Linear Temporal Logic Research Articles

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.

Symbolic model checking quantum circuits in Maude.

This article presents a symbolic approach to model checking quantum circuits using a set of laws from quantum mechanics and basic matrix operations with Dirac notation. We use Maude, a high-level specification/programming language based on rewriting logic, to implement our symbolic approach. As case studies, we use the approach to formally specify several quantum communication protocols in the early work of quantum communication and formally verify their correctness: Superdense Coding, Quantum Teleportation, Quantum Secret Sharing, Entanglement Swapping, Quantum Gate Teleportation, Two Mirror-image Teleportation, and Quantum Network Coding. We demonstrate that our approach/implementation can be a first step toward a general framework to formally specify and verify quantum circuits in Maude. The proposed way to formally specify a quantum circuit makes it possible to describe the quantum circuit in Maude such that the formal specification can be regarded as a series of quantum gate/measurement applications. Once a quantum circuit has been formally specified in the proposed way together with an initial state and a desired property expressed in linear temporal logic (LTL), the proposed model checking technique utilizes a built-in Maude LTL model checker to automatically conduct formal verification that the quantum circuit enjoys the property starting from the initial state.

Describing Weighted Safety with Weighted LTL over Product omega-valuation Monoids

We define the notion of k-safe infinitary series over idempotent ordered totally generalized product omega-valuation monoids that satisfy specific properties. For each element k of the underlying structure (different from the neutral elements of the additive, and the multiplicative operation) we determine two syntactic fragments of the weighted LTL with the property that the semantics of the formulas in these fragments are k-safe infinitary series. For specific idempotent ordered totally generalized product omega-valuation monoids we provide algorithms that given a weighted Buchi automaton and a weighted LTL formula in these fragments, decide whether the behavior of the automaton coincides with the semantics of the formula.

Temporally extended goal recognition in fully observable non-deterministic domain models

Goal Recognition is the task of discerning the intended goal that an agent aims to achieve, given a set of goal hypotheses, a domain model, and a sequence of observations (i.e., a sample of the plan executed in the environment). Existing approaches assume that goal hypotheses comprise a single conjunctive formula over a single final state and that the environment dynamics are deterministic, preventing the recognition of temporally extended goals in more complex settings. In this paper, we expand goal recognition to temporally extended goals in Fully Observable Non-Deterministic (fond) planning domain models, focusing on goals on finite traces expressed in Linear Temporal Logic (ltlf\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$_f$$\\end{document}) and Pure-Past Linear Temporal Logic (ppltl). We develop the first approach capable of recognizing goals in such settings and evaluate it using different ltlf\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$_f$$\\end{document} and ppltl goals over six fond planning domain models. Empirical results show that our approach is accurate in recognizing temporally extended goals in different recognition settings.

LTL-specification for development and verification of control programs

This work continues the series of articles on development and verification of control programs based on the LTL-specification. The essence of the approach is to describe the behavior of programs using formulas of linear temporal logic LTL of a special form. The developed LTL-specification can be directly verified by using a model checking tool. Next, according to the LTL-specification, the program code in the imperative programming language is unambiguously built. The translation of the specification into the program is carried out using a template. The novelty of the work consists in the proposal of two LTL-specifications of a new form — declarative and imperative, as well as in a more strict formal justification for this approach to program development and verification. A transition has been made to a more modern verification tool for finite and infinite systems — nuXmv. It is proposed to describe the behavior of control programs in a declarative style. For this purpose, a declarative LTL-specification is intended, which defines a labelled transition system as a formal model of program behavior. This method of describing behavior is quite expressive — the theorem on the Turing completeness of the declarative LTL-specification is proved. Next, to construct program code in an imperative language, the declarative LTL-specification is converted into an equivalent imperative LTL-specification. An equivalence theorem is proved, which guarantees that both specifications specify the same behavior. The imperative LTL-specification is translated into imperative program code according to the presented template. The declarative LTL-specification, which is subject to verification, and the control program built on it are guaranteed to specify the same behavior in the form of a corresponding transition system. Thus, during verification, a model is used that is adequate to the real behavior of the control program.

Quantitative Robustness for Signal Temporal Logic With Time-Freeze Quantifiers

Signal Temporal Logic (STL) is a variant of Metric Temporal Logic (MTL) which can express intricate temporal requirements over signals, and has found wide adoption for expressing requirements over complex control systems models. A key factor in the success of STL has been that of quantitative robustness, and the development of efficient algorithms for computing the robustness values over traces. The real-valued quantitative robustness of a signal with respect to an STL property quantifies the degree of satisfaction or violation of the property by the signal. In this work we introduce a notion of robustness for a more expressive logic, Timed Signal Temporal Logic (TSTL), which can be seen as Timed Propositional Temporal Logic (TPTL) with predicates defined over real-valued signals. This logic can express many natural engineering requirements that STL cannot. We also develop algorithms for computing this robustness value over traces in the pointwise semantics. While the robustness computation for general TSTL formulae is PSPACEhard due to the PSPACE-hardness of the monitoring problem for TPTL, for the special case of one variable TSTL, a fragment of TSTL which is still more expressive than STL, we develop an optimized algorithm which computes robustness in time linear in the length of the trace. Finally, we experimentally validate the tractability of our algorithms with our prototype tool in Matlab.

Realizability problem for constraint LTL

Constraint linear-time temporal logic (CLTL) is an extension of LTL that is interpreted on sequences of valuations of variables over an infinite domain. Atomic formulas can constrain valuations over a range of positions along a sequence, with the range being bounded by a parameter depending on the formula. The satisfiability and model checking problems for CLTL have been studied before. We consider the realizability problem for CLTL. The set of variables is partitioned into two parts, with each part controlled by a player. Players take turns to choose valuations for their variables, generating a sequence of valuations. The winning condition is specified by a CLTL formula — the first player wins if the sequence of valuations satisfies the specified formula. We prove that checking whether the first player has a winning strategy in the realizability game for a given CLTL formula is undecidable in general and identify decidable fragments.

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.

Weighted Linear Dynamic Logic

We introduce a weighted linear dynamic logic (weighted LDL for short) and show the expressive equivalence of its formulas to weighted rational expressions. This adds a new characterization for recognizable series to the fundamental Schützenberger theorem. Surprisingly, the equivalence does not require any restriction to our weighted LDL. Our results hold over arbitrary (resp. totally complete) semirings for finite (resp. infinite) words. As a consequence, the equivalence problem for weighted LDL formulas over fields is decidable in doubly exponential time. In contrast to classical logics, we show that our weighted LDL is expressively incomparable to weighted LTL for finite words. We determine a fragment of the weighted LTL such that series over finite and infinite words definable by LTL formulas in this fragment are definable also by weighted LDL formulas. This is an extended version of [17].

Change‐aware model checking for evolving concurrent programs based on Program Dependence Net

SummaryConcurrent software needs to be maintained over time, and the differences between continuous versions tend to be localized. The expense that simply reapplying standard model checking techniques to the new version as they evolve may be infeasible. The existing methods reuse partial state‐space to reduce the scope. However, it is obviously costly to analyze on the explosive interleaving space of the evolving concurrent programs. The conservative change‐impact analysis methods without considering the specific property and leveraging the verified result from the prior version often results in exploring redundant state‐space irrelevant to this property. Moreover, the impact of the deleted elements needs to be analyzed on old version, and their impact needs to be mapped to new version, bringing some dispensable costs. In this paper, we propose a change‐aware model checking method based on program dependence net (PDNet) for linear temporal logic (LTL). We first propose an incremental modeling method to construct a PDNet of new version by modification rules. Then, we propose a reuse checking algorithm to judge whether the verified result can be reused based on the PDNet slice. Finally, we implement change‐aware model checking tool (DAMER) and validate the advantages of our methods.

Measuring rule-based LTLf process specifications: A probabilistic data-driven approach

Declarative process specifications define the behavior of processes by means of rules based on Linear Temporal Logic on Finite Traces LTLf. In a mining context, these specifications are inferred from, and checked on, multi-sets of runs recorded by information systems (namely, event logs). To this end, being able to gauge the degree to which process data comply with a specification is key. However, existing mining and verification techniques analyze the rules in isolation, thereby disregarding their interplay. In this paper, we introduce a framework to devise probabilistic measures for declarative process specifications. Thereupon, we propose a technique that measures the degree of satisfaction of specifications over event logs. To assess our approach, we conduct an evaluation with real-world data, evidencing its applicability for diverse process mining tasks, including discovery, checking, and drift detection.

Safety Constraint-Guided Reinforcement Learning with Linear Temporal Logic

In the context of reinforcement learning (RL), ensuring both safety and performance is crucial, especially in real-world scenarios where mistakes can lead to severe consequences. This study aims to address this challenge by integrating temporal logic constraints into RL algorithms, thereby providing a formal mechanism for safety verification. We employ a combination of theoretical and empirical methods, including the use of temporal logic for formal verification and extensive simulations to validate our approach. Our results demonstrate that the proposed method not only maintains high levels of safety but also achieves comparable performance to traditional RL algorithms. Importantly, our approach fills a critical gap in the existing literature by offering a solution that is both mathematically rigorous and empirically validated. The study concludes that the integration of temporal logic into RL offers a promising avenue for developing algorithms that are both safe and efficient. This work lays the foundation for future research aimed at generalizing this approach to various complex systems and applications.

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.

Model Checking of Possibilistic Linear-Time Properties Based on Generalized Possibilistic Decision Processes

Model checking possibilistic linear-time properties was investigated by Li (2017). However, nondeterminism of the system is absent in previous studies. Therefore, in order to permit both possibilistic and nondeterministic choices, we use the generalized possibilistic decision process (GPDP) as a model of the system. First, the definition of GPDP describing the behavior of nondeterministic system is given in detail, the resolution of nondeterminism is performed by using the notion of schedulers, and the semantics of generalized possibilistic linear-temporal logic (GPoLTL) with schedulers are defined. Second, we study possibilistic model checking of some fuzzy linear-time properties under GPDP. Since there are many (infinite) schedulers satisfying a certain linear timing property in a given state of a GPDP, it is particularly critical to study the optimal strategy and its corresponding possible measure, which is called extremal possibility model checking. For some special fuzzy linear-time properties, such as constrained reachability, step-bounded constrained reachability, reachability, always reachability, repeated reachability, persistence reachability, we present complete solution to the optimal (including maximum and minimum cases) possibilistic model checking of the above reachability using the fixpoint techniques. We also introduce fuzzy !-regular properties in GPDP and show that their model checking can be simplified by repeated reachability. The algorithms for model checking are also provided. Additionally, an example is presented to illustrate the methods described in the paper.

Towards a model of human-cyber–physical automata and a synthesis framework for control policies

Advances in research and increasing applications of Cyber–Physical Systems (CPSs) show the need to consider factors of humans in the loop. This has led to the growing research focus on Human-Cyber–Physical Systems (HCPSs). In general, humans in an HCPS interact with both the cyber and physical systems, as well as among the humans themselves. For a better understanding, correct design, development, operation, and maintenance of HCPSs, a computational theory based on a computational model is required. This paper presents our initial work towards a model of human-cyber–physical automata (HCPA). We consider an HCPS as a combination of a human-physical system (HPS) and a CPS in which the control switches between the humans and the machines. We define an HCPA by connecting the automaton of the HPS and the automaton of the CPS through a switch control automaton. The switch control automaton makes switching decision in some critical states shared by the HPS and the CPS. Our theorem shows that the control switching between the HPS and the CPS increases the probability of satisfying a given property. We model the behaviour of a human in specified applications or even in carry out specific tasks, instead of general human intelligence. Therefore, a human can make mistakes to decision making and thus it is a probabilistic automaton with learning ability. The switching between the human and the machine is modelled by an oracle. The oracle learns about the human behaviour, the machine behaviour, as well as the environment to make the control decisions. To generate the control policies of the human and the oracle, we propose a synthesis framework to maximize the probability of the satisfaction of a property specified in Linear Temporal Logic (LTL) by the HCPA. We present a prototype implementation of the framework by extending the model-free reinforcement learning (RL) algorithm and model-free deep-RL algorithm, and our experiment shows that our synthesis framework is effective in obtaining switch policies.

Temporal logics with language parameters

We develop a generic framework to extend the logics LTL, CTL+ and CTL⁎ by automata-based connectives from formal language classes and analyse this framework with regard to regular languages, visibly pushdown languages, deterministic and non-deterministic context-free languages. More precisely, we consider how the use of different automata classes changes the expressive power of the logics and provide algorithms for the satisfiability and model checking problems induced by the use of different classes of automata. For the model checking problem, we treat not only finite Kripke transition systems, but also visibly pushdown systems and pushdown systems. We provide completeness or undecidability results in all cases and show that the extensions we consider can formulate properties not expressible in classical temporal logics or regular extensions thereof.

Mapping natural language procedures descriptions to linear temporal logic templates: an application in the surgical robotic domain

Natural language annotations and manuals can provide useful procedural information and relations for the highly specialized scenario of autonomous robotic task planning. In this paper, we propose and publicly release AUTOMATE, a pipeline for automatic task knowledge extraction from expert-written domain texts. AUTOMATE integrates semantic sentence classification, semantic role labeling, and identification of procedural connectors, in order to extract templates of Linear Temporal Logic (LTL) relations that can be directly implemented in any sufficiently expressive logic programming formalism for autonomous reasoning, assuming some low-level commonsense and domain-independent knowledge is available. This is the first work that bridges natural language descriptions of complex LTL relations and the automation of full robotic tasks. Unlike most recent similar works that assume strict language constraints in substantially simplified domains, we test our pipeline on texts that reflect the expressiveness of natural language used in available textbooks and manuals. In fact, we test AUTOMATE in the surgical robotic scenario, defining realistic language constraints based on a publicly available dataset. In the context of two benchmark training tasks with texts constrained as above, we show that automatically extracted LTL templates, after translation to a suitable logic programming paradigm, achieve comparable planning success in reduced time, with respect to logic programs written by expert programmers.

Preference-Based Multi-Robot Planning for Nuclear Power Plant Online Monitoring and Diagnostics

Current preventative maintenance paradigms in nuclear power plants carry several costly risks and challenges associated with component downtime and the need for human data collection. Preventative maintenance may be enabled by an online monitoring system that accurately assesses component condition and identifies potential faults. We present an approach for autonomous online monitoring and multiagent planning for robotic data collection. Under the occurrence of a fault, we utilize a machine learning model to form an initial guess of its nature, which we then refine by selectively measuring certain variables to gain additional information via a situation-aware variable selection model. To generate a multi-robot plan to conduct these measurements, we develop a preference-based planning framework within a linear temporal logic–based planning approach that prioritizes collecting data from the most important features. Finally, we demonstrate our approach on a case study using a simulated nuclear power plant circulating water system, showing fault diagnostic performance as well as simulated robot data collection.

Fairness, assumptions, and guarantees for extended bounded response LTL+P synthesis

Realizability and reactive synthesis from temporal logics are fundamental problems in formal verification. The complexity of these problems for linear temporal logic with past (LTL+P ) led to the identification of fragments with lower complexities and simpler algorithms. Recently, the logic of extended bounded response LTL+P (LTLEBR+P\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extsf {LTL} _{\ extsf {EBR} }\ extsf {{+}P} $$\\end{document} for short) has been introduced. It allows one to express safety languages definable in LTL+P and it is provided with an efficient, fully symbolic algorithm for reactive synthesis. This paper features four related contributions. First, we introduce GR-EBR , an extension of LTLEBR+P\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extsf {LTL} _{\ extsf {EBR} }\ extsf {{+}P} $$\\end{document} with fairness conditions, assumptions, and guarantees that, on the one hand, allows one to express properties beyond the safety fragment and, on the other, it retains the efficiency of LTLEBR+P\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extsf {LTL} _{\ extsf {EBR} }\ extsf {{+}P} $$\\end{document} in practice. Second, we the expressiveness of GR-EBR starting from the expressiveness of its fragments. In particular, we prove that: (1) LTLEBR+P\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extsf {LTL} _{\ extsf {EBR} }\ extsf {{+}P} $$\\end{document} is expressively complete with respect to the safety fragment of LTL+P , (2) the removal of past operators from LTLEBR+P\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extsf {LTL} _{\ extsf {EBR} }\ extsf {{+}P} $$\\end{document} results into a loss of expressive power, and (3) GR-EBR is expressively equivalent to the logic GR(1) of Bloem et al. Third, we provide a fully symbolic algorithm for the realizability problem from GR-EBR specifications, that reduces it to a number of safety subproblems. Fourth, to ensure soundness and completeness of the algorithm, we propose and exploit a general framework for safety reductions in the context of realizability of (fragments of) LTL+P . The experimental evaluation shows promising results.