Tractable interval temporal propositional and description logics

We design a tractable Horn fragment of the Halpern-Shaham temporal logic and extend it to interval-based temporal description logics, instance checking in which is P-complete for both combined and data complexity

An interval logic for higher-level temporal reasoning

Prior work explored temporal logics, based on classical modal logics, as a framework for specifying and reasoning about concurrent programs, distributed systems, and communications protocols, and reported on efforts using temporal reasoning primitives to express very high level abstract requirements that a program or system is to satisfy. Based on experience with those primitives, this report describes an Interval Logic that is more suitable for expressing such higher level temporal properties. The report provides a formal semantics for the Interval Logic, and several examples of its use. A description of decision procedures for the logic is also included

Interval-based Temporal Reasoning with General TBoxes

From the Motivation: „Description Logics (DLs) are a family of formalisms well-suited for the representation of and reasoning about knowledge. Whereas most Description Logics represent only static aspects of the application domain, recent research resulted in the exploration of various Description Logics that allow to, additionally, represent temporal information, see [4] for an overview. The approaches to integrate time differ in at least two important aspects: First, the basic temporal entity may be a time point or a time interval. Second, the temporal structure may be part of the semantics (yielding a multi-dimensional semantics) or it may be integrated as a so-called concrete domain. Examples for multi-dimensional point-based logics can be find in, e.g., [21;29], while multi-dimensional interval-based logics are used in, e.g., [23;2]. The concrete domain approach needs some more explanation. Concrete domains have been proposed by Baader and Hanschke as an extension of Description Logics that allows reasoning about 'concrete qualities' of the entities of the application domain such as sizes, length, or weights of real-worlds objects [5]. Description Logics with concrete domains do usually not use a fixed concrete domain; instead the concrete domain can be thought of as a parameter to the logic. As was first described in [16], if a 'temporal' concrete domain is employed, then concrete domains may be point-based, interval-based, or both. ...

Metric Temporal Description Logics with Interval-Rigid Names: Extended Version

In contrast to qualitative linear temporal logics, which can be used to state that some property will eventually be satisfied, metric temporal logics allow to formulate constraints on how long it may take until the property is satisfied. While most of the work on combining Description Logics (DLs) with temporal logics has concentrated on qualitative temporal logics, there has recently been a growing interest in extending this work to the quantitative case. In this paper, we complement existing results on the combination of DLs with metric temporal logics over the natural numbers by introducing interval-rigid names. This allows to state that elements in the extension of certain names stay in this extension for at least some specified amount of time

Dyck algebras, interval temporal logic and posets of intervals

We investigate a natural Heyting algebra structure on the set of Dyck paths of the same length. We provide a geometrical description of the operations of pseudocomplement and relative pseudocomplement, as well as of regular elements. We also find a logic-theoretic interpretation of such Heyting algebras, which we call Dyck algebras, by showing that they are the algebraic counterpart of a certain fragment of a classical interval temporal logic (also known as Halpern-Shoham logic). Finally, we propose a generalization of our approach, suggesting a similar study of the Heyting algebra arising from the poset of intervals of a finite poset using Birkh\"off duality. In order to illustrate this, we show how several combinatorial parameters of Dyck paths can be expressed in terms of the Heyting algebra structure of Dyck algebras together with a certain total order on the set of atoms of each Dyck algebra.Comment: 17 pages, 3 figure