Conjunctive Query Answering for Directional Rules Technical Report 3026 AIFB, Kalrsruhe Institute of Technology Markus Krötzsch 1 and Sebastian Rudolph 2 1 Dep. of Computer Science, University of Oxford, UK, markus.kroetzsch@cs.ox.ac.uk 2 Institute AIFB, Karlsruhe Institute of Technology, DE, sebastian.rudolph@kit.edu Abstract. This paper introduces Directional Rules, a new extension of Datalog with existential quantifiers in rule heads in the spirit of formalisms like tuple- generating dependencies, Datalog+/− and ∀∃-rules that have attracted new inter- est recently. As opposed to known decidable classes of such existential rules, Di- rectional Rules support complex join conditions as required for expressing tran- sitivity. Nonetheless, the new language suggests surprisingly simple algorithms for answering a broad class of conjunctive queries in polynomial time regarding data complexity. In contrast, answering unrestricted conjunctive queries is un- decidable, and we propose further restrictions and more complex algorithms for recovering decidability in the general case. Besides their immediate use for data integration and data exchange, Directional Rules are of particular interest since they can capture large real-world ontologies that could hitherto be modelled in description logics only, even though they are mostly used in database-driven ap- plications. 1 Introduction Datalog – the logical language of function-free first-order Horn clauses – has been widely studied and applied in both in the field of deductive databases and that of knowledge representation and reasoning. While pure Datalog can only be used to make statements and infer information about the active domain (i.e., the set of constants or database individuals that are given a priori), the capability to derive the existence of new individuals – a feature commonly called value invention [10, 2] – is considered a crucial prerequisite for the deployment of rule based paradigms for ontological knowl- edge representation [31]. This led to the introduction of extensions of Datalog featuring value invention. In the database community, the according logical fragment is usually referred to as tu- ple generating dependencies (TGDs, see, e.g., [1]) and has been employed in the area of data exchange and data integration [21]. The Datalog+/– formalism [12] which is based on TGDs has been shown to be able to capture lightweight ontological languages such as the DL-Lite family [15]. Coming from a parallel strand of research concerned with graph-based knowledge representation [17, 33] ∀∃-rules have been suggested for