The Complexity of Satisfiability for Sub-Boolean Fragments of ALC

The standard reasoning problem, concept satisfiability, in the basic description logic ALC is PSPACE-complete, and it is EXPTIME-complete in the presence of unrestricted axioms. Several fragments of ALC, notably logics in the FL, EL, and DL-Lite family, have an easier satisfiability problem; sometimes it is even tractable. All these fragments restrict the use of Boolean operators in one way or another. We look at systematic and more general restrictions of the Boolean operators and establish the complexity of the concept satisfiability problem in the presence of axioms. We separate tractable from intractable cases.Comment: 17 pages, accepted (in short version) to Description Logic Workshop 201

Loop-checking and the uniform word problem for join-semilattices with an inflationary endomorphism

We solve in polynomial time two decision problems that occur in type checking when typings depend on universe level constraints

Complexity of branching temporal description logics

We study branching-time temporal description logics (TDLs) based on the DLs ALC and EL and the temporal logics CTL and CTL*. The main contributions are algorithms for satisfiability that are more direct than existing approaches, and (mostly) tight elementary complexity bounds that range from PTIME to 2EXPTIME and 3EXPTIME. A careful use of tree automata techniques allows us to obtain transparent and uniform algorithms, avoiding to deal directly with the intricacies of CTL*

Unification in the Description Logic EL w.r.t. Cycle-Restricted TBoxes

Unification in Description Logics (DLs) has been proposed as an inference service that can, for example, be used to detect redundancies in ontologies. The inexpressive Description Logic EL is of particular interest in this context since, on the one hand, several large biomedical ontologies are defined using EL. On the other hand, unification in EL has recently been shown to be NP-complete, and thus of significantly lower complexity than unification in other DLs of similarly restricted expressive power. However, the unification algorithms for EL developed so far cannot deal with general concept inclusion axioms (GCIs). This paper makes a considerable step towards addressing this problem, but the GCIs our new unification algorithm can deal with still need to satisfy a certain cycle restriction

Uniform and Modular Sequent Systems for Description Logics

We introduce a framework that allows for the construction of sequent systems for expressive description logics extending ALC. Our framework not only covers a wide array of common description logics, but also allows for sequent systems to be obtained for extensions of description logics with special formulae that we call "role relational axioms." All sequent systems are sound, complete, and possess favorable properties such as height-preserving admissibility of common structural rules and height-preserving invertibility of rules

Topics in Knowledge Bases: Epistemic Ontologies and Secrecy-preserving Reasoning

Applications of ontologies/knowledge bases (KBs) in many domains (healthcare, national security, intelligence) have become increasingly important. In this dissertation, we focus on developing techniques for answering queries posed to KBs under the open world assumption (OWA). In the first part of this dissertation, we study the problem of query answering in KBs that contain epistemic information, i.e., knowledge of different experts. We study ALCKm, which extends the description logic ALC by adding modal operators of the basic multi-modal logic Km. We develop a sound and complete tableau algorithm for answering ALCKm queries w.r.t. an ALCKm knowledge base with an acyclic TBox. We then consider answering ALCKm queries w.r.t. an ALCKm knowledge base in which the epistemic operators correspond to those of classical multi-modal logic S4m and provide a sound and complete tableau algorithm. Both algorithms can be implemented in PSpace. In the second part, we study problems that allow autonomous entities or organizations (collectively called querying agents) to be able to selectively share information. In this scenario, the KB must make sure its answers are informative but do not disclose sensitive information. Most of the work in this area has focused on access control mechanisms that prohibit access to sensitive information (secrets). However, such an approach can be too restrictive in that it prohibits the use of sensitive information in answering queries against knowledge bases even when it is possible to do so without compromising secrets. We investigate techniques for secrecy-preserving query answering (SPQA) against KBs under the OWA. We consider two scenarios of increasing difficulty: (a) a KB queried by a single agent; and (b) a KB queried by multiple agents where the secrecy policies can differ across the different agents and the agents can selectively communicate the answers that they receive from the KB with each other subject to the applicable answer sharing policies. We consider classes of KBs that are of interest from the standpoint of practical applications (e.g., description logics and Horn KBs). Given a KB and secrets that need to be protected against the querying agent(s), the SPQA problem aims at designing a secrecy-preserving reasoner that answers queries without compromising secrecy under OWA. Whenever truthfully answering a query risks compromising secrets, the reasoner is allowed to hide the answer to the query by feigning ignorance, i.e., answering the query as Unknown . Under the OWA, the querying agent is not able to infer whether an Unknown answer to a query is obtained because of the incomplete information in the KB or because secrecy protection mechanism is being applied. In each scenario, we provide a general framework for the problem. In the single-agent case, we apply the general framework to the description logic EL and provide algorithms for answering queries as informatively as possible without compromising secrecy. In the multiagent case, we extend the general framework for the single-agent case. To model the communication between querying agents, we use a communication graph, a directed acyclic graph (DAG) with self-loops, where each node represents an agent and each edge represents the possibility of information sharing in the direction of the edge. We discuss the relationship between secrecy-preserving reasoners and envelopes (used to protect secrets) and present a special case of the communication graph that helps construct tight envelopes in the sense that removing any information from them will leave some secrets vulnerable. To illustrate our general idea of constructing envelopes, Horn KBs are considered

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