A Characterization Theorem for a Modal Description Logic

Modal description logics feature modalities that capture dependence of knowledge on parameters such as time, place, or the information state of agents. E.g., the logic S5-ALC combines the standard description logic ALC with an S5-modality that can be understood as an epistemic operator or as representing (undirected) change. This logic embeds into a corresponding modal first-order logic S5-FOL. We prove a modal characterization theorem for this embedding, in analogy to results by van Benthem and Rosen relating ALC to standard first-order logic: We show that S5-ALC with only local roles is, both over finite and over unrestricted models, precisely the bisimulation invariant fragment of S5-FOL, thus giving an exact description of the expressive power of S5-ALC with only local roles

A cookbook for temporal conceptual data modelling with description logic

We design temporal description logics suitable for reasoning about temporal conceptual data models and investigate their computational complexity. Our formalisms are based on DL-Lite logics with three types of concept inclusions (ranging from atomic concept inclusions and disjointness to the full Booleans), as well as cardinality constraints and role inclusions. In the temporal dimension, they capture future and past temporal operators on concepts, flexible and rigid roles, the operators `always' and `some time' on roles, data assertions for particular moments of time and global concept inclusions. The logics are interpreted over the Cartesian products of object domains and the flow of time (Z,<), satisfying the constant domain assumption. We prove that the most expressive of our temporal description logics (which can capture lifespan cardinalities and either qualitative or quantitative evolution constraints) turn out to be undecidable. However, by omitting some of the temporal operators on concepts/roles or by restricting the form of concept inclusions we obtain logics whose complexity ranges between PSpace and NLogSpace. These positive results were obtained by reduction to various clausal fragments of propositional temporal logic, which opens a way to employ propositional or first-order temporal provers for reasoning about temporal data models

Combining DL-LiteNbool with branching time: a gentle marriage

We study combinations of the description logic DL-Lite_{bool}^N with the branching temporal logics CTL* and CTL. We analyse two types of combinations, both with rigid roles: (i) temporal operators are applied to concepts and to ABox assertions, and (ii) temporal operators are applied to concepts and Boolean combinations of concept inclusions and ABox assertions. For the resulting logics, we present algorithms for the satisfiability problem and (mostly tight) complexity bounds ranging from ExpTime to 3ExpTime

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

LTL over Description Logic Axioms

Most of the research on temporalized Description Logics (DLs) has concentrated on the case where temporal operators can occur within DL concept descriptions. In this setting, reasoning usually becomes quite hard if rigid roles, i.e., roles whose interpretation does not change over time, are available. In this paper, we consider the case where temporal operators are allowed to occur only in front of DL axioms (i.e., ABox assertions and general concept inclusion axioms), but not inside of concepts descriptions. As the temporal component, we use linear temporal logic (LTL) and in the DL component we consider the basic DL ALC. We show that reasoning in the presence of rigid roles becomes considerably simpler in this setting

Evolving Objects in Temporal Information Systems

This paper presents a semantic foundation of temporal conceptual models used to design temporal information systems. We consider a modelling language able to express both timestamping and evolution constraints. We conduct a deeper investigation of evolution constraints, eventually devising a model-theoretic semantics for a full-fledged model with both timestamping and evolution constraints. The proposed formalization is meant both to clarify the meaning of the various temporal constructors that appeared in the literature and to give a rigorous definition, in the context of temporal information systems, to notions like satisfiability, subsumption and logical implication. Furthermore, we show how to express temporal constraints using a subset of first-order temporal logic, i.e. DLRUS, the description logic DLR extended with the temporal operators Since and Until. We show how DLRUS is able to capture the various modelling constraints in a succinct way and to perform automated reasoning on temporal conceptual models

Description logics of context

We introduce Description Logics of Context (DLCs)—an extension of Description Logics (DLs) for context-based reasoning. Our approach descends from J. McCarthy's tradition of treating contexts as formal objects over which one can quantify and express first-order properties. DLCs are founded in two-dimensional possible world semantics, where one dimension represents a usual object domain and the other a domain of contexts, and accommodate two interacting DL languages—the object and the context language—interpreted over their respective domains. Effectively, DLCs comprise a family of two-sorted , two-dimensional combinations of pairs of DLs. We argue that this setup ensures a well-grounded, generic framework for capturing and studying mechanisms of contextualization in the DL paradigm. As the main technical contribution, we prove 2ExpTime-completeness of the satisfiability problem in the maximally expressive DLC, based on the DL forumla . As an interesting corollary, we show that under certain conditions this result holds also for a range of two-dimensional DLs, including the prominent forumla

Probabilistic description logics for subjective uncertainty

We propose a family of probabilistic description logics (DLs) that are derived in a principled way from Halpern's probabilistic first-order logic. The resulting probabilistic DLs have a two-dimensional semantics similar to temporal DLs and are well-suited for representing subjective probabilities. We carry out a detailed study of reasoning in the new family of logics, concentrating on probabilistic extensions of the DLs ALC and EL, and showing that the complexity ranges from PTime via ExpTime and 2ExpTime to undecidable