Interval Temporal Logic Research Articles

Metric propositional neighborhood logics on natural numbers

Interval logics formalize temporal reasoning on interval structures over linearly (or partially) ordered domains, where time intervals are the primitive ontological entities and truth of formulae is defined relative to time intervals, rather than time points. In this paper, we introduce and study Metric Propositional Neighborhood Logic (MPNL) over natural numbers. MPNL features two modalities referring, respectively, to an interval that is "met by" the current one and to an interval that "meets" the current one, plus an infinite set of length constraints, regarded as atomic propositions, to constrain the length of intervals. We argue that MPNL can be successfully used in different areas of computer science to combine qualitative and quantitative interval temporal reasoning, thus providing a viable alternative to well-established logical frameworks such as Duration Calculus. We show that MPNL is decidable in double exponential time and expressively complete with respect to a well-defined sub-fragment of the two-variable fragment $${{\rm FO}^2[\mathbb{N},=,<,s]}$$ of first-order logic for linear orders with successor function, interpreted over natural numbers. Moreover, we show that MPNL can be extended in a natural way to cover full $${{\rm FO}^2[\mathbb{N},=,<,s]}$$ , but, unexpectedly, the latter (and hence the former) turns out to be undecidable.

Incremental verification of component-based timed systems

We are interested in the incremental development, by integration of components, of component-based timed systems, and in particular, in the preservation of their properties during such a development process. We model timed components with timed automata. Their composition is achieved with the classic parallel composition operator for timed automata. The specifications of these timed systems are expressed with the timed linear logic Metric Interval Temporal Logic (MITL). To guarantee the preservation of properties during an incremental development process, we propose to use τ-simulation relations, adapted for timed systems. First, we extend the classic notion of τ-simulation with timed aspects. As in the untimed case, this relation, called timed τ-simulation, preserves safety properties. To preserve more properties, in particular liveness ones, we present another relation, called divergence-sensitive and stability-respecting (DS) timed τ-simulation. This last relation preserves all MITL properties (and thus liveness ones), but also strong non-zenoness and deadlock-freedom. Moreover, as we put ourselves in a component-based framework, we study if the relations are appropriate to the use of the composition operator that we consider. For this purpose, we study if the relations are compatible with this operator, and if composability and compositionality hold. These three properties are a way to reduce the cost of the verification of the preservation, or even to get it for free. It results that the timed τ-simulation is appropriate with the classic operator since the properties hold without any assumption. However, this is not the case for the DS timed τ-simulation. We implemented the algorithmic verification of the simulations in a tool called Verification of Simulation for Timed Automata (VeSTA). The structure of the tool was inspired from the one of the OPEN-KRONOS tool. This allows, as additional feature, to connect the models considered in VeSTA to the modules of the verification platform OPEN-CAESAR. We show the interest of our method by applying it on a case study, concerning a production cell example.

A Formal Language for XML Authorisations Based on Answer Set Programming and Temporal Interval Logic Constraints

The Extensible Markup Language is susceptible to security breaches because it does not incorporate methods to protect the information it encodes. This work focuses on the development of a formal language that can provide role-based access control to information stored in XML formatted documents. This language has the capacity to reason whether access to an XML document should be allowed. The language, Axml(T), allows for the specification of authorisations on XML documents and distinguishes itself from other research with the inclusion of temporal interval reasoning and the XPath query language.

A contract-based approach to adaptivity

Adaptive systems are systems capable of adapting their behaviour to changes in their environment. Creating such systems is not an easy task, however. Especially, creating such a system as one monolithic software component taking all eventualities and environments into account bears the risk of bad system design. To circumvent this risk, an adaptive system can be composed from partial solutions handling only a subset of all possible circumstances and environments. Then, the system can be changed through reconfigurations as the environment evolves. In this paper, we propose an approach for the verification of systems using reconfiguration as means of adaptation. For the specification of such systems and their components we introduce reM itl which is based on Metric Interval Temporal Logic M itl and allows to express connectivity of components. Based on an example from the domain of pervasive computing, we show how a system undergoing reconfigurations can be verified to satisfy a global assume-guarantee contract expressed as a pair of reM itl formulas.

Security Management for Mobile Ad Hoc Network of Networks (MANoN)

Many military research efforts have concentrated on how to allow war-fighters to take advantage of all available information within the battlefield in a rapid and flexible manner. As a result, the development of the Global Information Grid (GIG) was the key enabler for this process; hence, adding to the development of the mobile networking part of the GIG, the concept of the Mobile Ad hoc Network of Networks (MANoN) is introduced. This article proposes a novel security management algorithm achieving the three management essentials: Security Administration; Prevention and Detection; and Containment and Recovery; based on the International Telecommunication Union’s recommendation M.3400 to manage securely the future of military Network-Centric Warfare (NCW). The authors will employ Interval Temporal Logic (ITL) as a method of handling both sequential and parallel composition in flexible timely constrains, in addition, this technique will be evaluated using the Network Simulator (NS-2) to provide and check whether security requirements are met in a comprehensive manner.

A duration calculus with neighborhood modalities

To reason about continuous processes in some areas of artificial intelligence and embedded systems one has to express real-time properties. For such purpose a real-time logic has to be considered. Various such logics have been proposed. Some of these formalisms interpret formulas over intervals of time. These are called interval logics. Zhou Chaochen and Michael Hansen have introduced one such first-order interval logic called Neighborhood Logic (NL) which has two expanding modalities ◊r and ◊l. They have shown the adequacy of these modalities by deriving other unary and binary modalities from them; in particular, they show that chop can be expressed in terms of these modalities. The logic is subsequently expanded to get a Duration Calculus (called DC/NL) by expressing temporal variables as integrals of state variables. Liveness and fairness properties of a computing system can now be expressed in DC/NL by means of neighborhood (expanding) modalities. We take up an investigation of DC/NL in detail. We present a proof system for it. We further show that most of the results that hold in the original Duration Calculus continue to hold in our logic with some modifications. In this process we discuss soundness, completeness, and decidability results for DC/NL. Thus this paper establishes DC/NL as a useful formalism for specifying and verifying properties of real-time computing systems.

Propositional interval neighborhood logics: Expressiveness, decidability, and undecidable extensions

In this paper, we investigate the expressiveness of the variety of propositional interval neighborhood logics (PNL), we establish their decidability on linearly ordered domains and some important subclasses, and we prove the undecidability of a number of extensions of PNL with additional modalities over interval relations. All together, we show that PNL form a quite expressive and nearly maximal decidable fragment of Halpern–Shoham’s interval logic HS.

Robustness of temporal logic specifications for continuous-time signals

In this paper, we consider the robust interpretation of Metric Temporal Logic (MTL) formulas over signals that take values in metric spaces. For such signals, which are generated by systems whose states are equipped with non-trivial metrics, for example continuous or hybrid, robustness is not only natural, but also a critical measure of system performance. Thus, we propose multi-valued semantics for MTL formulas, which capture not only the usual Boolean satisfiability of the formula, but also topological information regarding the distance, ε , from unsatisfiability. We prove that any other signal that remains ε -close to the initial one also satisfies the same MTL specification under the usual Boolean semantics. Finally, our framework is applied to the problem of testing formulas of two fragments of MTL, namely Metric Interval Temporal Logic (MITL) and closed Metric Temporal Logic (clMTL), over continuous-time signals using only discrete-time analysis. The motivating idea behind our approach is that if the continuous-time signal fulfills certain conditions and the discrete-time signal robustly satisfies the temporal logic specification, then the corresponding continuous-time signal should also satisfy the same temporal logic specification.

Complete and Terminating Tableau for the Logic of Proper Subinterval Structures Over Dense Orderings

We introduce special pseudo-models for the interval logic of proper subintervals over dense linear orderings. We prove finite model property with respect to such pseudo-models, and using that result we develop a decision procedure based on a sound, complete, and terminating tableau for that logic. The case of proper subintervals is essentially more complicated than the case of strict subintervals, for which we developed a similar tableau-based decision procedure in a recent work.

Exploration of the Dendritic Cell Algorithm with the Duration Calculus

As one of the newest members in Artificial Immune Systems (AIS), the Dendritic Cell Algorithm (DCA) has been applied to a range of problems. These applications mainly belong to the field of anomaly detection. However, real-time detection, a new challenge to anomaly detection, requires improvement on the real-time capability of the DCA. To assess such capability, formal methods in the research of real-time systems can be employed. The findings of the assessment can provide guideline for the future development of the algorithm. Therefore, in this paper we use an interval logic based method, named the Duration Calculus (DC), to specify a simplified single-cell model of the DCA. Based on the DC specifications with further induction, we find that each individual cell in the DCA can perform its function as a detector in real-time. Since the DCA can be seen as many such cells operating in parallel, it is potentially capable of performing real-time detection. However, the analysis process of the standard DCA constricts its real-time capability. As a result, we conclude that the analysis process of the standard DCA should be replaced by a real-time analysis component, which can perform periodic analysis for the purpose of real-time detection.

MILPRIT*

In this article, we consider a new kind of temporal pattern where both interval and punctual time representation are considered. These patterns, which we call temporal point-interval patterns, aim at capturing how events taking place during different time periods or at different time instants relate to each other. The datasets where these kinds of patterns may appear are temporal relational databases whose relations contain point or interval timestamps. We use a simple extension of Allen’s Temporal Interval Logic as a formalism for specifying these temporal patterns. We also present the algorithm MILPRIT* for mining temporal point-interval patterns, which uses variants of the classical levelwise search algorithms. In addition, MILPRIT* allows a broad spectrum of constraints to be incorporated into the mining process. An extensive set of experiments of MILPRIT* executed over synthetic and real data is presented, showing its effectiveness for mining temporal relational patterns.

Timed automata for metric interval temporal logic formulae in prototype verification system

Based on analysis of the syntax structure and semantics model of the metric interval temporal logic (MITL) formulas, it is shown how to transform a formula written in the real-time temporal logic MITL formula into a fair timed automaton (TA) that recognizes its satisfying models with prototype verification system (PVS) in this paper. Both the tabular construction’s principles and the PVS implementation details are given for the different type of MITL formula according to the corresponding semantics interpretations. After this transformation procedure, specifications expressed with MITL formula can be verified formally in the timed automata framework developed previously.

Application of consistency checking to evaluation of uncertainty in multiple replicate measurements

Use of repeated measurements in quantitative chemical analysis is common but leads to the problem of how to combine the measurement values and produce a result with an uncertainty following the GUM. There is often confusion between repeated indications or observations of an input quantity, for whose uncertainty the GUM prescribes a type A evaluation, and complete measurements repeated on multiple sub-samples, as considered here. A solution for combining repeated measurement results and their individual uncertainties based on simple interval logic is proposed here. The individual measurement values and their uncertainties are compared with the calculated average value to see if this implies that another, possibly unknown, source of uncertainty is present. The model of the individual results is modified for this possible between-replicate effect so that the repeated measurements are consistent. Lack of consistency is a strong indication that the measurement is not fully under control and needs further development or investigation. This is not always possible, however and the method given here is proposed to ensure that the values of the repeated measurements agree with each other. A simple numerical example is given showing how the method can be implemented in practice.

Interval rough mereology and description logic: An approach to formal treatment of imprecision in the Semantic Web ontologies

Learning and maintaining ontologies play critically important roles in the Semantic Web. Ontologies provide formal and explicit semantics to available knowledge and make possible shared understanding of a domain. A great deal of knowledge in the Semantic Web involves different sorts of imprecision. Common examples of imprecise knowledge include various statistical hypotheses, vagueness and ambiguities of natural language and many others. Consequently, ontologies must be expressive enough to handle imprecision. This paper presents an approach to formal handling of granularity as a special kind of imprecision in ontologies. It is based on Soft Computing techniques, in particular, Rough Set and Fuzzy Set theories. The approach also makes heavy use of Rough Mereology - a recently proposed paradigm for approximate reasoning - to compute degrees of imprecision of ontological concepts, relationships and assertions. It is demonstrated how the proposed mereological approach can be used together with the framework for representing uncertainty in Description Logic - family of popular logics for specifying formal ontologies. Finally, the paper introduces interval computations to the area of representing and reasoning with ontologies and shows how intervals can formally capture the imprecision incurred by computing rough inclusion (mereological) functions.

A decision procedure for propositional projection temporal logic with infinite models

This paper investigates the satisfiability of Propositional Projection Temporal Logic (PPTL) with infinite models. A decision procedure for PPTL formulas is given. To this end, Normal Form (NF) and Labeled Normal Form Graph (LNFG) for PPTL formulas are defined, and algorithms for transforming a formula to its normal form and constructing the LNFG for the given formula are presented. Further, the finiteness of LNFGs is proved in details. Moreover, the decision procedure is extended to check the satisfiability of the formulas of Propositional Interval Temporal Logic. In addition, examples are also given to illustrate how the decision procedure works.

Semiring Neighbours: An Algebraic Embedding and Extension of Neighbourhood Logic

In 1996 Zhou and Hansen proposed a first-order interval logic called Neighbourhood Logic (NL) for specifying liveness and fairness of computing systems and defining notions of real analysis in terms of expanding modalities. After that, Roy and Zhou developed a sound and relatively complete Duration Calculus as an extension of NL. We present an embedding of NL into an idempotent semiring of intervals. This embedding allows us to extend NL from single intervals to sets of intervals as well as to extend the approach to arbitrary idempotent semirings. We show that most of the required properties follow directly from Galois connections, hence we get many properties for free. As one important result we obtain that some of the axioms which were postulated for NL can be dropped since they are theorems in our generalisation. Furthermore, we discuss other interval operations like Allen's 13 relations between intervals and their relationship to semiring neighbours. Then we present some possible interpretations for neighbours beyond interval settings. Here we discuss for example reachability in graphs and applications to hybrid systems. At the end of the paper we add finite and infinite iteration to NL and extend idempotent semirings to Kleene algebras and ω algebras. These extensions are useful for formulating properties of repetitive procedures like loops.

Probabilistic Interval Temporal Logic and Duration Calculus with Infinite Intervals: Complete Proof Systems

The paper presents probabilistic extensions of interval temporal logic (ITL) and duration calculus (DC) with infinite intervals and complete Hilbert-style proof systems for them. The completeness results are a strong completeness theorem for the system of probabilistic ITL with respect to an abstract semantics and a relative completeness theorem for the system of probabilistic DC with respect to real-time semantics. The proposed systems subsume probabilistic real-time DC as known from the literature. A correspondence between the proposed systems and a system of probabilistic interval temporal logic with finite intervals and expanding modalities is established too.

Using Temporal Logic to Analyse Temporal Logic: A Hierarchical Approach Based on Intervals

Temporal logic has been extensively utilized in academia and industry to formally specify and verify behavioural properties of numerous kinds of hardware and software. We present a novel way to apply temporal logic to the study of a version of itself, namely, propositional linear-time temporal logic (PTL). This involves a hierarchical framework for obtaining standard results for PTL, including a small model property, decision procedures and axiomatic completeness. A large number of the steps involved are expressed in a propositional version of Interval Temporal Logic (ITL) which is referred to as PITL. It is a natural generalization of PTL and includes operators for reasoning about periods of time and sequential composition. Versions of PTL with finite time and infinite time are both considered and one benefit of the framework is the ability to systematically reduce infinite-time reasoning to finite-time reasoning. The treatment of PTL with the operator until and past time naturally reduces to that for PTL without either one. The interval-oriented methodology differs from other analyses of PTL which typically use sets of formulas and sequences of such sets for canonical models. Instead, we represent models as time intervals expressible in PITL. The analysis furthermore relates larger intervals with smaller ones. Being an interval-based formalism, PITL is well suited for sequentially combining and decomposing the relevant formulas. Consequently, we can articulate issues of equal significance in more conventional analyses of PTL but normally only considered at the metalevel. A good example of this is the existence of bounded models with periodic suffixes for PTL formulas which are satisfiable in infinite time. We also describe decision procedures based on binary decision diagrams and exploit some links with finite-state automata. Beyond the specific issues involving PTL, the research is a significant application of ITL and interval-based reasoning and illustrates a general approach to formally reasoning about sequential and parallel behaviour in discrete linear time. The work also includes some interesting representation theorems. In addition, it has relevance to hardware description and verification since the specification languages PSL/Sugar (IEEE Standard 1850) and ‘temporal e’ (part of IEEE Standard 1647) both contain temporal constructs concerning intervals of time as does the related SystemVerilog Assertion language contained in SystemVerilog (IEEE Standard 1800), an extension of the IEEE 1364–2001 Verilog language.

A New Modal Approach to the Logic of Intervals

In Artificial Intelligence there is a need for reasoning about continuous processes, where assertions refer to time intervals rather than time points. Taking our lead from van Benthem’s treatment of interval temporal structures and Halpern and Shoham’s work on intervals, we present a interval temporal logic with two binary relations, precedence and inclusion. We study the logic in its full generality without making any assumptions about the underlying nature of time, be it discrete or dense, linear or branching. We identify two general classes of interval temporal structures, minimal interval structures and van Benthem minimal interval structures. We show that in our interval temporal language, the two classes in fact have the same logic. We go on to prove that the logic of minimal interval structures is complete and decidable, possessing the finite model property, and that the satisfiability problem is PSPACEcomplete. In order to establish the complexity result we extend the tableau method introduced by Horrocks et al., which treats transitive and inverse relations only, to also incorporate interaction between relations. We go on to identify some important limitations in the expressive power of our logic, before concluding the paper by raising a number of interesting questions that follow from our work.

GENERATING PROPERTIES FOR RUNTIME MONITORING FROM SOFTWARE SPECIFICATION PATTERNS

This paper presents an approach to support run-time verification of software systems that combines two existing tools, Prospec and Java-MaC, into a single framework. Prospec can be used to clarify natural language specifications for sequential, concurrent, and nondeterministic behavior. In addition, Prospec assists the user in reading, writing, and understanding formal specifications through the use of property patterns and visual abstractions. Prospec automatically generates specifications written in Future Interval Logic (FIL). Java-MaC monitors Java programs at runtime to ensure adherence to a set of formally specified properties. Safety properties of a program are specified in the formal language Meta-Event Definition Language (MEDL). Java-MaC generates runtime components from specifications. The components are used to instrument the target program and determine whether the execution of the program violates any of the safety properties. This paper describes an algorithm for translating FIL formulas into MEDL formulas. It provides the transformation rules used by this algorithm, and it demonstrates the general correctness of the translation.