Towards Log-Linear Logics with Concrete Domains

We present $\mathcal{MEL}^{++}$ (M denotes Markov logic networks) an extension of the log-linear description logics $\mathcal{EL}^{++}$-LL with concrete domains, nominals, and instances. We use Markov logic networks (MLNs) in order to find the most probable, classified and coherent $\mathcal{EL}^{++}$ ontology from an $\mathcal{MEL}^{++}$ knowledge base. In particular, we develop a novel way to deal with concrete domains (also known as datatypes) by extending MLN's cutting plane inference (CPI) algorithm.Comment: StarAI201

Using Model Theory to Find Decidable and Tractable Description Logics with Concrete Domains

Concrete domains have been introduced in the area of Description Logic (DL) to enable reference to concrete objects (such as numbers) and predefined predicates on these objects (such as numerical comparisons) when defining concepts. Unfortunately, in the presence of general concept inclusions (GCIs), which are supported by all modern DL systems, adding concrete domains may easily lead to undecidability. To regain decidability of the DL ALC in the presence of GCIs, quite strong restrictions, called ω-admissibility, were imposed on the concrete domain. On the one hand, we generalize the notion of ω-admissibility from concrete domains with only binary predicates to concrete domains with predicates of arbitrary arity. On the other hand, we relate ω-admissibility to well-known notions from model theory. In particular, we show that finitely bounded homogeneous structures yield ω-admissible concrete domains. This allows us to show ω-admissibility of concrete domains using existing results from model theory. When integrating concrete domains into lightweight DLs of the EL family, achieving decidability of reasoning is not enough. One wants the resulting DL to be tractable. This can be achieved by using so-called p-admissible concrete domains and restricting the interaction between the DL and the concrete domain. We investigate p-admissibility from an algebraic point of view. Again, this yields strong algebraic tools for demonstrating p-admissibility. In particular, we obtain an expressive numerical p-admissible concrete domain based on the rational numbers. Although ω-admissibility and p-admissibility are orthogonal conditions that are almost exclusive, our algebraic characterizations of these two properties allow us to locate an infinite class of p-admissible concrete domains whose integration into ALC yields decidable DLs. DL systems that can handle concrete domains allow their users to employ a fixed set of predicates of one or more fixed concrete domains when modelling concepts. They do not provide their users with means for defining new predicates, let alone new concrete domains. The good news is that finitely bounded homogeneous structures offer precisely that. We show that integrating concrete domains based on finitely bounded homogeneous structures into ALC yields decidable DLs even if we allow predicates specified by first-order formulas. This class of structures also provides effective means for defining new ω-admissible concrete domains with at most binary predicates. The bad news is that defining ω-admissible concrete domains with predicates of higher arities is computationally hard. We obtain two new lower bounds for this meta-problem, but leave its decidability open. In contrast, we prove that there is no algorithm that would facilitate defining p-admissible concrete domains already for binary signatures.:1. Introduction . . . 1 2. Preliminaries . . . 5 3. Description Logics with Concrete Domains . . . 9 3.1. Basic definitions and undecidability results . . . 9 3.2. Decidable and tractable DLs with concrete domains . . . 16 4. A Model-Theoretic Analysis of ω-Admissibility . . . 23 4.1. Homomorphism ω-compactness via ω-categoricity . . . 23 4.2. Patchworks via homogeneity . . . 24 4.3. JDJEPD via decomposition into orbits . . . 27 4.4. Upper bounds via finite boundedness . . . 28 4.5. ω-admissible finitely bounded homogeneous structures . . . 32 4.6. ω-admissible homogeneous cores with a decidable CSP . . . 34 4.7. Coverage of the developed sufficient conditions . . . 36 4.8. Closure properties: homogeneity & finite boundedness . . . 39 5. A Model-Theoretic Analysis of p-Admissibility . . . 47 5.1. Convexity via square embeddings . . . 47 5.2. Convex ω-categorical structures . . . 50 5.3. Convex numerical structures . . . 52 5.4. Ages defined by forbidden substructures . . . 54 5.5. Ages defined by forbidden homomorphic images . . . 56 5.6. (Non-)closure properties of convexity . . . 59 6. Towards user-definable concrete domains . . . 61 6.1. A proof-theoretic perspective . . . 65 6.2. Universal Horn sentences and the JEP . . . 66 6.3. Universal sentences and the AP: the Horn case . . . 77 6.4. Universal sentences and the AP: the general case . . . 90 7. Conclusion . . . 99 7.1. Contributions and future outlook . . . 99 A. Concrete Domains without Equality . . . 103 Bibliography . . . 107 List of figures . . . 115 Alphabetical Index . . . 11

Using model theory to find w-admissible concrete domains

Concrete domains have been introduced in the area of Description Logic to enable reference to concrete objects (such as numbers) and predefined predicates on these objects (such as numerical comparisons) when defining concepts. Unfortunately, in the presence of general concept inclusions (GCIs), which are supported by all modern DL systems, adding concrete domains may easily lead to undecidability. One contribution of this paper is to strengthen the existing undecidability results further by showing that concrete domains even weaker than the ones considered in the previous proofs may cause undecidability. To regain decidability in the presence of GCIs, quite strong restrictions, in sum called w-admissiblity, need to be imposed on the concrete domain. On the one hand, we generalize the notion of w-admissiblity from concrete domains with only binary predicates to concrete domains with predicates of arbitrary arity. On the other hand, we relate w-admissiblity to well-known notions from model theory. In particular, we show that finitely bounded, homogeneous structures yield w-admissible concrete domains. This allows us to show w-admissibility of concrete domains using existing results from model theory

A Tableau Algorithm for DLs with Concrete Domains and GCIs

We identify a general property of concrete domains that is sufficient for proving decidability of DLs equipped with them and GCIs. We show that some useful concrete domains, such as temporal one based on the Allen relations and a spatial one based on the RCC-8 relations, have this property. Then, we present a tableau algorithm for reasoning in DLs equipped with such concrete domains

Keys, Nominals, and Concrete Domains

Many description logics (DLs) combine knowledge representation on an abstract, logical level with an interface to 'concrete' domains such as numbers and strings with built-in predicates such as <, +, and prefix-of. These hybrid DLs have turned out to be quite useful for reasoning about conceptual models of information systems, and as the basis for expressive ontology languages. We propose to further extend such DLs with key constraints that allow the expression of statements like 'US citizens are uniquely identified by their social security number'. Based on this idea, we introduce a number of natural description logics and perform a detailed analysis of their decidability and computational complexity. It turns out that naive extensions with key constraints easily lead to undecidability, whereas more careful extensions yield NEXPTIME-complete DLs for a variety of useful concrete domains

An Algebraic View on p-Admissible Concrete Domains for Lightweight Description Logics: Extended Version

Concrete domains have been introduced in Description Logics (DLs) to enable reference to concrete objects (such as numbers) and predefined predicates on these objects (such as numerical comparisons) when defining concepts. To retain decidability when integrating a concrete domain into a decidable DL, the domain must satisfy quite strong restrictions. In previous work, we have analyzed the most prominent such condition, called w-admissibility, from an algebraic point of view. This provided us with useful algebraic tools for proving w-admissibility, which allowed us to find new examples for concrete domains whose integration leaves the prototypical expressive DL ALC decidable. When integrating concrete domains into lightweight DLs of the EL family, achieving decidability is not enough. One wants reasoning in the resulting DL to be tractable. This can be achieved by using so-called p-admissible concrete domains and restricting the interaction between the DL and the concrete domain. In the present paper, we investigate p-admissibility from an algebraic point of view. Again, this yields strong algebraic tools for demonstrating p-admissibility. In particular, we obtain an expressive numerical padmissible concrete domain based on the rational numbers. Although w-admissibility and p-admissibility are orthogonal conditions that are almost exclusive, our algebraic characterizations of these two properties allow us to locate an infinite class of p-admissible concrete domains whose integration into ALC yields decidable DLs