A decidable multi-modal logic of context

We give a logic for formulas Á¡± Ã, with the informal reading ”à is true in the context described by Á”. These are interpreted as binary modalities, by quantification over an enumerable set of unary modalities c¡± Ã, meaning ”à is true in context c”. The logic allows arbitrary nesting of contexts. A corresponding axiomatic presentation is given, and proven to be decidable, sound, and complete. Previously, quantificational logic of context restricted the nesting of contexts, and was only known to be decidable in very special cases

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

A uniform quantificational logic for algebraic notions ofcontext

A quantificational framework of formal reasoning is proposed, which emphasises the pattern of entering and exiting context. Contexts are modelled by an algebraic structure which reflects the order and manner in which context is entered into and exited from. The equations of the algebra partitions context terms into equivalence classes. A formal semantics is defined, containing models that map equivalence classes of certain context terms to sets of first order structures. The corresponding Hilbert system incorporates the algebraic equations as axioms asserted in context. In this way a uniform logic for arbitrary algebras of context is obtained. Soundness and completeness are proved. In semigroups of contexts, where combination of contexts is associative, finite ground algebraic equations correspond to contingent equivalence between certain logical formulas. Systems for sets and multisets of contexts are obtained by presenting their respective algebras as associativity plus finite ground equations. Some contextual reasoning systems in the literature are inherently associative, and we present those as special cases

ALC_ALC: A context description logic

We develop a novel description logic (DL) for representing and reasoning with contextual knowledge. Our approach descends from McCarthy’s tradition of treating contexts as formal objects over which one can quantify and express first-order properties. As a foundation we consider several common product-like combinations of DLs with multimodal logics and adopt the prominent (Kn)ALC. We then extend it with a second sort of vocabulary for describing contexts, i.e., objects of the second dimension. In this way, we obtain a two-sorted, two-dimensional combination of a pair of DLs ALC, called ALCALC. As our main technical result, we show that the satisfiability problem in this logic, as well as in its proper fragment (Kn)ALC with global TBoxes and local roles, is 2ExpTime- complete. Hence, the surprising conclusion is that the significant increase in the expressiveness of ALCALC due to adding the vocabulary comes for no substantial price in terms of its worst-case complexity

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

Reasoning with Contexts in Description Logics

Harmelen, F.A.H. van [Promotor]Schlobach, K.S. [Copromotor