formulae consists in finding the best explanation for how two sets of execution traces differ, in the form of the shortest formula that separates the two sets. We approach the problem by implementing an exhaustive search algorithm optimized for execution speed. We apply it to the use-case of a robot moving in an unstructured environment as its battery discharges, both in simulation and in the real world. The results of our experiments confirm that our approach can learn temporal formulas explaining task failures in a case of practical interest.

A Specification-Guided Framework for Temporal Logic Control of Nonlinear Systems

This article proposes a specification-guided framework for control of nonlinear systems with linear temporal logic (LTL) specifications. In contrast with well-known abstraction-based methods, the proposed framework directly characterizes the winning set, i.e., the set of initial conditions from which a given LTL formula can be realized, over the continuous state space of the system via a monotonic operator. Following this characterization, an algorithm is proposed to practically approximate the operator via an adaptive interval subdivision scheme, which yields a finite-memory control strategy. We show that the proposed algorithm is sound for full LTL specifications, and <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">robustly complete</i> for specifications recognizable by deterministic Büchi automaton (DBA), the latter in the sense that control strategies can be found whenever the given specification can be satisfied with additional bounded disturbances. Without having to compute and store the abstraction and the resulting product system with the DBA, the proposed method is more memory efficient, which is demonstrated by complexity analysis and performance tests. A preprocessing stage is also devised to reduce computational cost via a decomposition of the specification. We show that the proposed method can effectively solve real-world control problems, such as jet engine compressor control and motion planning for manipulators and mobile robots.

Overcoming Exploration: Deep Reinforcement Learning for Continuous Control in Cluttered Environments From Temporal Logic Specifications

Model-free continuous control for robot navigation tasks using Deep Reinforcement Learning (DRL) that relies on noisy policies for exploration is sensitive to the density of rewards. In practice, robots are usually deployed in cluttered environments, containing many obstacles and narrow passageways. Designing dense effective rewards is challenging, resulting in exploration issues during training. Such a problem becomes even more serious when tasks are described using temporal logic specifications. This work presents a deep policy gradient algorithm for controlling a robot with unknown dynamics operating in a cluttered environment when the task is specified as a Linear Temporal Logic (LTL) formula. To overcome the environmental challenge of exploration during training, we propose a novel path planning-guided reward scheme by integrating sampling-based methods to effectively complete goal-reaching missions. To facilitate LTL satisfaction, our approach decomposes the LTL mission into sub-goal-reaching tasks that are solved in a distributed manner. Our framework is shown to significantly improve performance (effectiveness, efficiency) and exploration of robots tasked with complex missions in large-scale cluttered environments.

Safe reinforcement learning under temporal logic with reward design and quantum action selection

This paper proposes an advanced Reinforcement Learning (RL) method, incorporating reward-shaping, safety value functions, and a quantum action selection algorithm. The method is model-free and can synthesize a finite policy that maximizes the probability of satisfying a complex task. Although RL is a promising approach, it suffers from unsafe traps and sparse rewards and becomes impractical when applied to real-world problems. To improve safety during training, we introduce a concept of safety values, which results in a model-based adaptive scenario due to online updates of transition probabilities. On the other hand, a high-level complex task is usually formulated via formal languages, including Linear Temporal Logic (LTL). Another novelty of this work is using an Embedded Limit-Deterministic Generalized Büchi Automaton (E-LDGBA) to represent an LTL formula. The obtained deterministic policy can generalize the tasks over infinite and finite horizons. We design an automaton-based reward, and the theoretical analysis shows that an agent can accomplish task specifications with the maximum probability by following the optimal policy. Furthermore, a reward shaping process is developed to avoid sparse rewards and enforce the RL convergence while keeping the optimal policies invariant. In addition, inspired by quantum computing, we propose a quantum action selection algorithm to replace the existing varepsilon-greedy algorithm for the balance of exploration and exploitation during learning. Simulations demonstrate how the proposed framework can achieve good performance by dramatically reducing the times to visit unsafe states while converging optimal policies.

Optimal Probabilistic Motion Planning With Potential Infeasible LTL Constraints

This paper studies optimal motion planning subject to motion and environment uncertainties. By modeling the system as a probabilistic labeled Markov decision process (PL-MDP), the control objective is to synthesize a finite-memory policy, under which the agent satisfies complex high-level tasks expressed as linear temporal logic (LTL) with desired satisfaction probability. In particular, the cost optimization of the trajectory that satisfies infinite horizon tasks is considered, and the trade-off between reducing the expected mean cost and maximizing the probability of task satisfaction is analyzed. The LTL formulas are converted to limit-deterministic Büchi automata (LDBA) with a reachability acceptance condition and a compact graph structure. The novelty of this work lies in considering the cases where LTL specifications can be potentially infeasible and developing a relaxed product MDP between PL- MDP and LDBA. The relaxed product MDP allows the agent to revise its motion plan whenever the task is not fully feasible and quantify the revised plan’s violation measurement. A multi- objective optimization problem is then formulated to jointly consider the probability of task satisfaction, the violation with respect to original task constraints, and the implementation cost of the policy execution. The formulated problem can be solved via coupled linear programs. This work first bridges the gap between probabilistic planning revision of potential infeasible LTL specifications and optimal control synthesis of both plan prefix and plan suffix of the trajectory over the infinite horizons. Experimental results are provided to demonstrate the effectiveness of the proposed framework.

Failure-Aware Self-Diagnostic Task Planning under Temporal Logic Specifications*

This paper investigates the problem of task planning for high-level specifications described by linear temporal logic (LTL) formulae. Existing works on this topic mainly based on the assumption that the functionalities of the system are always correct during the execution. In this work, we consider the scenario where the system is subject to internal failures that cannot be measured directly but may be inferred by a sequence of actions. The objective is to design a failure-aware task plan such that (i) the system will achieve the LTL task when there is no failure; and (ii) along the designed plan, any potential failure can be detected within a bounded number of steps. We provide a framework for modeling the behavior of the system with potential internal failures. Furthermore, an effective algorithm is designed to synthesize an optimal self-diagnostic plan, in the form of the prefix-suffix structure, such that these two requirements are satisfied. We illustrate the proposed framework by a case study of production task planning.