Interval Temporal Logic Research Articles

The addition of temporal neighborhood makes the logic of prefixes and sub-intervals EXPSPACE-complete

A classic result by Stockmeyer gives a non-elementary lower bound to the emptiness problem for star-free generalized regular expressions. This result is intimately connected to the satisfiability problem for interval temporal logic, notably for formulas that make use of the so-called chop operator. Such an operator can indeed be interpreted as the inverse of the concatenation operation on regular languages, and this correspondence enables reductions between non-emptiness of star-free generalized regular expressions and satisfiability of formulas of the interval temporal logic of chop under the homogeneity assumption. In this paper, we study the complexity of the satisfiability problem for suitable weakenings of the chop interval temporal logic, that can be equivalently viewed as fragments of Halpern and Shoham interval logic. We first consider the logic $\mathsf{BD}_{hom}$ featuring modalities $B$, for \emph{begins}, corresponding to the prefix relation on pairs of intervals, and $D$, for \emph{during}, corresponding to the infix relation. The homogeneous models of $\mathsf{BD}_{hom}$ naturally correspond to languages defined by restricted forms of regular expressions, that use union, complementation, and the inverses of the prefix and infix relations. Such a fragment has been recently shown to be PSPACE-complete . In this paper, we study the extension $\mathsf{BD}_{hom}$ with the temporal neighborhood modality $A$ (corresponding to the Allen relation \emph{Meets}), and prove that it increases both its expressiveness and complexity. In particular, we show that the resulting logic $\mathsf{BDA}_{hom}$ is EXPSPACE-complete.

Expressive completeness by separation for discrete time interval temporal logic with expanding modalities

Recently we established an analog of Gabbay's separation theorem about linear temporal logic (LTL) for the extension of Moszkowski's discrete time propositional Interval Temporal Logic (ITL) by two sets of expanding modalities, namely the unary neighbourhood modalities and the binary weak inverses of ITL's chop operator. One of the many useful applications of separation in LTL is the concise proof of LTL's expressive completeness wrt the monadic first-order theory of 〈ω,<〉 it enables. In this paper we show how our separation theorem about ITL facilitates a similar proof of the expressive completeness of ITL with expanding modalities wrt the monadic first- and second-order theories of 〈Z,<〉.

Pspace-completeness of the temporal logic of sub-intervals and suffixes

In this paper, we prove Pspace-completeness of the finite satisfiability and model checking problems for the fragment of Halpern and Shoham interval logic with modality 〈E〉, for the “suffix” relation on pairs of intervals, and modality 〈D〉, for the “sub-interval” relation, under the homogeneity assumption. The result significantly improves the Expspace upper bound recently established for the same fragment, and proves the rather surprising fact that the complexity of the considered problems does not change when we add either the modality for suffixes (〈E〉) or, symmetrically, the modality for prefixes (〈B〉) to the logic of sub-intervals (featuring only 〈D〉).

From Non-punctuality to Non-adjacency: A Quest for Decidability of Timed Temporal Logics with Quantifiers

Metric Temporal Logic (MTL) and Timed Propositional Temporal Logic (TPTL) are prominent real-time extensions of Linear Temporal Logic (LTL). In general, the satisfiability checking problem for these extensions is undecidable when both the future (Until, U) and the past (Since, S) modalities are used (denoted by MTL[U,S] and TPTL[U,S]). In a classical result, the satisfiability checking for Metric Interval Temporal Logic (MITL[U,S]), a non-punctual fragment of MTL[U,S], is shown to be decidable with EXPSPACE complete complexity. A straightforward adoption of non-punctuality does not recover decidability in the case of TPTL[U,S]. Hence, we propose a more refined notion called non-adjacency for TPTL[U,S] and focus on its 1-variable fragment, 1-TPTL[U,S]. We show that non-adjacent 1-TPTL[U,S] is strictly more expressive than MITL. As one of our main results, we show that the satisfiability checking problem for non-adjacent 1-TPTL[U,S] is decidable with EXPSPACE complete complexity. Our decidability proof relies on a novel technique of anchored interval word abstraction and its reduction to a non-adjacent version of the newly proposed logic called PnEMTL. We further propose an extension of MSO [&lt;] (Monadic Second Order Logic of Orders) with Guarded Metric Quantifiers (GQMSO) and show that it characterizes the expressiveness of PnEMTL. That apart, we introduce the notion of non-adjacency in the context of GQMSO (NA-GQMSO), which is a syntactic generalization of logic Q2MLO due to Hirshfeld and Rabinovich and show the decidability of satisfiability checking for NA-GQMSO.

Robust Satisfaction of Metric Interval Temporal Logic Objectives in Adversarial Environments

This paper studies the synthesis of controllers for cyber-physical systems (CPSs) that are required to carry out complex time-sensitive tasks in the presence of an adversary. The time-sensitive task is specified as a formula in the metric interval temporal logic (MITL). CPSs that operate in adversarial environments have typically been abstracted as stochastic games (SGs); however, because traditional SG models do not incorporate a notion of time, they cannot be used in a setting where the objective is time-sensitive. To address this, we introduce durational stochastic games (DSGs). DSGs generalize SGs to incorporate a notion of time and model the adversary’s abilities to tamper with the control input (actuator attack) and manipulate the timing information that is perceived by the CPS (timing attack). We define notions of spatial, temporal, and spatio-temporal robustness to quantify the amounts by which system trajectories under the synthesized policy can be perturbed in space and time without affecting satisfaction of the MITL objective. In the case of an actuator attack, we design computational procedures to synthesize controllers that will satisfy the MITL task along with a guarantee of its robustness. In the presence of a timing attack, we relax the robustness constraint to develop a value iteration-based procedure to compute the CPS policy as a finite-state controller to maximize the probability of satisfying the MITL task. A numerical evaluation of our approach is presented on a signalized traffic network to illustrate our results.

Temporal Analysis and Classification of Sensor Signals

Understanding the behaviour of sensors, and in particular, the specifications of multisensor systems, are complex problems. The variables that need to be taken into consideration include, inter alia, the application domain, the way sensors are used, and their architectures. Various models, algorithms, and technologies have been designed to achieve this goal. In this paper, a new interval logic, referred to as Duration Calculus for Functions (DC4F), is applied to precisely specify signals originating from sensors, in particular sensors and devices used in heart rhythm monitoring procedures, such as electrocardiograms. Precision is the key issue in case of safety critical system specification. DC4F is a natural extension of the well-known Duration Calculus, an interval temporal logic used for specifying the duration of a process. It is suitable for describing complex, interval-dependent behaviours. Said approach allows one to specify temporal series, describe complex interval-dependent behaviours, and evaluate the corresponding data within a unifying logical framework. The use of DC4F allows one, on the one hand, to precisely specify the behaviour of functions modelling signals generated by different sensors and devices. Such specifications can be used for classifying signals, functions, and diagrams; and for identifying normal and abnormal behaviours. On the other hand, it allows one to formulate and frame a hypothesis. This is a significant advantage over machine learning algorithms, since the latter are capable of learning different patterns but fail to allow the user to specify the behaviour of interest.

Interval temporal logic for visibly pushdown systems

In this paper, we introduce and investigate an extension of Halpern and Shoham’s interval temporal logic HS for the specification and verification of branching-time context-free requirements of pushdown systems under a state-based semantics over Kripke structures enforcing visibility of the pushdown operations. The proposed logic, called nested BHS , supports branching-time both in the past and in the future, and is able to express non-regular properties of linear and branching behaviours of procedural contexts in a natural way. It strictly subsumes well-known linear time context-free extensions of LTL such as CaRet [4] and NWTL [2]. The main result is the decidability of the visibly pushdown model-checking problem against nested BHS . The proof exploits a non-trivial automata-theoretic construction.

The voice of COVID-19: Breath and cough recording classification with temporal decision trees and random forests

Symbolic learning is the logic-based approach to machine learning, and its mission is to provide algorithms and methodologies to extract logical information from data and express it in an interpretable way. Interval temporal logic has been recently proposed as a suitable tool for symbolic learning, specifically via the design of an interval temporal logic decision tree extraction algorithm. In order to improve their performances, interval temporal decision trees can be embedded into interval temporal random forests, mimicking the corresponding schema at the propositional level. In this article we consider a dataset of cough and breath sample recordings of volunteer subjects, labeled with their COVID-19 status, originally collected by the University of Cambridge. By interpreting such recordings as multivariate time series, we study the problem of their automated classification using interval temporal decision trees and forests. While this problem has been approached with the same dataset as well as with other datasets, in all cases, non-symbolic learning methods (usually, deep learning-based) have been applied to solve it; in this article we apply a symbolic approach, and show that it does not only outperform the state-of-the-art obtained with the same dataset, but its results are also superior to those of most non-symbolic techniques applied on other datasets. As an added bonus, thanks to the symbolic nature of our approach, we are also able to extract explicit knowledge to help physicians characterize typical COVID-positive cough and breath.

Improving Temporal Event Scheduling through STEP Perpetual Learning

Currently, most machine learning applications follow a one-off learning process: given a static dataset and a learning algorithm, generate a model for a task. These applications can neither adapt to a dynamic and changing environment, nor accomplish incremental task performance improvement continuously. STEP perpetual learning, by continuous knowledge refinement through sequential learning episodes, emphasizes the accomplishment of incremental task performance improvement. In this paper, we describe how a personalized temporal event scheduling system SmartCalendar, can benefit from STEP perpetual learning. We adopt the interval temporal logic to represent events’ temporal relationships and determine if events are temporally inconsistent. To provide strategies that approach user preferences for handling temporal inconsistencies, we propose SmartCalendar to recognize, resolve and learn from temporal inconsistencies based on STEP perpetual learning. SmartCalendar has several cornerstones: similarity measures for temporal inconsistency; a sparse decomposition method to utilize historical data; and a loss function based on cross-entropy to optimize performance. The experimental results on the collected dataset show that SmartCalendar incrementally improves its scheduling performance and substantially outperforms comparison methods.

STAn: analysis of data traces using an event-driven interval temporal logic

The increasing integration of systems into people’s daily routines, especially smartphones, requires ensuring correctness of their functionality and even some performance requirements. Sometimes, we can only observe the interaction of the system (e.g. the smartphone) with its environment at certain time points; that is, we only have access to the data traces produced due to this interaction. This paper presents the tool STAn, which performs runtime verification on data traces that combine timestamped discrete events and sampled real-valued magnitudes. STAn uses the Spin model checker as the underlying execution engine, and analyzes traces against properties described in the so-called event-driven interval temporal logic (eLTL) by transforming each eLTL formula into a network of concurrent automata, written in Promela, that monitors the trace. We present two different transformations for online and offline monitoring, respectively. Then, Spin explores the state space of the automata network and the trace to return a verdict about the corresponding property. We use the proposal to analyze data traces obtained during mobile application testing in different network scenarios.

Reactive and Risk-Aware Control for Signal Temporal Logic

The deployment of autonomous systems in uncertain and dynamic environments has raised fundamental questions. Addressing these is pivotal to build fully autonomous systems and requires a systematic integration of planning and control. We first propose reactive risk signal interval temporal logic (ReRiSITL) as an extension of signal temporal logic (STL) to formulate complex spatiotemporal specifications. Unlike STL, ReRiSITL allows to consider uncontrollable propositions that may model humans as well as random environmental events such as sensor failures. Additionally, ReRiSITL allows to incorporate risk measures, such as (but not limited to) the conditional value-at-risk, to measure the risk of violating certain spatial specifications. Second, we propose an algorithm to check if an ReRiSITL specification is satisfiable. For this purpose, we abstract the ReRiSITL specification into a timed signal transducer and devise a game-based approach. Third, we propose a reactive planning and control framework for dynamical control systems under ReRiSITL specifications.

Parametric Interval Temporal Logic over Infinite Words

Model checking for Halpern and Shoham's interval temporal logic HS has been recently investigated in a systematic way, and it is known to be decidable under three distinct semantics. Here, we focus on the trace-based semantics, where the infinite execution paths (traces) of the given (finite) Kripke structure are the main semantic entities. In this setting, each finite infix of a trace is interpreted as an interval, and a proposition holds over an interval if and only if it holds over each component state (homogeneity assumption). In this paper, we introduce a quantitative extension of HS over traces, called parametric HS (PHS). The novel logic allows to express parametric timing constraints on the duration (length) of the intervals. We show that checking the existence of a parameter valuation for which a Kripke structure satisfies a PHS formula (model checking), or a PHS formula admits a trace as a model under the homogeneity assumption (satisfiability) is decidable. Moreover, we identify a fragment of PHS which subsumes parametric LTL and for which model checking and satisfiability are shown to be EXPSPACE-complete.

Computational complexity of hybrid interval temporal logics

Interval logics are very expressive temporal formalisms, but reasoning with them is often undecidable or has high computational complexity. As a result, a vast number of approaches limiting their expressive power—in order to obtain better computational behaviour—have been introduced. Unfortunately, due to such restrictions, interval logics often lose referentiality, that is, the capacity to refer to specific time intervals, which is crucial for temporal representation and reasoning. The computational price that needs to be paid in order to regain referentiality is not well studied and our research aims to fill this gap. In particular we study the main interval temporal logic, called the Halpern-Shoham logic, and its low complexity modifications. To regain referentiality in these modifications, we extend the language with the hybrid machinery—nominals and satisfaction operators—and classify the obtained logics according to their computational complexity. We show that such a hybridisation often makes tractable logics intractable but not undecidable. This allows us to construct hybrid interval temporal logics which are referential as well as maintain a good compromise between expressiveness and complexity; it makes them valuable formalisms for temporal knowledge representation. We also introduce a class of models which, due to a specific interplay between the interpretation of modal operators and a structure of time, makes reasoning in interval logics computationally hard even in the absence of the hybrid machinery.

Fuzzy Halpern and Shoham's interval temporal logics

The most representative interval temporal logic, called HS, was introduced by Halpern and Shoham in the nineties. Recently, HS has been proposed as a suitable formalism for modern artificial intelligence applications; however, when dealing with real-life data one is not always able to express temporal relations and propositional labels in a definite, crisp way. In this paper, following the seminal ideas of Fitting and Zadeh, we present a fuzzy generalization of HS, called FHS, that partially solves such problems of expressive power. We study FHS from both a theoretical and an application standpoint: first, we discuss its syntax, semantics, expressive power, and satisfiability problem; then, we define and solve the time series FHS finite model checking problem, to serve as the basis of future applications.

A separation theorem for discrete-time interval temporal logic

Gabbay's separation theorem about linear temporal logic with past has proved to be one of the most useful theoretical results in temporal logic. In this paper, we establish an analogous statement about Moszkowski's discrete-time propositional Interval Temporal Logic () with two sets of expanding modalities, namely the unary neighbourhood modalities and the binary weak inverses of 's chop operator. We prove that separation holds for both with and without its loop construct chop-star.

Online Motion Planning With Soft Metric Interval Temporal Logic in Unknown Dynamic Environment

Motion planning of an autonomous system with high-level specifications has wide applications. However, research of formal languages involving timed temporal logic is still under investigation. Furthermore, many existing results rely on a key assumption that user-specified tasks are feasible in the given environment. Challenges arise when the operating environment is dynamic and unknown since the environment can be found prohibitive, leading to potentially conflicting tasks where pre-specified timed missions cannot be fully satisfied. Such issues become even more challenging when considering time-bound requirements. To address these challenges, this work proposes a control framework that considers hard constraints to enforce safety requirements and soft constraints to enable task relaxation. The metric interval temporal logic (MITL) specifications are employed to deal with time-bound constraints. By constructing a relaxed timed product automaton, an online motion planning strategy is synthesized with a receding horizon controller to generate policies, achieving multiple objectives in decreasing order of priority 1) formally guarantee the satisfaction of hard safety constraints; 2) mostly fulfill soft timed tasks; and 3) collect time-varying rewards as much as possible. Another novelty of the relaxed structure is to consider violations of both time and tasks for infeasible cases. Simulation results are provided to validate the proposed approach.

Reactive synthesis from interval temporal logic specifications

In this paper, we deal with the synthesis problem for Halpern and Shoham's interval temporal logic HS extended with an equivalence relation ∼ over time points (HS▪ for short). The definition of the problem is analogous to that for MSO(ω,<). Given an HS▪ formula φ and a finite set Σφ⋄ of proposition letters and temporal requests, it consists of determining, whether or not, for all possible valuations of elements in Σφ⋄ in every interval structure, there is a valuation of the remaining proposition letters and temporal requests such that the resulting structure is a model for φ.We focus on the decidability and complexity of the problem for some meaningful fragments of HS▪, whose modalities are drawn from the set {A(meets),A¯(metby),B(begunby),B¯(begins)} interpreted over finite linear orders and N. We prove that, over finite linear orders, the problem is decidable (Ackermann-hard) for ▪ and undecidable for AA¯BB¯. Moreover, we show that if we replace finite linear orders by N, then it becomes undecidable even for ABB¯. Finally, we study the generalization of ▪ to ▪, where k is the number of distinct equivalence relations. Despite the fact that the satisfiability problem for ▪, with k>1, over finite linear orders, is already undecidable, we prove that, under a natural semantic restriction (refinement condition), the synthesis problem turns out to be decidable.

Verifying Stochastic Hybrid Systems with Temporal Logic Specifications via Model Reduction

We present a scalable methodology to verify stochastic hybrid systems for inequality linear temporal logic (iLTL) or inequality metric interval temporal logic (iMITL). Using the Mori–Zwanzig reduction method, we construct a finite-state Markov chain reduction of a given stochastic hybrid system and prove that this reduced Markov chain is approximately equivalent to the original system in a distributional sense. Approximate equivalence of the stochastic hybrid system and its Markov chain reduction means that analyzing the Markov chain with respect to a suitably strengthened property allows us to conclude whether the original stochastic hybrid system meets its temporal logic specifications. Based on this, we propose the first statistical model checking algorithms to verify stochastic hybrid systems against correctness properties, expressed in iLTL or iMITL. The scalability of the proposed algorithms is demonstrated by a case study.

A complete robust control network based on skewed temporal logic

The robust control network for nonlinear large-scale systems with parametric uncertainties also considers the uncertain robust stabilization problem for controlled networks. In heterogeneous populations, hybrid regression models are the most important statistical analysis tools. To aim of the study is to conduct a more in-depth analysis of the existing completive robust control networks relying on biased temporal logic. Compared with the symmetric distribution, the skewed distribution can obtain accurate and effective information. Therefore, a time-series logic model under skewed distribution is proposed. The temporal logic under skew state is applied to describe the normative language of fuzzy systems. Firstly, the mixed nonlinear regression model under skewed distribution data is introduced to test whether the temporal logic formula can be realized under the skew state. Secondly, through the method of reduction, the control flow interval logic CFITL is studied, and the time series logic sequence is used to describe the measurement output loss. The sufficient conditions for the control network system to satisfy the exponential stability and H ∞ performance index are given. The linear matrix inequality obtains the completeness control network to be designed, and the effectiveness of the proposed method is verified by stochastic simulation experiments. Finally, the method is verified to be practical and feasible based on actual data. The maximum recognition rates of nearest neighbor classification, nearest subspace classification and biased distribution temporal logic classification reached 0.9019, 0.9622 and 0.9304, respectively.

Adding the Relation Meets to the Temporal Logic of Prefixes and Infixes makes it EXPSPACE-Complete

The choice of the right trade-off between expressiveness and complexity is the main issue in interval temporal logic. In their seminal paper, Halpern and Shoham showed that the satisfiability problem for HS (the temporal logic of Allen's relations) is highly undecidable over any reasonable class of linear orders. In order to recover decidability, one can restrict the set of temporal modalities and/or the class of models. In the following, we focus on the satisfiability problem for HS fragments under the homogeneity assumption, according to which any proposition letter holds over an interval if only if it holds at all its points. The problem for full HS with homogeneity has been shown to be non-elementarily decidable, but its only known lower bound is EXPSPACE (in fact, EXPSPACE-hardness has been shown for the logic of prefixes and suffixes BE, which is a very small fragment of it. The logic of prefixes and infixes BD has been recently shown to be PSPACE-complete. In this paper, we prove that the addition of the Allen relation Meets to BD makes it EXPSPACE-complete.