On the Complexity and Expressiveness of Description Logics with Counting

Simple counting quantifiers that can be used to compare the number of role successors of an individual or the cardinality of a concept with a fixed natural number have been employed in Description Logics (DLs) for more than two decades under the respective names of number restrictions and cardinality restrictions on concepts. Recently, we have considerably extended the expressivity of such quantifiers by allowing to impose set and cardinality constraints formulated in the quantifier-free fragment of Boolean Algebra with Presburger Arithmetic (QFBAPA) on sets of role successors and concepts, respectively. We were able to prove that this extension does not increase the complexity of reasoning. In the present paper, we investigate the expressive power of the DLs obtained in this way, using appropriate bisimulation characterizations and 0–1 laws as tools to differentiate between the expressiveness of different logics. In particular, we show that, in contrast to most classical DLs, these logics are no longer expressible in first-order predicate logic (FOL), and we characterize their first-order fragments. In most of our previous work on DLs with QFBAPA-based set and cardinality constraints we have employed finiteness restrictions on interpretations to ensure that the obtained sets are finite, as required by the standard semantics for QFBAPA. Here we dispense with these restrictions to ease the comparison with classical DLs, where one usually considers arbitrary models rather than finite ones, easier. It turns out that doing so does not change the complexity of reasoning

Concept Descriptions with Set Constraints and Cardinality Constraints

We introduce a new description logic that extends the well-known logic ALCQ by allowing the statement of constraints on role successors that are more general than the qualified number restrictions of ALCQ. To formulate these constraints, we use the quantifier-free fragment of Boolean Algebra with Presburger Arithmetic (QFBAPA), in which one can express Boolean combinations of set constraints and numerical constraints on the cardinalities of sets. Though our new logic is considerably more expressive than ALCQ, we are able to show that the complexity of reasoning in it is the same as in ALCQ, both without and with TBoxes.The first version of this report was put online on April 6, 2017. The current version, containing more information on related work, was put online on July 13, 2017. This is an extended version of a paper published in the proceedings of FroCoS 2017

Integrating Reasoning Services for Description Logics with Cardinality Constraints with Numerical Optimization Techniques

Recent research in the field of Description Logic (DL) investigated the complexity of the satisfiability problem for description logics that are obtained by enriching the well-known DL ALCQ with more complex set and cardinality constraints over role successors. The algorithms that have been proposed so far, despite providing worst-case optimal decision procedures for the concept satisfiability problem (both without and with a terminology) lack the efficiency needed to obtain usable implementations. In particular, the algorithm for the case without terminology is non-deterministic and the one for the case with a terminology is also best-case exponential. The goal of this thesis is to use well-established techniques from the field of numerical optimization, such as column generation, in order to obtain more practical algorithms. As a starting point, efficient approaches for dealing with counting quantifiers over unary predicates based on SAT-based column generation should be considered.:1. Introduction 2. Preliminaries 2.1. First-order logic 2.2. Linear Programming 2.3. The description logic ALCQ 2.4. Extending ALCQ with expressive role successor constraints 2.4.1. The logic QFBAPA 2.4.2 The description logic ALCSCC 3. The description logic ALCCQU 3.1. A normal form for ALCCQU 3.2. ALCQt as an equivalent formulation of ALCCQU 3.2.1. ALCQt is a sublogic of ALCCQU 3.2.2. ALCCQU is a sublogic of ALCQt 3.3. Model-theoretic characterization of ALCQt 3.3.1. ALCQt-bisimulation and invariance for ALCQt 3.3.2. Characterization of ALCQt concept descriptions 3.4. Expressive power 3.4.1. Relative expressivity of ALCQ and ALCCQU 3.4.2. Relative expressivity of ALCCQU and ALCSCC 3.5. ALCCQU as the first-order fragment of ALCSCC 4. Concept satisfiability in ALCCQU 4.1. The first-order fragment CQU 4.2. Column generation with SAT oracle 4.2.1. Column generation and CQU 4.2.2. From linear inequalities to propositional formulae 4.2.3. Column generation and ALCCQU 4.3. Branch-and-Price for ALCCQU concept satisfiability 4.4. Correctness of ALCCQU-BB 4.4.1. Complexity of ALCCQU-BB 5. Conclusion - Bibliograph

Pseudo-contractions as Gentle Repairs

Updating a knowledge base to remove an unwanted consequence is a challenging task. Some of the original sentences must be either deleted or weakened in such a way that the sentence to be removed is no longer entailed by the resulting set. On the other hand, it is desirable that the existing knowledge be preserved as much as possible, minimising the loss of information. Several approaches to this problem can be found in the literature. In particular, when the knowledge is represented by an ontology, two different families of frameworks have been developed in the literature in the past decades with numerous ideas in common but with little interaction between the communities: applications of AGM-like Belief Change and justification-based Ontology Repair. In this paper, we investigate the relationship between pseudo-contraction operations and gentle repairs. Both aim to avoid the complete deletion of sentences when replacing them with weaker versions is enough to prevent the entailment of the unwanted formula. We show the correspondence between concepts on both sides and investigate under which conditions they are equivalent. Furthermore, we propose a unified notation for the two approaches, which might contribute to the integration of the two areas