BDD-Tableau :

Description Logics (DLs) [1] are a family of well-understood logic-based knowledge representation frameworks. They have a wide range of applications, but are most widely known as the basis for ontology languages such as OWL. DLs are usually designed in such a way that relevant reasoning tasks (e.g. concept subsumption) are decidable. Respective reasoning algorithms have been studied, implemented, and optimized [4,6] during the last decades. Recently, [7] pointed out that the increasing use of DL-based ontologies in areas such as e-Science and the Semantic Web however is already stretching the capabilities of existing DL reasoning systems, and brings with it a range of challenges for future research on reasoning methods for DL. Key issues here are the provision of efficient algorithms that allow (advanced) applications (i) to scale up to knowledge bases of practical relevance and (ii) to leverage expressive languages for capturing domain knowledge. However, expressiveness of DLs comes at a price: the theoretically high (worst-case) complexity of relevant reasoning tasks. Hence, it is unlikely, that there is a single method, that performs well in all possible cases. Rather, one can expect that specific techniques perform well one particular classes of problems. Therefore, we consider the development of novel techniques in DL reasoning. As a first step, we propose to design and study new techniques for DL reasoning based on a prominent data structure that has been applied very successfully in various domains in computer science where one has to face the efficient representation and processing of large scale problems: Binary Decision Diagrams (BDDs) [2]. In particular, BDDs have been used very successfully for reasoning in propositional logics and model checking, and have been lifted to the level of first-order logics, too. In both cases, they provide a rich semantic structure to guide proof search. Therefore, we believe that (i) BDDs are interesting to study in the context of reasoning for a logic of intermediate expressivity (such as DLs) and (ii) that they provide a fertile ground for the design of novel efficient methods for reasoning in particular expressive DLs. However, it is not straightforward how to actually use BDDs in the context of DL reasoning: while in propositional logics BDDs embody canonical representations of formulae (which immediately gives a procedure for propositional reasoning), standard BDDs essentially lack the possibility to represent the modal aspects of DL concept expressions and formulae. In this talk, we address the most fundamental reasoning task of checking concept satisfiability and discuss a novel semantically-rich datastructure to represent knowledge during proof search. We present a first, rather natural and elegant attempt to extend standard propositional BDDs to a knowledge structure for DL reasoning that we call BDD-Tableau. BDD-Tableau combine the representational capabilities of BDDs for the purely propositional dimension of a DL knowledge base and the tableau data structure that is used in tableau-decision procedures to capture a (partial) modal interpretation of the input concept to cover the modal dimension. On top of this knowledge structure, we define the BDD-Tableau calculus as a decision procedure for concept satisfiability for the simple DL ALC and discuss relevant properties, such as such as correctness, completeness and termination. We then consider the specifically important case of checking satisfiability wrt. a general TBox. We argue that BDD-Tableau can be used for any DL where a standard tableau method is already known, in particular for very expressive DLs such as SHIQ, SHOIN or SROIQ which are underlying OWL-DL and lack the finite model and the tree-model property. Finally, we compare the BDD-Tableau calculus with various state-of-the-art methods, such as [5,8,3], and argue that BDD-Tableau might be an interesting tool for better understanding the relation between different DL inference procedures. A preliminary prototype implementation might be used for illustration purposes.