Two-dimensional description logics of context

Description Logics (DLs) provide a clear and broadly accepted paradigm for rea- soning about terminological knowledge. Under the standard Kripkean semantics, a DL ontology forces a unique, global view on the represented world, in which the ontology axioms are interpreted as universally true. This philosophy is well- suited as long as everyone can share the same conceptual perspective on the domain or there is no need for considering alternative viewpoints. Alas, this is hardly ever the case since a domain can be modeled dierently depending on the intended use of an ontology. Consequently, eective representation and reasoning about knowledge pertaining to such multiple, heterogenous viewpoints becomes the primary objective for many practical applications [1,2]. The challenges above resemble clearly those problems that originally inspired J. McCarthy to introduce a theory of formalizing contexts in knowledge repre- sentation systems, as a way of granting them more generality [3,4]. The gist of his proposal is to replace logical formulas ', as the basic knowledge carriers, with assertions ist(c; ') stating that ' is true in c, where c denotes an abstract first- order entity called a context, which on its own can be described in a first-order language

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

Towards More Useful Description Logics of Time, Change and Context

Description Logics (DLs) are a family of logic-based formalisms for the representation of and reasoning about knowledge. Classical DLs are fragments of first-order logic and therefore aim at capturing static knowledge. Alas, the lack of means of DLs to capture dynamic aspects of knowledge has been often criticized because many important DL applications depend on this kind of knowledge. As a reaction to this shortcoming of DLs, two-dimensional extensions of DLs with capabilities to represent and reason about dynamic knowledge were introduced. We further, in this thesis, the understanding and utility of two-dimensional DLs. We particularly focus on identifying two-dimensional DLs providing the right expressive power to model more accurately temporal and contextual aspects of knowledge required by certain DL applications, or providing better computational properties than other possible alternatives. We pursue three lines of research: we study branching-time temporal DLs that emerge from the combination of classical DLs with the classical temporal logics CTL* and CTL; we study description logics of change that emerge from the combination of classical DLs with the modal logic S5; we study description logics of context that emerge from the combination of classical DLs with multi-modal logics. We investigate temporal and contextual DLs based on the classical DL ALC and on members of the EL-family of DLs. Our main technical contributions are algorithms for satisfiability and subsumption, and (mostly) tight complexity bounds

Towards More Useful Description Logics of Time, Change and Context

Description Logics (DLs) are a family of logic-based formalisms for the representation of and reasoning about knowledge. Classical DLs are fragments of first-order logic and therefore aim at capturing static knowledge. Alas, the lack of means of DLs to capture dynamic aspects of knowledge has been often criticized because many important DL applications depend on this kind of knowledge. As a reaction to this shortcoming of DLs, two-dimensional extensions of DLs with capabilities to represent and reason about dynamic knowledge were introduced. We further, in this thesis, the understanding and utility of two-dimensional DLs. We particularly focus on identifying two-dimensional DLs providing the right expressive power to model more accurately temporal and contextual aspects of knowledge required by certain DL applications, or providing better computational properties than other possible alternatives. We pursue three lines of research: we study branching-time temporal DLs that emerge from the combination of classical DLs with the classical temporal logics CTL* and CTL; we study description logics of change that emerge from the combination of classical DLs with the modal logic S5; we study description logics of context that emerge from the combination of classical DLs with multi-modal logics. We investigate temporal and contextual DLs based on the classical DL ALC and on members of the EL-family of DLs. Our main technical contributions are algorithms for satisfiability and subsumption, and (mostly) tight complexity bounds

On the uniform one-dimensional fragment

The uniform one-dimensional fragment of first-order logic, U1, is a recently introduced formalism that extends two-variable logic in a natural way to contexts with relations of all arities. We survey properties of U1 and investigate its relationship to description logics designed to accommodate higher arity relations, with particular attention given to DLR_reg. We also define a description logic version of a variant of U1 and prove a range of new results concerning the expressivity of U1 and related logics

Description Logics of Context with Rigid Roles Revisited

To represent and reason about contextualized knowledge often two-dimensional Description Logics (DLs) are employed, where one DL is used to describe contexts (or possible worlds) and the other DL is used to describe the objects, i.e. the relational structure of the specific contexts. Previous approaches for DLs of context that combined pairs of DLs resulted in undecidability in those cases where so-called rigid roles are admitted, i.e. if parts of the relational structure are the same in all contexts. In this paper, we present a novel combination of pairs of DLs and show that reasoning stays decidable even in the presence of rigid roles. We give complexity results for various combinations of DLs involving ALC, SHOQ, and EL