Linear Temporal Logic Formulas Research Articles

Learning Interpretable Models Expressed in Linear Temporal Logic

We examine the problem of learning models that characterize the high-level behavior of a system based on observation traces. Our aim is to develop models that are human interpretable. To this end, we introduce the problem of learning a Linear Temporal Logic (LTL) formula that parsimoniously captures a given set of positive and negative example traces. Our approach to learning LTL exploits a symbolic state representation, searching through a space of labeled skeleton formulae to construct an alternating automaton that models observed behavior, from which the LTL can be read off. Construction of interpretable behavior models is central to a diversity of applications related to planning and plan recognition. We showcase the relevance and significance of our work in the context of behavior description and discrimination: i) active learning of a human-interpretable behavior model that describes observed examples obtained by interaction with an oracle; ii) passive learning of a classifier that discriminates individual agents, based on the human-interpretable signature way in which they perform particular tasks. Experiments demonstrate the effectiveness of our symbolic model learning approach in providing human-interpretable models and classifiers from reduced example sets.

Adaptive Teaching of Temporal Logic Formulas to Preference-based Learners

Machine teaching is an algorithmic framework for teaching a target hypothesis via a sequence of examples or demonstrations. We investigate machine teaching for temporal logic formulas—a novel and expressive hypothesis class amenable to time-related task specifications. In the context of teaching temporal logic formulas, an exhaustive search even for a myopic solution takes exponential time (with respect to the time span of the task). We propose an efficient approach for teaching parametric linear temporal logic formulas. Concretely, we derive a necessary condition for the minimal time length of a demonstration to eliminate a set of hypotheses. Utilizing this condition, we propose an efficient myopic teaching algorithm by solving a sequence of integer programming problems. We further show that, under two notions of teaching complexity, the proposed algorithm has near-optimal performance. We evaluate our algorithm extensively under different classes of learners (i.e., learners with different preferences over hypotheses) and interaction protocols (e.g., non-adaptive and adaptive). Our results demonstrate the effectiveness of the proposed algorithm in teaching temporal logic formulas; in particular, we show that there are significant gains of teaching efficacy when the teacher adapts to feedback of the learner, or adapts to a (non-myopic) oracle.

RT-MOBS: A compositional observer semantics of time Petri net for real-time property specification language based on μ-calculus

• Defines a new verification method supporting real-time property specification. • Focuses on the composability of expressiveness and the performance of verification. • Achieves several orders of magnitude improvement in the verification cost. We define a new verification method, called RT-MOBS, for checking real-time requirements based on Time Petri nets (TPN). Our approach supports requirements specified using a very expressive pattern language, the Property Specification Language (PSL) of Autili, Grusnke et al., and relies on marking observers' verification. RT-MOBS has many distinctive features, such as a focus on performances, a compositional method for deriving the observer and the target property directly from the structure of the specification pattern, and the ability to deal with the whole real-time fragment of PSL. We demonstrate the effectiveness of our approach from three industrial use cases: a mobile ad-hoc network system; the model of a flight management system, which is realistic with respect to the industry use during the architecture evaluation phase; and a model of an order to cash smart contract. Our experimental results show that we can achieve performances that are several orders of magnitude better than with methods based on an interpretation of patterns into Linear Temporal Logic (LTL) formulas.

BeBoSy: Behavior Examples Meet Bounded Synthesis

Bounded synthesis generates a state machine satisfying given temporal specification in linear temporal logic (LTL) while bounding the number of states of the target solution. Bounded synthesis methods currently do not support finite-length behavior examples, which are often helpful when formulating the specification of the desired state machine. In this work we show how to incorporate behavior examples into bounded synthesis methods and exemplify the approach by presenting BeBoSy ( Behavior Examples meet BoSy ), an extension of the bounded synthesis tool BoSy based on reductions to Boolean satisfiability (SAT) and Quantified Boolean formula satisfiability (QSAT) problems. The proposed approach is to augment the encodings used in BoSy with additional constraints that ensure the compliance of the generated state machine with the given behavior examples. Experiments with openly available data from the annual reactive synthesis competition SYNTCOMP augmented with random behavior examples show that the efficiency of the proposed approach is superior to both BoSy (with naive representation of behavior examples with LTL formulas) and a state-of-the-art counterexample-guided approach EFSM-tools.

ROCS 2.0: An Integrated Temporal Logic Control Synthesis Tool for Nonlinear Dynamical Systems

Abstract This paper introduces ROCS 2.0, a control synthesis tool for nonlinear systems with control objectives given as temporal logic formulas. In addition to the basic invariance, reachability, Buchi, and co-Buchi specifications that can be handled in the previous version of ROCS, ROCS 2.0 provides a major upgrade to support the general class of linear temporal logic formulas that can be translated to deterministic Buchi automata. Moreover, ROCS 2.0 not only maintains and accelerates the kernel of its previous version—the engine based on the specification-guided control method—by more efficient implementation, but also integrates a second engine that implements the abstraction-based control method, which is optimized to gain time and memory efficiency. Such a feature gives the user the freedom to choose the control synthesis method that is more suitable for a specific control problem.

Receding Horizon Control-Based Motion Planning With Partially Infeasible LTL Constraints

This work considers online optimal motion planning of an autonomous agent subject to linear temporal logic (LTL) constraints. The environment is dynamic in the sense of containing mobile obstacles and time-varying areas of interest (i.e., time-varying reward and workspace properties) to be visited by the agent. Since user-specified tasks may not be fully realized (i.e., partially infeasible), this work considers hard and soft LTL constraints, where hard constraints enforce safety requirement (e.g. avoid obstacles) while soft constraints represent tasks that can be relaxed to not strictly follow user specifications. The motion planning of the agent is to generate policies, in decreasing order of priority, to 1) formally guarantee the satisfaction of safety constraints; 2) mostly satisfy soft constraints (i.e., minimize the violation cost if desired tasks are partially infeasible); and 3) optimize the objective of rewards collection (i.e., visiting dynamic areas of more interests). To achieve these objectives, a relaxed product automaton, which allows the agent to not strictly follow the desired LTL constraints, is constructed. A utility function is developed to quantify the differences between the revised and the desired motion plan, and the accumulated rewards are designed to bias the motion plan towards those areas of more interests. Receding horizon control is synthesized with an LTL formula to maximize the accumulated utilities over a finite horizon, while ensuring that safety constraints are fully satisfied and soft constraints are mostly satisfied. Simulation and experiment results are provided to demonstrate the effectiveness of the developed motion strategy.

Satisfaction of Linear Temporal Logic Specifications Through Recurrence Tools for Hybrid Systems

In this work we formulate the problem of satisfying a linear temporal logic formula on a linear plant with output feedback, through a recent hybrid systems formalism. We relate this problem to the notion of recurrence introduced for the considered formalism, and we then extend Lyapunov-like conditions for recurrence of an open, unbounded set. One of the proposed relaxed conditions allows certifying recurrence of a suitable set, and this guarantees that the high-level evolution of the plant satisfies the formula, without relying on discretizations of the plant. Simulations illustrate the proposed approach.

A Weakness Measure for GR(1) Formulae

When dealing with unrealizable specifications in reactivesynthesis, finding the weakest environment assumptions that ensurerealizability is often considered a desirable property. However,little effort has been dedicated to defining or evaluating thenotion of weakness of assumptions formally. The question of whetherone assumption is weaker than another is commonly interpreted byconsidering the implication relationship between the two or,equivalently, their language inclusion. This interpretation fails toprovide any insight into the weakness of the assumptions whenimplication (or language inclusion) does not hold. To ourknowledge, the only measure that is capable of comparing twoformulae in this case is entropy, but even it cannot distinguish theweakness of assumptions expressed as fairness properties. In thispaper, we propose a refined measure of weakness based on combiningentropy with Hausdorff dimension, a concept that captures the notionof size of the omega -language satisfying a linear temporal logicformula. We focus on a special subset of linear temporal logicformulae which is of particular interest in reactive synthesis,called GR(1). We identify the conditions under which this measureis guaranteed to distinguish between weaker and stronger GR(1)formulae, and propose a refined measure to cover cases when twoformulae are strictly ordered by implication but have the sameentropy and Hausdorff dimension. We prove the consistency betweenour weakness measure and logical implication, that is, if oneformula implies another, the latter is weaker than the formeraccording to our measure. We evaluate our proposed weaknessmeasure in two contexts. The first is in computing GR(1) assumptionrefinements where our weakness measure is used as a heuristic todrive the refinement search towards weaker solutions. The second isin the context of quantitative model checking where it is used tomeasure the size of the language of a model violating a lineartemporal logic formula.

A Unified Translation of Linear Temporal Logic to ω-Automata

We present a unified translation of linear temporal logic (LTL) formulas into deterministic Rabin automata (DRA), limit-deterministic Büchi automata (LDBA), and nondeterministic Büchi automata (NBA). The translations yield automata of asymptotically optimal size (double or single exponential, respectively). All three translations are derived from one single Master Theorem of purely logical nature. The Master Theorem decomposes the language of a formula into a positive Boolean combination of languages that can be translated into ω-automata by elementary means. In particular, Safra’s, ranking, and breakpoint constructions used in other translations are not needed. We further give evidence that this theoretical clean and compositional approach does not lead to large automata per se and in fact in the case of DRAs yields significantly smaller automata compared to the previously known approach using determinisation of NBAs.

A Theory of Distributed Markov Chains

We present the theory of distributed Markov chains (DMCs). A DMC consists of a collection of communicating probabilistic agents in which the synchronizations determine the probability distribution for the next moves of the participating agents. The key feature of a DMC is that the synchronizations are deterministic, in the sense that any two simultaneously enabled synchronizations involve disjoint sets of agents. Using our theory of DMCs we show how one can analyze the behavior using the interleaved semantics of the model. A key point is, the transition system which defines the interleaved semantics is—except in degenerate cases—not a Markov chain. Hence one must develop new techniques to analyze these behaviors exhibiting both concurrency and stochasticity. After establishing the core theory we develop a statistical model checking procedure which verifies the dynamical properties of the trajectories generated by the the model. The specifications consist of Boolean combinations of component-wise bounded linear time temporal logic formulas. We also provide a probabilistic Petri net representation of DMCs and use it to derive a probabilistic event structure semantics.

SLA-Driven modeling and verifying cloud systems: A Bigraphical reactive systems-based approach

We propose a formal approach based on Bigraphical Reactive Systems (BRSs) and model checking techniques for modeling and verifying the interaction behaviours of SLA-based cloud computing systems. In the first phase of this approach, we address the modeling of the static structure and the dynamic behavior of cloud systems using BRSs. We show how bigraphs enable the description of the different cloud entities, including cloud actors, cloud services, Service Level Agreements (SLAs), the diversity of their properties, and the complex interactions and dependencies among them. Furthermore, we propose a four-stages SLA lifecycle, and define a set of bigraphical reaction rules to abstract these stages and model the dynamic nature of the cloud. The second phase of this approach verifies that the behavior of services and cloud actors will cope with the agreed SLA terms during the lifecycle of the SLA. We map the proposed bigraphical models into SMV descriptions. Then, we express the interaction behaviors as a set of liveness and safety properties using Linear Temporal Logic (LTL) and Computation Tree Logic (CTL) formulas, and we use the NuSMV model checker to verify them. Finally, we define a case study on which we illustrate the application of our proposed approach.

On How Bit-Vector Logic Can Help Verify LTL-Based Specifications

This paper studies how bit-vector logic (bv logic) can help improve the efficiency of verifying specifications expressed in Linear Temporal Logic (LTL). First, it exploits the notion of Bounded Satisfiability Checking to propose an improved encoding of LTL formulae into formulae of bv logic, which can be formally verified by means of Satisfiability Modulo Theories (SMT) solvers. To assess the gain in efficiency, we compare the proposed encoding, implemented in our tool <inline-formula><tex-math notation="LaTeX">$\mathbb {Z}$</tex-math></inline-formula> ot, against three well-known encodings available in the literature: the classic bounded encoding and the optimized, incremental one, as implemented in both NuSMV and nuXmv, and the encoding optimized for metric temporal logic, which was the “standard” implementation provided by <inline-formula><tex-math notation="LaTeX">$\mathbb {Z}$</tex-math></inline-formula> ot. We also compared the newly proposed solution against five additional efficient algorithms proposed by nuXmv, which is the state-of-the-art tool for verifying LTL specifications. The experiments show that the new encoding provides significant benefits with respect to existing tools. Since the first set of experiments only used Z3 as SMT solver, we also wanted to assess whether the benefits were induced by the specific solver or were more general. This is why we also embedded different SMT solvers in <inline-formula><tex-math notation="LaTeX">$\mathbb {Z}$</tex-math></inline-formula> ot. Besides Z3, we also carried out experiments with CVC4, Mathsat, Yices2, and Boolector, and compared the results against the first and second best solutions provided by either NuSMV or nuXmv. Obtained results witness that the benefits of the bv logic encoding are independent of the specific solver. Bv logic-based solutions are better than traditional ones with only a few exceptions. It is also true that there is no particular SMT solver that outperformed the others. Boolector is often the best as for memory usage, while Yices2 and Z3 are often the fastest ones.

Modeling and Verification Methods for Application Design in Heterogeneous Architectures

A methodology of application design for massive parallelism systems on the example of GPGPU systems focused on the algorithmic design stage is proposed. Two stages of design are considered: creation of a formal specification and its research and verification. For the first stage, the use of mathematical apparatus of the system of algorithmic algebras/modified system of algorithmic algebras and transition systems is proposed. For the second stage, the use of network and automatic models is analyzed, and the advantages of each model are given. In particular, the computational model in NVIDIA CUDA using Petri nets, as well as the linear-temporal logic formulas and automatic model, was studied.

Automated temporal equilibrium analysis: Verification and synthesis of multi-player games

In the context of multi-agent systems, the rational verification problem is concerned with checking which temporal logic properties will hold in a system when its constituent agents are assumed to behave rationally and strategically in pursuit of individual objectives. Typically, those objectives are expressed as temporal logic formulae which the relevant agent desires to see satisfied. Unfortunately, rational verification is computationally complex, and requires specialised techniques in order to obtain practically useable implementations. In this paper, we present such a technique. This technique relies on a reduction of the rational verification problem to the solution of a collection of parity games. Our approach has been implemented in the Equilibrium Verification Environment (EVE) system. The EVE system takes as input a model of a concurrent/multi-agent system represented using the Simple Reactive Modules Language (SRML), where agent goals are represented as Linear Temporal Logic (▪) formulae, together with a claim about the equilibrium behaviour of the system, also expressed as an ▪ formula. EVE can then check whether the ▪ claim holds on some (or every) computation of the system that could arise through agents choosing Nash equilibrium strategies; it can also check whether a system has a Nash equilibrium, and synthesise individual strategies for players in the multi-player game. After presenting our basic framework, we describe our new technique and prove its correctness. We then describe our implementation in the EVE system, and present experimental results which show that EVE performs favourably in comparison to other existing tools that support rational verification.

Reactive sampling-based path planning with temporal logic specifications

We develop a sampling-based motion planning algorithm that combines long-term temporal logic goals with short-term reactive requirements. The mission specification has two parts: (1) a global specification given as a linear temporal logic (LTL) formula over a set of static service requests that occur at the regions of a known environment, and (2) a local specification that requires servicing a set of dynamic requests that can be sensed locally during the execution. The proposed computational framework consists of two main ingredients: (a) an off-line sampling-based algorithm for the construction of a global transition system that contains a path satisfying the LTL formula; and (b) an on-line sampling-based algorithm to generate paths that service the local requests, while making sure that the satisfaction of the global specification is not affected. The off-line algorithm has four main features. First, it is incremental, in the sense that the procedure for finding a satisfying path at each iteration scales only with the number of new samples generated at that iteration. Second, the underlying graph is sparse, which implies low complexity for the overall method. Third, it is probabilistically complete. Fourth, under some mild assumptions, it has the best possible complexity bound. The on-line algorithm leverages ideas from LTL monitoring and potential functions to ensure progress towards the satisfaction of the global specification while servicing locally sensed requests. Examples and experimental trials illustrating the usefulness and the performance of the framework are included.

On the complexity of linear temporal logic with team semantics

A specification given as a formula in linear temporal logic (LTL) defines a system by its set of traces. However, certain features such as information flow security constraints are rather modeled as so-called hyperproperties, which are sets of sets of traces. One logical approach to this is team logic, which is a logical framework for the specification of dependence and independence of information. LTL with team semantics has recently been discovered as a logic for hyperproperties.We study the complexity theoretic aspects of LTL with so-called synchronous team semantics and Boolean negation, and prove that both its model checking and satisfiability problems are highly undecidable, and equivalent to the decision problem of third-order arithmetic. Furthermore, we prove that this complexity already appears at small temporal depth and with only the “future” modality F. Finally, we also introduce a team-semantical generalization of stutter-invariance.

Point-Based Methods for Model Checking in Partially Observable Markov Decision Processes

Autonomous systems are often required to operate in partially observable environments. They must reliably execute a specified objective even with incomplete information about the state of the environment. We propose a methodology to synthesize policies that satisfy a linear temporal logic formula in a partially observable Markov decision process (POMDP). By formulating a planning problem, we show how to use point-based value iteration methods to efficiently approximate the maximum probability of satisfying a desired logical formula and compute the associated belief state policy. We demonstrate that our method scales to large POMDP domains and provides strong bounds on the performance of the resulting policy.

Planning With Uncertain Specifications (PUnS)

Reward engineering is crucial to high performance in reinforcement learning systems. Prior research into reward design has largely focused on Markovian functions representing the reward. While there has been research into expressing non-Markov rewards as linear temporal logic (LTL) formulas, this has focused on task specifications directly defined by the user. However, in many real-world applications, task specifications are ambiguous, and can only be expressed as a belief over LTL formulas. In this paper, we introduce planning with uncertain specifications (PUnS), a novel formulation that addresses the challenge posed by non-Markovian specifications expressed as beliefs over LTL formulas. We present four criteria that capture the semantics of satisfying a belief over specifications for different applications, and analyze the qualitative implications of these criteria within a synthetic domain. We demonstrate the existence of an equivalent Markov decision process (MDP) for any instance of PUnS. Finally, we demonstrate our approach on the real-world task of setting a dinner table automatically with a robot that inferred task specifications from human demonstrations.

Reinforcement Learning of Control Policy for Linear Temporal Logic Specifications Using Limit-Deterministic Generalized Büchi Automata

This letter proposes a novel reinforcement learning method for the synthesis of a control policy satisfying a control specification described by a linear temporal logic formula. We assume that the controlled system is modeled by a Markov decision process (MDP). We convert the specification to a limit-deterministic generalized Büchi automaton (LDGBA) with several accepting sets that accepts all infinite sequences satisfying the formula. The LDGBA is augmented so that it explicitly records the previous visits to accepting sets. We take a product of the augmented LDGBA and the MDP, based on which we define a reward function. The agent gets rewards whenever state transitions are in an accepting set that has not been visited for a certain number of steps. Consequently, sparsity of rewards is relaxed and optimal circulations among the accepting sets are learned. We show that the proposed method can learn an optimal policy when the discount factor is sufficiently close to one.

Reachability-based Human-in-the-Loop Control with Uncertain Specifications

Abstract We propose a shared autonomy approach for implementing human operator decisions onto an automated system during multi-objective missions, while guaranteeing safety and mission completion. A mission is specified as a set of linear temporal logic (LTL) formulae. Then, using a novel correspondence between LTL and reachability analysis, we synthesize a set of controllers for assisting the human operator to complete the mission, while guaranteeing that the system maintains specified spatial and temporal properties. We assume the human operator’s exact preference of how to complete the mission is unknown. Instead, we use a data-driven approach to infer and update the automated system’s internal belief of which specified objective the human intends to complete. If, while the human is operating the system, she provides inputs that violate any of the invariances prescribed by the LTL formula, our verified controller will use its internal belief of the human operator’s intended objective to guide the operator back on track. Moreover, we show that as long as the specifications are initially feasible, our controller will stay feasible and can guide the human to complete the mission despite some unexpected human errors. We illustrate our approach with a simple, but practical, experimental setup where a remote operator is parking a vehicle in a parking lot with multiple parking options. In these experiments, we show that our approach is able to infer the human operator’s preference over parking spots in real-time and guarantee that the human will park in the spot safely.