Minimal Definition Signatures: Computation and Application to Ontology Alignment

In computer science, ontologies define a domain to facilitate knowledge representation and sharing, in a machine processable way. Ontologies approximate an actual world representation, and thus ontologies will differ for many reasons. Therefore knowledge sharing, and in general semantic interoperability, is inherently hindered or even precluded between heterogenous ontologies. Ontology matching addresses this fundamental issue by producing alignments, i.e. sets of correspondences that describe relations between semantically related entities of different ontologies. However, alignments are typically incomplete. In order to support and improve ontology alignment, and semantic interoperability in general, this thesis exploits the notion of implicit definability. Implicit definability is a semantic property of ontologies, signatures, and concepts (and roles) stating that whenever the signature is fixed under a given ontology then the definition of a particular concept (or role) is also fixed. This thesis introduces the notion of minimal definition signature (MDS) from which a given entity is implicitly definable, and presents a novel approach that provides an efficient way to compute in practice all MDSs of the definable entities. Furthermore, it investigates the application of MDSs in the context of alignment generation, evaluation, and negotiation (whereby agents cooperatively establish a mutually acceptable alignment to support opportunistic communication within open environments). As implicit definability permits defined entities to be removed without semantic loss, this thesis argues, that if the meaning of the defined entity is wholly fixed by the terms of its definition, only the terms in the definition are required to be mapped in order to map the defined entity itself; thus implicit definability entails a new type of definability-based correspondence correspondence. Therefore this thesis defines and explores the properties of definability- based correspondences, and extends several ontology alignment evaluation metrics in order to accommodate their assessment. As task signature coverage is a prerequisite of many knowledge-based tasks (e.g. service invocation), a definability-based, efficient approximation approach to obtaining minimal signature cover sets is presented. Moreover, this thesis outlines a specific alignment negotiation approach and shows that by considering definability, agents are better equipped to: (i) determine whether an alignment provides the necessary coverage to achieve a particular task (align the whole ontology, formulate a message or query); (ii) adhere to privacy and confidentiality constraints; and (iii) minimalise the cardinality of the resulting mutual alignment

Living without Beth and Craig: Definitions and Interpolants in the Guarded and Two-Variable Fragments

In logics with the Craig interpolation property (CIP) the existence of an interpolant for an implication follows from the validity of the implication. In logics with the projective Beth definability property (PBDP), the existence of an explicit definition of a relation follows from the validity of a formula expressing its implicit definability. The two-variable fragment, FO2, and the guarded fragment, GF, of first-order logic both fail to have the CIP and the PBDP. We show that nevertheless in both fragments the existence of interpolants and explicit definitions is decidable. In GF, both problems are 3ExpTime-complete in general, and 2ExpTime-complete if the arity of relation symbols is bounded by a constant c not smaller than 3. In FO2, we prove a coN2ExpTime upper bound and a 2ExpTime lower bound for both problems. Thus, both for GF and FO2 existence of interpolants and explicit definitions are decidable but harder than validity (in case of FO2 under standard complexity assumptions)

Living Without Beth and Craig: Definitions and Interpolants in Description and Modal Logics with Nominals and Role Inclusions

The Craig interpolation property (CIP) states that an interpolant for an implication exists iff it is valid. The projective Beth definability property (PBDP) states that an explicit definition exists iff a formula stating implicit definability is valid. Thus, the CIP and PBDP reduce potentially hard existence problems to entailment in the underlying logic. Description (and modal) logics with nominals and/or role inclusions do not enjoy the CIP nor the PBDP, but interpolants and explicit definitions have many applications, in particular in concept learning, ontology engineering, and ontology-based data management. In this article we show that, even without Beth and Craig, the existence of interpolants and explicit definitions is decidable in description logics with nominals and/or role inclusions such as $\mathcal {ALCO}$ , $\mathcal {ALCH}$ and $\mathcal {ALCHOI}$ and corresponding hybrid modal logics. However, living without Beth and Craig makes these problems harder than entailment: the existence problems become 2ExpTime-complete in the presence of an ontology or the universal modality, and coNExpTime-complete otherwise. We also analyze explicit definition existence if all symbols (except the one that is defined) are admitted in the definition. In this case the complexity depends on whether one considers individual or concept names. Finally, we consider the problem of computing interpolants and explicit definitions if they exist and turn the complexity upper bound proof into an algorithm computing them, at least for description logics with role inclusions. </jats:p

Enumerating Query Plans via Conditional Tableau Interpolation

Database query optimization studies the problem of finding equivalent and efficient query execution plans for user queries under schema constraints. Logic-based approaches to query optimization leverage automated theorem proving and Craig interpolation to enumerate query plans that are correct and performance-optimal. In this thesis, we investigate and improve one of the state-of-the-art logic-based query optimizers – the Interpolation Test Bed (ITB). We begin by formally capturing the physical data independence framework and query optimization problem with first-order logic. Then, we give a gentle introduction to the classical results from logic that form the basis of logic-based query optimizers. We re-establish the correctness of ITB’s conditional tableau interpolation mechanism by reduction to free-variable tableau interpolation. To facilitate the reduction proof, we introduce interpolation rules for the free-variable tableau and prove the correctness of interpolation. Then we show the correctness of conditional tableau interpolation by reduction. We investigate a limitation of ITB’s forward chaining design, which causes missing optimal plans. To address this limitation, we propose a rewriting procedure inspired by Magic Set Transformation (MST), to extend the plan space for the current ITB system. We show that the propose rewriting procedure effectively generates the missing query plans, which are otherwise not found, while accommodating the existing forward chaining design

Algebraic proof theory for LE-logics

In this paper we extend the research programme in algebraic proof theory from axiomatic extensions of the full Lambek calculus to logics algebraically captured by certain varieties of normal lattice expansions (normal LE-logics). Specifically, we generalise the residuated frames in [16] to arbitrary signatures of normal lattice expansions (LE). Such a generalization provides a valuable tool for proving important properties of LE-logics in full uniformity. We prove semantic cut elimination for the display calculi D.LE associated with the basic normal LE-logics and their axiomatic extensions with analytic inductive axioms. We also prove the finite model property (FMP) for each such calculus D.LE, as well as for its extensions with analytic structural rules satisfying certain additional properties