Linear-time Logic Research Articles

Linear-time logics -- a coalgebraic perspective

We describe a general approach to deriving linear-time logics for a wide variety of state-based, quantitative systems, by modelling the latter as coalgebras whose type incorporates both branching and linear behaviour. Concretely, we define logics whose syntax is determined by the type of linear behaviour, and whose domain of truth values is determined by the type of branching behaviour, and we provide two semantics for them: a step-wise semantics akin to that of standard coalgebraic logics, and a path-based semantics akin to that of standard linear-time logics. The former semantics is useful for model checking, whereas the latter is the more natural semantics, as it measures the extent with which qualitative properties hold along computation paths from a given state. Our main result is the equivalence of the two semantics. We also provide a semantic characterisation of a notion of logical distance induced by these logics. Instances of our logics support reasoning about the possibility, likelihood or minimal cost of exhibiting a given linear-time property.

Streamlining Temporal Formal Verification over Columnar Databases

Recent findings demonstrate how database technology enhances the computation of formal verification tasks expressible in linear time logic for finite traces (LTLf). Human-readable declarative languages also help the common practitioner to express temporal constraints in a straightforward and accessible language. Notwithstanding the former, this technology is in its infancy, and therefore, few optimization algorithms are known for dealing with massive amounts of information audited from real systems. We, therefore, present four novel algorithms subsuming entire LTLf expressions while outperforming previous state-of-the-art implementations on top of KnoBAB, thus postulating the need for the corresponding, leading to the formulation of novel xtLTLf-derived algebraic operators.

Probabilistic temporal logic with countably additive semantics

This work presents a proof-theoretical and model-theoretical approach to probabilistic temporal logic. We present two novel logics; each of them extends both the language of linear time logic (LTL) and the language of probabilistic logic with polynomial weight formulas. The first logic is designed for reasoning about probabilities of temporal events, allowing statements like “the probability that A will hold in next moment is at least the probability that B will always hold” and conditional probability statements like “probability that A will always hold, given that B holds, is at least one half”, where A and B are arbitrary statements. We axiomatize this logic, provide corresponding sigma additive semantics and prove that the axiomatization is sound and strongly complete. We show that the satisfiability problem for our logic is decidable, by presenting a procedure which runs in polynomial space. We also present a logic with much richer language, in which probabilities are not attached only to temporal events, but the language allows arbitrary nesting of probability and temporal operators, allowing statements like “probability that tomorrow the chance of rain will be less than 80% is at least a half”. For this logic we prove a decidability result.

Reinforcement Learning for Omega-Regular Specifications on Continuous-Time MDP

Continuous-time Markov decision processes (CTMDPs) are canonical models to express sequential decision-making under dense-time and stochastic environments. When the stochastic evolution of the environment is only available via sampling, model-free reinforcement learning (RL) is the algorithm-of-choice to compute optimal decision sequence. RL, on the other hand, requires the learning objective to be encoded as scalar reward signals. Since doing such translations manually is both tedious and error-prone, a number of techniques have been proposed to translate high-level objectives (expressed in logic or automata formalism) to scalar rewards for discrete-time Markov decision processes. Unfortunately, no automatic translation exists for CTMDPs. We consider CTMDP environments against the learning objectives expressed as omega-regular languages. Omega-regular languages generalize regular languages to infinite-horizon specifications and can express properties given in popular linear-time logic LTL. To accommodate the dense-time nature of CTMDPs, we consider two different semantics of omega-regular objectives: 1) satisfaction semantics where the goal of the learner is to maximize the probability of spending positive time in the good states, and 2) expectation semantics where the goal of the learner is to optimize the long-run expected average time spent in the ''good states'' of the automaton. We present an approach enabling correct translation to scalar reward signals that can be readily used by off-the-shelf RL algorithms for CTMDPs. We demonstrate the effectiveness of the proposed algorithms by evaluating it on some popular CTMDP benchmarks with omega-regular objectives.

An abstraction-refinement framework for verifying strategic properties in multi-agent systems with imperfect information

We investigate the verification of Multi-Agent Systems against strategic properties expressed in Alternating-time Temporal Logic under the assumptions of imperfect information and perfect recall. To this end, we develop a three-valued semantics for concurrent game structures upon which we define an abstraction method. We prove that concurrent game structures with imperfect information admit perfect information abstractions that preserve three-valued satisfaction. Furthermore, to deal with cases in which the value of a specification is undefined, we develop a novel automata-theoretic technique for the linear-time logic (LTL), then apply it to finding “failure” states. The latter can then be fed into a refinement procedure, thus providing a sound, albeit incomplete, verification method. We illustrate the overall procedure in a variant of the Train Gate Controller scenario and a simple voting protocol under imperfect information and perfect recall. We also present an implementation of our procedure and provide preliminary experimental results.

Learning-Based Path Planning Under Co-Safe Temporal Logic Specifications

This paper presents a path planning algorithm for efficiently generating low-cost trajectories that meet mission requirements specified in Linear Time Logic (LTL), where cost functions are defined throughout the configuration space. The main idea of the paper is to increase efficiency by adding learning-based extensions to sampling-based path planning algorithms. The proposed method includes two layers: a high-level layer that determines how to expand the search tree to satisfy logical specifications and a low-level layer that extends the search tree to search for low-cost trajectories. To efficiently find low-cost trajectories, it is learned how to extend the search tree from the data via deep learning. By leveraging the conditional variational autoencoder, we learn the ideal search tree extension distribution in a given situation, which increases solution search efficiency. Simulations show that the proposed method finds a low-cost trajectory while meeting a given mission specification. Furthermore, it is confirmed that the proposed approach performs better than the existing methods.

Decentralized Asynchronous Crash-resilient Runtime Verification

Runtime verification is a lightweight method for monitoring the formal specification of a system during its execution. It has recently been shown that a given state predicate can be monitored consistently by a set of crash-prone asynchronous distributed monitors observing the system, only if each monitor can emit verdicts taken from a large enough finite set. We revisit this impossibility result in the concrete context of linear-time logic ( ltl ) semantics for runtime verification, that is, when the correctness of the system is specified by an ltl formula on its execution traces. First, we show that monitors synthesized based on the 4-valued semantics of ltl ( rv-ltl ) may result in inconsistent distributed monitoring, even for some simple ltl formulas. More generally, given any ltl formula φ, we relate the number of different verdicts required by the monitors for consistently monitoring φ, with a specific structural characteristic of φ called its alternation number . Specifically, we show that, for every k ≥ 0 , there is an ltl formula φ with alternation number k that cannot be verified at runtime by distributed monitors emitting verdicts from a set of cardinality smaller than k + 1. On the positive side, we define a family of logics, called distributed ltl (abbreviated as dltl ), parameterized by k ≥ 0, which refines rv-ltl by incorporating 2k + 4 truth values. Our main contribution is to show that, for every k ≥ 0, every ltl formula φ with alternation number k can be consistently monitored by distributed monitors, each running an automaton based on a (2 ⌈ k /2 ⌉ +4)-valued logic taken from the dltl family.

A Bounded Model Checker for Timed Automata and Its Application to LTL Properties

Model checking with a time aspect is often used in verification on hardware and embedded systems. Timed automata are often used for such models. UPPAAL is a world-wide famous model checking tool for timed automata; however, UPPAAL is a Computational Tree Logic (CTL)-based model checking tool and cannot use Linear Temporal Logic (LTL) properties. Bounded model checking uses LTL (Linear Time Logic) for a checking formula. Bounded model checking specifies a boundary k and obtains counterexamples by searching from the initial state of a system to states reachable by k-steps. There are several studies on bounded model checking. Sorea has proposed a concrete algorithm for a timed automaton. There are, however, no clear details on how to implement bounded model checking tools for timed automata, and study the performance. Another problem is that the timed automaton covered by the method does not support general variables except for clock variables. The objective of this study is to implement a bounded model checking tool using LTL for timed automata. We also improve Sorea's method so that it can handle extended timed automata that handle general variables. This paper also presents some LTL examples from texts on requirement specifications for embedded systems and the results of applying the tool to them.

Foundations for Restraining Bolts: Reinforcement Learning with LTLf/LDLf Restraining Specifications

In this work we investigate on the concept of “restraining bolt”, envisioned in Science Fiction. Specifically we introduce a novel problem in AI. We have two distinct sets of features extracted from the world, one by the agent and one by the authority imposing restraining specifications (the “restraining bolt”). The two sets are apparently unrelated since of interest to independent parties, however they both account for (aspects of) the same world. We consider the case in which the agent is a reinforcement learning agent on the first set of features, while the restraining bolt is specified logically using linear time logic on finite traces LTLf/LDLf over the second set of features. We show formally, and illustrate with examples, that, under general circumstances, the agent can learn while shaping its goals to suitably conform (as much as possible) to the restraining bolt specifications.

Trace-Length Independent Runtime Monitoring of Quantitative Policies

Metric linear-time logic (MTL) has been widely used to specify runtime policies. Traditionally this use of MTL is to capture the qualitative aspects of the monitored systems, but recent developments in its extensions with aggregate operators allow some quantitative policies to be specified. Our interest in MTL-based policy languages is driven by applications in runtime malware or intrusion detection in platforms like Android and autonomous vehicles, which requires the monitoring algorithm to be independent of the length of the system event traces so that its performance does not degrade as the traces grow. We propose a policy language based on a past-time variant of MTL, extended with an aggregate operator called the metric temporal counting quantifier to specify a policy based on the number of times some sub-policies are satisfied in the specified past time interval. We show that a broad class of policies, but not all policies, specified with our language can be monitored in a trace-length independent way, and provide a concrete algorithm to do so. We implement and test our algorithm in both an existing Android monitoring framework and an autonomous vehicle simulation platform, and show that our approach can effectively specify and monitor quantitative policies drawn from real-world studies.

ANALYSIS OF APPROACHES TO THE SIMULATION AND VERIFICATION OF CYBER-PHYSICAL SYSTEMS

Context. Current trends in the performance and complexity of system requirements require fundamentally new approaches to design, in which cybernetic and physical components are integrated at different stages. Cyber-physical systems are systems that provide close interaction between physical and cybernetic components, integration of computing, physical processes and networks. In such systems, software and physical subsystems operate in different temporal and spatial dimensions, interacting in different ways. Cyberphysical systems surround humans in almost every area of existence, from housing and transportation to medical devices and interregional power grids. Therefore, verification and validation of such systems is an urgent task today. Approaches to verification of cyber-physical systems are considered. The object of research is the process of verification of cyber-physical systems, the subject is the methods of verification of cyber-physical systems, models and logic used in formal verification.Objective. The purpose of the work is to analyze approaches to the verification of cyber-physical systems, detailing the individual steps, such as the selection of models, verification tools, and, in fact, verification methods.Method. The main methods outlined in the paper are methods of formal verification of cyber-physical systems, namely simulation, theorem proving, symbolic execution, and model checking. In addition, the methodology of the model checking method – the Kripke structure and temporal logics: logic of computational trees and linear time logic is discussed in detail. Modeling using finite state machines is also performed. Results. The paper deals with modeling of the cyber-physical system in the form of creation of the Kripke structure that allowed to describe all states of the system necessary for executing of formal verification.Conclusions. The paper describes the characteristics of cyber-physical systems, analyzes the methods of verification of such systems. After analysis the conclusion is made about the most promising method of model verification, for which the basic methodology is considered. Characteristics of Kripke structure and temporal logics are described as the main elements of the model checking method. Following the review, the shortcomings of the standard methodology most relevant to the modeling stage of cyber-physical systems are concluded. The possibility of using finite state machines, namely Kripke structures, for modeling elements of a cyber-physical system is shown. The scientific novelty of the work is that models of cyber-physical systems have been developed, which, unlike existing ones, are based on Kripke structures, which allow to make a detailed description of all states of the system, which, in turn, is an important step to verify such a system. The practical value of the work is the developed models of the independent power station of alternative energy, which will automate the process of charging electric vehicles. Digital duplicates have been implemented, which allow modeling the processes of an independent energy station of alternative energy. The developed duplicates are used in the study of disciplines in the preparation of bachelors and masters in 121 computer science.

Restraining Bolts for Reinforcement Learning Agents

In this work we have investigated the concept of “restraining bolt”, inspired by Science Fiction. We have two distinct sets of features extracted from the world, one by the agent and one by the authority imposing some restraining specifications on the behaviour of the agent (the “restraining bolt”). The two sets of features and, hence the model of the world attainable from them, are apparently unrelated since of interest to independent parties. However they both account for (aspects of) the same world. We have considered the case in which the agent is a reinforcement learning agent on a set of low-level (subsymbolic) features, while the restraining bolt is specified logically using linear time logic on finite traces f/f over a set of high-level symbolic features. We show formally, and illustrate with examples, that, under general circumstances, the agent can learn while shaping its goals to suitably conform (as much as possible) to the restraining bolt specifications.1

Branching-time logic [formula omitted] and its tree-style one-pass tableau: Extending fairness expressibility of [formula omitted

Temporal logic has become essential for various areas in computer science, most notably for the specification and verification of hardware and software systems. For the specification purposes rich temporal languages are required that, in particular, can express fairness constraints. For linear-time logics which deal with fairness in the linear-time setting, one-pass and two-pass tableau methods have been developed. In the repository of the CTL-type branching-time setting, the well-known logics ECTL and ECTL+ were developed to explicitly deal with fairness. However, due to the syntactical restrictions, these logics can only express restricted versions of fairness. The logic CTL⋆, often considered as ‘the full branching-time logic’ overcomes these restrictions on expressing fairness. However, CTL⋆ is extremely challenging for the application of verification techniques, and the tableau technique, in particular. For example, there is no one-pass tableau construction for CTL⋆, while one-pass tableau has an additional benefit enabling the formulation of dual sequent calculi that are often treated as more ‘natural’ being more friendly for human understanding. These two considerations lead to the following problem - are there logics that have richer expressiveness than ECTL+, allowing the formulation of a new range of fairness constraints with ‘until’ operator, yet ‘simpler’ than CTL⋆, and for which a one-pass tableau can be developed? Here we give a positive answer to this question, introducing a sub-logic of CTL⋆ called ECTL#, its tree-style one-pass tableau, and an algorithm for obtaining a systematic tableau, for any given admissible branching-time formulae. We prove the termination, soundness and completeness of the method. As tree-shaped one-pass tableaux are well suited for the automation and are amenable for the implementation and for the formulation of sequent calculi. Our results also open a prospect of relevant developments of the automation and implementation of the tableau method for ECTL#, and of a dual sequent calculi.

Ontology-Based Query Answering for Probabilistic Temporal Data

We investigate ontology-based query answering for data that are both temporal and probabilistic, which might occur in contexts such as stream reasoning or situation recognition with uncertain data. We present a framework that allows to represent temporal probabilistic data, and introduce a query language with which complex temporal and probabilistic patterns can be described. Specifically, this language combines conjunctive queries with operators from linear time logic as well as probability operators. We analyse the complexities of evaluating queries in this language in various settings. While in some cases, combining the temporal and the probabilistic dimension in such a way comes at the cost of increased complexity, we also determine cases for which this increase can be avoided.

From Real-time Logic to Timed Automata

We show how to construct temporal testers for the logic MITL, a prominent linear-time logic for real-time systems. A temporal tester is a transducer that inputs a signal holding the Boolean value of atomic propositions and outputs the truth value of a formula along time. Here we consider testers over continuous-time Boolean signals that use clock variables to enforce duration constraints, as in timed automata. We first rewrite the MITL formula into a “simple” formula using a limited set of temporal modalities. We then build testers for these specific modalities and show how to compose testers for simple formulae into complex ones. Temporal testers can be turned into acceptors, yielding a compositional translation from MITL to timed automata. This construction is much simpler than previously known and remains asymptotically optimal. It supports both past and future operators and can easily be extended.

Temporal Equilibrium Logic with past operators

In this paper, we study the introduction of modal past temporal operators in Temporal Equilibrium Logic (TEL), an hybrid formalism that mixes linear-time modalities and logic programs interpreted under stable models and their characterisation in terms of Equilibrium Logic. We show that Kamp’s translation can also be used to translate the new extension of TEL with past operators into Quantified Equilibrium Logic. Additionally, we provide a method for removing past operators that consists in replacing past-time subformulas by fresh auxiliary atoms, obtaining an equivalent formula, modulo the original alphabet.

In All But Finitely Many Possible Worlds: Model-Theoretic Investigations on ‘Overwhelming Majority’ Default Conditionals

Defeasible conditionals are statements of the form `ifAthen normallyB'. One plausible interpretation introduced in nonmonotonic reasoning dictates that ($$A\Rightarrow B$$AźB) is true iff B is true in `most' A-worlds. In this paper, we investigate defeasible conditionals constructed upon a notion of `overwhelming majority', defined as `truth in a cofinite subset of$$\omega $$ź', the first infinite ordinal. One approach employs the modal logic of the frame $$(\omega , <)$$(ź,<), used in the temporal logic of discrete linear time. We introduce and investigate conditionals, defined modally over $$(\omega , <)$$(ź,<); several modal definitions of the conditional connective are examined, with an emphasis on the nonmonotonic ones. An alternative interpretation of `majority' as sets cofinal (in $$\omega $$ź) rather than cofinite (subsets of $$\omega $$ź) is examined. For these modal approaches over $$(\omega , <)$$(ź,<), a decision procedure readily emerges, as the modal logic $${\mathbf {K4DLZ}}$$K4DLZ of this frame is well-known and a translation of the conditional sentences can be mechanically checked for validity; this allows also for a quick proof of $$\mathsf {NP}$$NP-completeness of the satisfiability problem for these logics. A second approach employs the conditional version of Scott-Montague semantics, in the form of $$\omega $$ź-many possible worlds, endowed with neighborhoods populated by collections of cofinite subsets of $$\omega $$ź. This approach gives rise to weak conditional logics, as expected. The relative strength of the conditionals introduced is compared to (the conditional logic `equivalent' of) KLM logics and other conditional logics in the literature.

Quantifying conformance using the Skorokhod metric

The conformance testing problem for dynamical systems asks, given two dynamical models (e.g., as Simulink diagrams), whether their behaviors are “close” to each other. In the semi-formal approach to conformance testing, the two systems are simulated on a large set of tests, and a metric, defined on pairs of real-valued, real-timed trajectories, is used to determine a lower bound on the distance. We show how the Skorokhod metric on continuous dynamical systems can be used as the foundation for conformance testing of complex dynamical models. The Skorokhod metric allows for both state value mismatches and timing distortions, and is thus well suited for checking conformance between idealized models of dynamical systems and their implementations. We demonstrate the robustness of the metric by proving a transference theorem: trajectories close under the Skorokhod metric satisfy “close” logical properties in the timed linear time logic FLTL (Freeze LTL) containing a rich class of temporal and spatial constraint predicates involving time and value freeze variables. We provide efficient window-based streaming algorithms to compute the Skorokhod metric for both piecewise affine and piecewise constant traces, and use these as a basis for a conformance testing tool for Simulink. We experimentally demonstrate the effectiveness of our tool in finding discrepant behaviors on a set of control system benchmarks, including an industrial challenge problem.

PROBLEMÁTICAS EXISTENCIAIS E MODERNIDADE DE HÚMUS DE RAUL BRANDÃO

O presente ensaio apresenta uma releitura de Húmus de Raul Brandão centrada nas problemáticas existenciais que atravessam o mencionado texto, sublinhando o trajeto de modernidade que o mesmo prefigura, articulando-o com temáticas e questionamentos de ordem filosófico-antropológica que o século XX desenvolveria. Assim, o pendor metafísico da obra desdobra um conjunto de intuições próprias das filosofias ditas da existência, radicando na ambígua relação com o divino, gerando o pressentimento de uma angústia existencial que implica uma autoexigência de configuração de um sentido que oblitere ou supere a consciência agónica do tempo, sentido que a relação erótica (não) concretizaria. A fragmentação textual, a propensão repetitiva e circular da narrativa, a presentificação, a polifonia, as dilações e omissões discursivas constituem estratégias textuais de desafio à lógica destrutiva do tempo linear tendente ao nada que esta ordem metafísica instaura. O dito desafio, sendo invariavelmente fracassado, acaba ainda assim por derivar numa proposta ética da resiliência face à morte, da elevação da vida como valor, que justifica o cunho pré-existencialista de Raul Brandão e dá o tom da sua modernidade.

Polynomial Algorithm of Verication for Subset of PLTL Logic

In this article a polynomial algorithm is described of verification of dynamic properties of Markov chains described by formulas of a subset of temporal logic PLTL (propositional temporal logic of linear time). The algorithm allows to find probability of the validity of the formula on the Markov chain, and also set of trajectories on which the verified formula is true.