We investigate fundamental properties of three-valued semantics for abstract dialectical frameworks (ADFs). In particular, we deal with realizability, i.e., the question whether there exists an ADF that has a given set of interpretations as its semantics. We provide necessary and sufficient conditions that hold for a set of three-valued interpretations whenever there is an ADF realizing it under admissible, complete, grounded, or preferred semantics. Moreover, we discuss how to construct such an ADF in case of realizability. Our results lay the ground for studying the expressiveness of ADFs under three-valued semantics. As a first application we study implications of our results on the existence of certain join operators on ADFs.

The classical probabilistic entailment problem is to We apply the family of minimal violation measures from determine upper and lower bounds on the probability [Potyka, 2014] since they allow us to extend the classical notion of formulas, given a consistent set of probabilistic of models of a probabilistic knowledge base to inconsistent assertions. We generalize this problem ones. Intuitively, the generalized models are those probability by omitting the consistency assumption and, thus, functions that minimally violate the knowledge base provide a general framework for probabilistic reasoning [Potyka and Thimm, 2014]. We incorporate integrity constraints under inconsistency. To do so, we utilize and study a family of generalized entailment problems inconsistency measures to determine probability for probabilistic knowledge bases. More specifically, functions that are closest to satisfying the knowledge the contributions of this work are as follows: base. We illustrate our approach on several 1. We introduce the computational problem of generalized examples and show that it has both nice formal and entailment with integrity constraints in probabilistic logics computational properties.

Parameterized complexity is a well recognized vehicle for understanding the multitude of complexity AI problems typically exhibit. However, the prominent problem of belief revision has not undergone a systematic investigation in this direction yet. This is somewhat surprising, since by its very nature of involving a knowledge base and a revision formula, this problem provides a perfect playground for investigating novel parameters. Among our results on the parameterized complexity of revision is thus a versatile fpt algorithm which is based on the parameter of the number of atoms shared by the knowledge base and the revision formula. Towards identifying the frontier between parameterized tractability and intractability, we also give hardness results for classes such as co-W[1], para-Theta 2 P and FPT NP[f(k)]

The AGM paradigm of belief change studies the dynamics of belief states in light of new information. Finding, or even approximating, dependent or relevant beliefs to a change is valuable because, for example, it can narrow the set of beliefs considered during belief change operations. Gärdenfors' preservation criterion (GPC) suggests that formulas independent of a belief change should remain intact. GPC allows to build dependence relations that are theoretically linked with belief change. Such dependence relations can in turn be used as a theoretical benchmark against which to evaluate other approximate dependence or relevance relations. There are already some studies, based on GPC, on the parallelism between belief change and dependence. One study offers a dependence relation parallel to AGM contraction for belief sets. Another study links base dependence relation to a more general belief base contraction, saturated kernel contraction. Here we offer yet a more general parallelism between kernel contraction and base dependence. At this level of generalization, different types of base dependence emerge. We prove that this differentiation of base dependence types is a result of possible redundancy in the base. This provides a theoretical means to distinguish between redundant and informative parts of a belief base.

Materialisation precomputes all consequences of a set of facts and a datalog program so that queries can be evaluated directly (i.e., independently from the program). Rewriting optimises materialisation for datalog programs with equality by replacing all equal constants with a single representative; and incremental maintenance algorithms can efficiently update a materialisation for small changes in the input facts. Both techniques are critical to practical applicability of datalog systems; however, we are unaware of an approach that combines rewriting and incremental maintenance. In this paper we present the first such combination, and we show empirically that it can speed up updates by several orders of magnitude compared to using either rewriting or incremental maintenance in isolation.

In the context of ontology-based data access with description logics (DLs), we study ontology-mediated queries in which selected predicates can be closed (OMQCs). In particular, we contribute to the classification of the data complexity of such queries in several relevant DLs. For the case where only concept names can be closed, we tightly link this question to the complexity of surjective CSPs. When also role names can be closed, we show that a full complexity classification is equivalent to classifying the complexity of all problems in coNP, thus currently out of reach. We also identify a class of OMQCs based on ontologies formulated in DL-LiteR that are guaranteed to be tractable and even FO-rewritable.

We address the issue of Ontology-Based Query Answering (OBQA), which seeks to exploit knowledge expressed in ontologies when querying data. Ontologies are represented in the framework of existential rules (aka Datalog+/-). A commonly used technique consists in rewriting queries into unions of conjunctive queries (UCQs). However, the obtained queries can be prohibitively large in practice. A well-known source of combinatorial explosion are very simple rules, typically expressing taxonomies and relation signatures. We propose a rewriting technique, which consists in compiling these rules into a preorder on atoms and embedding this preorder into the rewriting process. This allows to compute compact rewritings that can be considered as ``pivotal'' representations, in the sense that they can be evaluated by different kinds of database systems. The provided algorithm computes a sound, complete and minimal UCQ rewriting, if one exists. Experiments show that this technique leads to substantial gains, in terms of size and runtime, and scales on very large ontologies. We also compare to other tools for OBQA with existential rules and related lightweight description logics.

Description Logic (DL) based ontologies and non-monotonic rules provide complementary features whose combination is crucial in many applications. In hybrid knowledge bases (KBs), which combine both formalisms, for large real-world applications, often integrating knowledge originating from different sources, inconsistencies can easily occur. These commonly trivialize standard reasoning and prevent us from drawing any meaningful conclusions. When restoring consistency by changing the KB is not possible, paraconsistent reasoning offers an alternative by allowing us to obtain meaningful conclusions from its consistent part. In this paper, we address the problem of efficiently obtaining meaningful conclusions from (possibly inconsistent) hybrid KBs. To this end, we define two paraconsistent semantics for hybrid KBs which, beyond their differentiating properties, are faithful to well-known paraconsistent semantics as well as the non-paraconsistent logic they extend, and tractable if reasoning in the DL component is.

We study the problem of rewriting an ontology O1 expressed in a DL L1 into an ontology O2 in a Horn DL L2 such that O1 and O2 are equisatisfiable when extended with an arbitrary dataset.  Ontologies that admit such rewritings are amenable to reasoning techniques ensuring tractability in data complexity.  After showing undecidability whenever L1 extends ALCF , we focus on devising efficiently checkable conditions that ensure existence of a Horn rewriting.  By lifting existing techniques for rewriting Disjunctive Datalog programs into plain Datalog to the case of arbitrary first-order programs with function symbols, we identify a class of ontologies that admit Horn rewritings of polynomial size.  Our experiments indicate that many real-world ontologies satisfy our sufficient conditions and thus admit polynomial Horn rewritings.

Large formal mathematical knowledge bases encode considerable parts of advanced mathematics and exact science, allowing deep semantic computer assistance and verification of complicated theories down to the atomic logical rules. An essential part of automated reasoning over such large theories are methods learning selection of relevant knowledge from the thousands of proofs in the corpora. Such methods in turn rely on efficiently computable features characterizing the highly structured and inter-related mathematical statements.  In this work we (i) propose novel semantic features characterizing the statements in such large semantic knowledge bases, (ii) propose and carry out their efficient implementation using deductive-AI data-structures such as substitution trees and discrimination nets, and (iii) show that they significantly improve the strength of existing knowledge selection methods and automated reasoning methods over the large formal knowledge bases. In particular, on a standard large-theory benchmark we improve the average predicted rank of a mathematical statement needed for a proof by 22% in comparison with state of the art. This allows us to prove 8% more theorems in comparison with state of the art.