Fragment Of Linear Temporal Logic Research Articles

A first-order logic characterization of safety and co-safety languages

Linear Temporal Logic (LTL) is one of the most popular temporal logics, that comes into play in a variety of branches of computer science. Among the various reasons of its widespread use there are its strong foundational properties: LTL is equivalent to counter-free omega-automata, to star-free omega-regular expressions, and (by Kamp's theorem) to the First-Order Theory of Linear Orders (FO-TLO). Safety and co-safety languages, where a finite prefix suffices to establish whether a word does not belong or belongs to the language, respectively, play a crucial role in lowering the complexity of problems like model checking and reactive synthesis for LTL. SafetyLTL (resp., coSafetyLTL) is a fragment of LTL where only universal (resp., existential) temporal modalities are allowed, that recognises safety (resp., co-safety) languages only. The main contribution of this paper is the introduction of a fragment of FO-TLO, called SafetyFO, and of its dual coSafetyFO, which are expressively complete with respect to the LTL-definable safety and co-safety languages. We prove that they exactly characterize SafetyLTL and coSafetyLTL, respectively, a result that joins Kamp's theorem, and provides a clearer view of the characterization of (fragments of) LTL in terms of first-order languages. In addition, it gives a direct, compact, and self-contained proof that any safety language definable in LTL is definable in SafetyLTL as well. As a by-product, we obtain some interesting results on the expressive power of the weak tomorrow operator of SafetyLTL, interpreted over finite and infinite words. Moreover, we prove that, when interpreted over finite words, SafetyLTL (resp. coSafetyLTL) devoid of the tomorrow (resp., weak tomorrow) operator captures the safety (resp., co-safety) fragment of LTL over finite words.

Perception-Based Temporal Logic Planning in Uncertain Semantic Maps

In this article, we address a multi-robot planning problem in environments with partially unknown semantics. The environment is assumed to have a known geometric structure (e.g., walls) and to be occupied by static labeled landmarks with uncertain positions and classes. This modeling approach gives rise to an uncertain semantic map generated by semantic simultaneous localization and mapping algorithms. Our goal is to design control policies for robots equipped with noisy perception systems so that they can accomplish collaborative tasks captured by global temporal logic specifications. To specify missions that account for environmental and perceptual uncertainty, we employ a fragment of linear temporal logic (LTL), called co-safe LTL, defined over perception-based atomic predicates modeling probabilistic satisfaction requirements. The perception-based LTL planning problem gives rise to an optimal control problem, solved by a novel sampling-based algorithm, that generates open-loop control policies that are updated online to adapt to a continuously learned semantic map. We provide extensive experiments to demonstrate the efficiency of the proposed planning architecture.

Expressiveness of Extended Bounded Response LTL

Extended Bounded Response LTL with Past (LTLEBR+P) is a safety fragment of Linear Temporal Logic with Past (LTL+P) that has been recently introduced in the context of reactive synthesis. The strength of LTLEBR+P is a fully symbolic compilation of formulas into symbolic deterministic automata. Its syntax is organized in four levels. The first three levels feature (a particular combination of) future temporal modalities, the last one admits only past temporal operators. At the base of such a structuring there are algorithmic motivations: each level corresponds to a step of the algorithm for the automaton construction. The complex syntax of LTLEBR+P made it difficult to precisely characterize its expressive power, and to compare it with other LTL+P safety fragments. In this paper, we first prove that LTLEBR+P is expressively complete with respect to the safety fragment of LTL+P, that is, any safety language definable in LTL+P can be formalized in LTLEBR+P, and vice versa. From this, it follows that LTLEBR+P and Safety-LTL are expressively equivalent. Then, we show that past modalities play an essential role in LTLEBR+P: we prove that the future fragment of LTLEBR+P is strictly less expressive than full LTLEBR+P.

Interactive synthesis of temporal specifications from examples and natural language

Motivated by applications in robotics, we consider the task of synthesizing linear temporal logic (LTL) specifications based on examples and natural language descriptions. While LTL is a flexible, expressive, and unambiguous language to describe robotic tasks, it is often challenging for non-expert users. In this paper, we present an interactive method for synthesizing LTL specifications from a single example trace and a natural language description. The interaction is limited to showing a small number of behavioral examples to the user who decides whether or not they exhibit the original intent. Our approach generates candidate LTL specifications and distinguishing examples using an encoding into optimization modulo theories problems. Additionally, we use a grammar extension mechanism and a semantic parser to generalize synthesized specifications to parametric task descriptions for subsequent use. Our implementation in the tool LtlTalk starts with a domain-specific language that maps to a fragment of LTL and expands it through example-based user interactions, thus enabling natural language-like robot programming, while maintaining the expressive power and precision of a formal language. Our experiments show that the synthesis method is precise, quick, and asks only a few questions to the users, and we demonstrate in a case study how LtlTalk generalizes from the synthesized tasks to other, yet unseen, tasks.

Control of Mobile Robots Using Barrier Functions Under Temporal Logic Specifications

In this article, we propose a framework for the control of mobile robots subject to temporal logic specifications using barrier functions. Complex task specifications can be conveniently encoded using linear temporal logic. In particular, we consider a fragment of linear temporal logic, which encompasses a large class of motion planning specifications for a robotic system. Control barrier functions have recently emerged as a convenient tool to guarantee reachability and safety for a system. In addition, they can be encoded as affine constraints in a quadratic program. In this article, a fully automatic framework that translates a user defined specification in temporal logic to a sequence of barrier function based quadratic programs is presented. In addition, with the aim of alleviating infeasibility scenarios, we propose methods for composition of barrier functions as well as a prioritization-based control method to guarantee feasibility of the controller. We prove that the resulting system trajectory synthesized by the proposed controller satisfies the given specification. Robotic simulation and experimental results are provided in addition to the theoretical framework.

On-Line Permissive Supervisory Control of Discrete Event Systems for scLTL Specifications

We propose an on-line supervisory control scheme for discrete event systems (DESs), where a control specification is described by a fragment of linear temporal logic. On the product automaton of the DES and an acceptor for the specification, we define a ranking function that returns the minimum number of steps required to reach an accepting state from each state. In addition, we introduce a permissiveness function that indicates a time-varying permissive level. At each step during the on-line control scheme, the supervisor refers to the permissiveness function as well as the ranking function in order to guarantee the control specification while handling the tradeoff between its permissiveness and acceptance of the specification. The proposed scheme is demonstrated in a surveillance problem for a mobile robot.

On-Line Synthesis of Permissive Supervisors for Partially Observed Discrete Event Systems under scLTL Constraints

Abstract We consider a supervisory control problem of a discrete event system (DES) under partial observation, where a control specification is given by a fragment of linear temporal logic We design an on-line supervisor that dynamically computes its control action with the complete information of the product automaton of the DES and an acceptor for the specification. The concepts of controllability and observability are defined by means of a ranking function defined on the product automaton, which decreases its value if an accepting state of the produc automaton is being approached. The proposed on-line control scheme leverages the ranking function and a permissiveness function, which represents a time-varying permissiveness level As a result, the on-line supervisor achieves the specification, being aware of the tradeoff between its permissiveness and acceptance of the specification, if the product automaton is controllable and observable.

Backdoors for Linear Temporal Logic

In the present paper, we introduce the backdoor set approach into the field of temporal logic for the global fragment of linear temporal logic. We study the parameterized complexity of the satisfiability problem parameterized by the size of the backdoor. We distinguish between backdoor detection and evaluation of backdoors into the fragments of Horn and Krom formulas. Here we classify the operator fragments of globally-operators for past/future/always, and the combination of them. Detection is shown to be fixed-parameter tractable whereas the complexity of evaluation behaves differently. We show that for Krom formulas the problem is paraNP-complete. For Horn formulas, the complexity is shown to be either fixed parameter tractable or paraNP-complete depending on the considered operator fragment.

Nonuniform abstractions, refinement and controller synthesis with novel BDD encodings

This paper presents a control synthesis algorithm for dynamical systems to satisfy specifications given in a fragment of linear temporal logic. It is based on an abstraction-refinement scheme with nonuniform partitions of the state space. A novel encoding of the resulting transition system is proposed that uses binary decision diagrams for efficiency. We discuss several factors affecting scalability and present some benchmark results demonstrating the effectiveness of the new encodings. These ideas are also being implemented on a publicly available prototype tool, ARCS, that we briefly introduce in the paper.

Mode-Target Games: Reactive Synthesis for Control Applications

In this paper, we introduce a class of linear temporal logic (LTL) specifications for which the problem of synthesizing controllers can be solved in polynomial time. The new class of specifications is an LTL fragment that we term Mode Target (MT) and is inspired by numerous control applications where there are modes and corresponding (possibly multiple) targets for each mode. We formulate the problem of synthesizing a controller enforcing an MT specification as a game and provide an algorithm that requires $O(\sum _i t_i n^2)$ symbolic steps, where $n$ is the number of states in the game graph, and $t_i$ is the number of targets corresponding to mode $i$ .

Advances in probabilistic model checking with PRISM: variable reordering, quantiles and weak deterministic Büchi automata

The popular model checker PRISM has been successfully used for the modeling and analysis of complex probabilistic systems. As one way to tackle the challenging state explosion problem, PRISM supports symbolic storage and manipulation using multi-terminal binary decision diagrams for representing the models and in the computations. However, it lacks automated heuristics for variable reordering, even though it is well known that the order of BDD variables plays a crucial role for compact representations and efficient computations. In this article, we present a collection of extensions to PRISM. First, we provide support for automatic variable reordering within the symbolic engines of PRISM and allow users to manually control the variable ordering at a fine-grained level. Second, we provide extensions in the realm of reward-bounded properties, namely symbolic computations of quantiles in Markov decision processes and, for both the explicit and symbolic engines, the approximative computation of quantiles for continuous-time Markov chains as well as support for multi-reward-bounded properties. Finally, we provide an implementation for obtaining minimal weak deterministic Büchi automata for the obligation fragment of linear temporal logic (LTL), with applications for expected accumulated reward computations with a finite horizon given by a co-safe LTL formula.

Temporal logic control for stochastic linear systems using abstraction refinement of probabilistic games

We consider the problem of computing the set of initial states of a dynamical system such that there exists a control strategy to ensure that the trajectories satisfy a temporal logic specification with probability 1 (almost-surely). We focus on discrete-time, stochastic linear dynamics and specifications given as formulas of the Generalized Reactivity(1) fragment of Linear Temporal Logic over linear predicates in the states of the system. We propose a solution based on iterative abstraction-refinement, and turn-based 2-player probabilistic games. While the theoretical guarantee of our algorithm after any finite number of iterations is only a partial solution, we show that if our algorithm terminates, then the result is the set of all satisfying initial states. Moreover, for any (partial) solution our algorithm synthesizes witness control strategies to ensure almost-sure satisfaction of the temporal logic specification. While the proposed algorithm guarantees progress and soundness in every iteration, it is computationally demanding. We offer an alternative, more efficient solution for the reachability properties that decomposes the problem into a series of smaller problems of the same type. All algorithms are demonstrated on an illustrative case study.

Synthesizing a Lego Forklift Controller in GR(1): A Case Study

Reactive synthesis is an automated procedure to obtain a correct-by-construction reactive system from a given specification. GR(1) is a well-known fragment of linear temporal logic (LTL) where synthesis is possible using a polynomial symbolic algorithm. We conducted a case study to learn about the challenges that software engineers may face when using GR(1) synthesis for the development of a reactive robotic system. In the case study we developed two variants of a forklift controller, deployed on a Lego robot. The case study employs LTL specification patterns as an extension of the GR(1) specification language, an examination of two specification variants for execution scheduling, traceability from the synthesized controller to constraints in the specification, and generated counter strategies to support understanding reasons for unrealizability. We present the specifications we developed, our observations, and challenges faced during the case study.

A multi-paradigm language for reactive synthesis

This paper proposes a language for describing reactive synthesis problems that integrates imperative and declarative elements. The semantics is defined in terms of two-player turn-based infinite games with full information. Currently, synthesis tools accept linear temporal logic (LTL) as input, but this description is less structured and does not facilitate the expression of sequential constraints. This motivates the use of a structured programming language to specify synthesis problems. Transition systems and guarded commands serve as imperative constructs, expressed in a syntax based on that of the modeling language Promela. The syntax allows defining which player controls data and control flow, and separating a program into assumptions and guarantees. These notions are necessary for input to game solvers. The integration of imperative and declarative paradigms allows using the paradigm that is most appropriate for expressing each requirement. The declarative part is expressed in the LTL fragment of generalized reactivity(1), which admits efficient synthesis algorithms, extended with past LTL. The implementation translates Promela to input for the Slugs synthesizer and is written in Python. The AMBA AHB bus case study is revisited and synthesized efficiently, identifying the need to reorder binary decision diagrams during strategy construction, in order to prevent the exponential blowup observed in previous work.

BDD-versus SAT-based bounded model checking for the existential fragment of linear temporal logic with knowledge: algorithms and their performance

The paper deals with symbolic approaches to bounded model checking (BMC) for the existential fragment of linear temporal logic extended with the epistemic component (ELTLK), interpreted over interleaved interpreted systems. Two translations of BMC for ELTLK to SAT and to operations on BDDs are presented. The translations have been implemented, tested, and compared with each other as well as with another tool on several benchmarks for MAS. Our experimental results reveal advantages and disadvantages of SAT- versus BDD-based BMC for ELTLK.

Temporal logic robot control based on automata learning of environmental dynamics

We develop a technique to automatically generate a control policy for a robot moving in an environment that includes elements with unknown, randomly changing behavior. The robot is required to achieve a surveillance mission, in which a certain request needs to be serviced repeatedly, while the expected time inbetween consecutive services is minimized and additional temporal logic constraints are satisfied. We define a fragment of linear temporal logic to describe such a mission and formulate the problem as a temporal logic game. Our approach is based on two main ideas. First, we extend results in automata learning to detect patterns of the unknown behavior of the elements in the environment. Second, we employ an automata–theoretic method to generate the control policy. We show that the obtained control policy converges to an optimal one when the partially unknown behavior patterns are fully learned. In addition, we illustrate the method in an experimental setup, in which an unmanned ground vehicle, with the help of a cooperating unmanned aerial vehicle (UAV), satisfies a temporal logic requirement in a partitioned environment whose regions are controlled by barriers with unknown behavior.

Measuring Progress of Probabilistic LTL Model Checking

Recently, Zhang and Van Breugel introduced the notion of a progress measure for a probabilistic model checker. Given a linear-time property P and a description of the part of the system that has already been checked, the progress measure returns a real number in the unit interval. The real number captures how much progress the model checker has made towards verifying P. If the progress is zero, no progress has been made. If it is one, the model checker is done. They showed that the progress measure provides a lower bound for the measure of the set of execution paths that satisfy P. They also presented an algorithm to compute the progress measure when P is an invariant. In this paper, we present an algorithm to compute the progress measure when P is a formula of a positive fragment of linear temporal logic. In this fragment, we can express invariants but also many other interesting properties. The algorithm is exponential in the size of P and polynomial in the size of that part of the system that has already been checked. We also present an algorithm to compute a lower bound for the progress measure in polynomial time.

Almost linear Büchi automata

We introduce a new fragment of linear temporal logic (LTL) called LIO and a new class of Büchi automata (BA) called almost linear Büchi automata (ALBA). We provide effective translations between LIO and ALBA showing that the two formalisms are expressively equivalent. As we expect there to be applications of our results in model checking, we use two standard sources of specification formulae, namely Spec Patterns and BEEM, to study the practical relevance of the LIO fragment, and to compare our translation of LIO to ALBA with two standard translations of LTL to BA using alternating automata. Finally, we demonstrate that the LIO to ALBA translation can be much faster than the standard translation, and the resulting automata can be substantially smaller.

Safety alternating automata on data words

A data word is a sequence of pairs of a letter from a finite alphabet and an element from an infinite set, where the latter can only be compared for equality. Safety one-way alternating automata with one register on infinite data words are considered, their nonemptiness is shown to be ExpSpace-complete, and their inclusion decidable but not primitive recursive. The same complexity bounds are obtained for satisfiability and refinement, respectively, for the safety fragment of linear temporal logic with freeze quantification. Dropping the safety restriction, adding past temporal operators, or adding one more register, each causes undecidability.

Property specifications for workflow modelling

Previously we provided two formal behavioural semantics for the Business Process Modelling Notation (BPMN) in the process algebra CSP. By exploiting CSP’s refinement orderings, developers may formally compare their BPMN models. However, BPMN is not a specification language, and it is difficult and sometimes impossible to use it to construct behavioural properties against which other BPMN models may be verified. This paper considers a pattern-based approach to expressing behavioural properties. We describe a property specification language P L for capturing a generalisation of Dwyer et al.’s Property Specification Patterns, and present a translation from P L into a bounded, positive fragment of linear temporal logic, which can then be automatically translated into CSP for simple refinement checking. We present a detailed example studying the behavioural properties of an airline ticket reservation business process. Using the same example we also describe some recent results on expressing behavioural compatibility within our semantic models. These results lead to a compositional approach for ensuring deadlock freedom of interacting business processes.