Constructive interval temporal logic in Alf

This paper gives an implementation of an interval temporal logic in a constructive type theory, using the Alf proof system. After explaining the constructive approach, its relevance to interval temporal logic and potential applications of our work, we explain the fundamentals of the Alf system. We then present the implementation of the logic and give a number of examples of its use. We conclude by exploring how the work can be extended in the future

A Tableau Method for Interval Temporal Logic

In this paper we present a complete tableau method for interval temporal logic including the projection operator. Central to our strategy is the identification of normal forms for all the operators of our logic. In effect, these normal forms give inductive definitions of the ITL operators. Then, in the style of Wolper, we define a tableau decision procedure to check satisfiability of our logic. For simplicity of presentation we work in the propositional setting

Mexitl: Multimedia in Executable Interval Temporal Logic

This paper explores a formalism for describing a wide class of multimedia document constraints, based on an interval temporal logic. We describe the requirements on temporal logic specification that arise from the multimedia documents application area. In particular, we highlight a canonical specification example. Then we present the temporal logic formalism that we use. This extends existing interval temporal logic with a number of new features: actions, framing of actions, past operators, a projection-like operator called filter and a new handling of interval length. A model theory, logic and satisfaction relation are defined for the notation, a specification of the canonical example is presented, and a proof system for the logic is introduced

Prompt interval temporal logic

Interval temporal logics are expressive formalisms for temporal representation and reasoning, which use time intervals as primitive temporal entities. They have been extensively studied for the past two decades and successfully applied in AI and computer science. Unfortunately, they lack the ability of expressing promptness conditions, as it happens with the commonly-used temporal logics, e.g., LTL: whenever we deal with a liveness request, such as \u201csomething good eventually happens\u201d, there is no way to impose a bound on the delay with which it is fulfilled. In the last years, such an issue has been addressed in automata theory, game theory, and temporal logic. In this paper, we approach it in the interval temporal logic setting. First, we introduce PROMPT-PNL, a prompt extension of the well-studied interval temporal logic PNL, and we prove the undecidability of its satisfiability problem; then, we show how to recover decidability (NEXPTIME-completeness) by imposing a natural syntactic restriction on it

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.Comment: 43 page

Runtime validation using interval temporal logic

Formal specifications are one of the design choices in reactive and/or real-time systems as a number of notations exist to formally define parts of the system. However, defining the system formally is not enough to guarantee correctness thus the specifications are used as execution monitors over the system. A number of projects are around that provides a framework to define execution monitors in Interval Temporal Logic (ITL), such as Temporal-Rover, EAGLE Flier, and D3CA framework. This paper briefly describes the D3CA framework, consisting in the adaptation of Quantified Discrete-Time Duration Calculus to monitoring assertions. The D3CA framework uses the synchronous data-flow programming language Lustre as a generic platform for defining the notation. Additionally, Lustre endows the framework with the ability to predetermine the space and time requirements of the monitoring system. After defining the notation framework the second part of the paper presents two case studies - a mine pump and an answering machine. The case studies illustrate the power endowed by using ITL observers in a reactive or event-driven system.peer-reviewe

Begin, After, and Later: a Maximal Decidable Interval Temporal Logic

Interval temporal logics (ITLs) are logics for reasoning about temporal statements expressed over intervals, i.e., periods of time. The most famous ITL studied so far is Halpern and Shoham's HS, which is the logic of the thirteen Allen's interval relations. Unfortunately, HS and most of its fragments have an undecidable satisfiability problem. This discouraged the research in this area until recently, when a number non-trivial decidable ITLs have been discovered. This paper is a contribution towards the complete classification of all different fragments of HS. We consider different combinations of the interval relations Begins, After, Later and their inverses Abar, Bbar, and Lbar. We know from previous works that the combination ABBbarAbar is decidable only when finite domains are considered (and undecidable elsewhere), and that ABBbar is decidable over the natural numbers. We extend these results by showing that decidability of ABBar can be further extended to capture the language ABBbarLbar, which lays in between ABBar and ABBbarAbar, and that turns out to be maximal w.r.t decidability over strongly discrete linear orders (e.g. finite orders, the naturals, the integers). We also prove that the proposed decision procedure is optimal with respect to the complexity class

From TimeML to TPL

This paper describes a subset of the temporal mark-up language TimeML, and explains its relation to various formalisms found in the literature on interval temporal logic. The subset of TimeML we describe can be viewed as an interval temporal logic with a tractable satisfiability problem, but very limited expressive power. Most crucially, that logic does not permit quantification over events. The contribution of this paper is to point out that, by choosing an appropriate interval temporal logic, it is possible to introduce quantification into representations of event-structure without sacrificing decidability