Algebraic foundations for qualitative calculi and networks

A qualitative representation $\phi$ is like an ordinary representation of a relation algebra, but instead of requiring $(a; b)^\phi = a^\phi | b^\phi$, as we do for ordinary representations, we only require that $c^\phi\supseteq a^\phi | b^\phi \iff c\geq a ; b$, for each $c$ in the algebra. A constraint network is qualitatively satisfiable if its nodes can be mapped to elements of a qualitative representation, preserving the constraints. If a constraint network is satisfiable then it is clearly qualitatively satisfiable, but the converse can fail. However, for a wide range of relation algebras including the point algebra, the Allen Interval Algebra, RCC8 and many others, a network is satisfiable if and only if it is qualitatively satisfiable. Unlike ordinary composition, the weak composition arising from qualitative representations need not be associative, so we can generalise by considering network satisfaction problems over non-associative algebras. We prove that computationally, qualitative representations have many advantages over ordinary representations: whereas many finite relation algebras have only infinite representations, every finite qualitatively representable algebra has a finite qualitative representation; the representability problem for (the atom structures of) finite non-associative algebras is NP-complete; the network satisfaction problem over a finite qualitatively representable algebra is always in NP; the validity of equations over qualitative representations is co-NP-complete. On the other hand we prove that there is no finite axiomatisation of the class of qualitatively representable algebras.Comment: 22 page

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

Spatial reasoning with RCC8 and connectedness constraints in Euclidean spaces

The language RCC8 is a widely-studied formalism for describing topological arrangements of spatial regions. The variables of this language range over the collection of non-empty, regular closed sets of n-dimensional Euclidean space, here denoted RC+(R^n), and its non-logical primitives allow us to specify how the interiors, exteriors and boundaries of these sets intersect. The key question is the satisfiability problem: given a finite set of atomic RCC8-constraints in m variables, determine whether there exists an m-tuple of elements of RC+(R^n) satisfying them. These problems are known to coincide for all n ≥ 1, so that RCC8-satisfiability is independent of dimension. This common satisfiability problem is NLogSpace-complete. Unfortunately, RCC8 lacks the means to say that a spatial region comprises a ‘single piece’, and the present article investigates what happens when this facility is added. We consider two extensions of RCC8: RCC8c, in which we can state that a region is connected, and RCC8c0, in which we can instead state that a region has a connected interior. The satisfiability problems for both these languages are easily seen to depend on the dimension n, for n ≤ 3. Furthermore, in the case of RCC8c0, we show that there exist finite sets of constraints that are satisfiable over RC+(R^2), but only by ‘wild’ regions having no possible physical meaning. This prompts us to consider interpretations over the more restrictive domain of non-empty, regular closed, polyhedral sets, RCP+(R^n). We show that (a) the satisfiability problems for RCC8c (equivalently, RCC8c0) over RC+(R) and RCP+(R) are distinct and both NP-complete; (b) the satisfiability problems for RCC8c over RC+(R^2) and RCP+(R^2) are identical and NP-complete; (c) the satisfiability problems for RCC8c0 over RC+(R^2) and RCP+(R^2) are distinct, and the latter is NP-complete. Decidability of the satisfiability problem for RCC8c0 over RC+(R^2) is open. For n ≥ 3, RCC8c and RCC8c0 are not interestingly different from RCC8. We finish by answering the following question: given that a set of RCC8c- or RCC8c0-constraints is satisfiable over RC+(R^n) or RCP+(R^n), how complex is the simplest satisfying assignment? In particular, we exhibit, for both languages, a sequence of constraints Φ_n, satisfiable over RCP+(R^2), such that the size of Φ_n grows polynomially in n, while the smallest configuration of polygons satisfying Φ_n cuts the plane into a number of pieces that grows exponentially. We further show that, over RC+(R^2), RCC8c again requires exponentially large satisfying diagrams, while RCC8c0 can force regions in satisfying configurations to have infinitely many components

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