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On the Complexity of Axiom Pinpointing in the EL Family of Description Logics

AAAI Conferences

We investigate the computational complexity of axiom pinpointing, which is the task of finding minimal subsets of a Description Logic knowledge base that have a given consequence. We consider the problems of enumerating such subsets with and without order, and show hardness results that already hold for the propositional Horn fragment, or for the Description Logic EL. We show complexity results for several other related decision and enumeration problems for these fragments that extend to more expressive logics. In particular we show that hardness of these problems depends not only on expressivity of the fragment but also on the shape of the axioms used.


Efficient Rule-Based Inferencing for OWL EL

AAAI Conferences

We review recent results on inferencing for SROEL(×), a description logic that subsumes the main features of the W3C recommendation OWL EL. Rule-based deduction systems are developed for various reasoning tasks and logical sublanguages. Certain feature combinations lead to increased space upper bounds for materialisation, suggesting that efficient implementations are easier to obtain for suitable fragments of OWL EL.


Deciding Monotone Duality and Identifying Frequent Itemsets in Quadratic Logspace

arXiv.org Artificial Intelligence

The monotone duality problem is defined as follows: Given two monotone formulas f and g in iredundant DNF, decide whether f and g are dual. This problem is the same as duality testing for hypergraphs, that is, checking whether a hypergraph H consists of precisely all minimal transversals of a simple hypergraph G. By exploiting a recent problem-decomposition method by Boros and Makino (ICALP 2009), we show that duality testing for hypergraphs, and thus for monotone DNFs, is feasible in DSPACE[log^2 n], i.e., in quadratic logspace. As the monotone duality problem is equivalent to a number of problems in the areas of databases, data mining, and knowledge discovery, the results presented here yield new complexity results for those problems, too. For example, it follows from our results that whenever for a Boolean-valued relation (whose attributes represent items), a number of maximal frequent itemsets and a number of minimal infrequent itemsets are known, then it can be decided in quadratic logspace whether there exist additional frequent or infrequent itemsets.


Bayesian Networks Specified Using Propositional and Relational Constructs: Combined, Data, and Domain Complexity

AAAI Conferences

We examine the inferential complexity of Bayesian networks specified through logical constructs. We first consider simple propositional languages, and then move to relational languages. We examine both the combined complexity of inference (as network size and evidence size are not bounded) and the data complexity of inference (where network size is bounded); we also examine the connection to liftability through domain complexity. Combined and data complexity of several inference problems are presented, ranging from polynomial to exponential classes.


Computing Datalog Rewritings beyond Horn Ontologies

arXiv.org Artificial Intelligence

Rewriting-based approaches for answering queries over an OWL 2 DL ontology have so far been developed mainly for Horn fragments of OWL 2 DL. In this paper, we study the possibilities of answering queries over non-Horn ontologies using datalog rewritings. We prove that this is impossible in general even for very simple ontology languages, and even if PTIME = NP. Furthermore, we present a resolution-based procedure for $\SHI$ ontologies that, in case it terminates, produces a datalog rewriting of the ontology. Our procedure necessarily terminates on DL-Lite_{bool}^H ontologies---an extension of OWL 2 QL with transitive roles and Boolean connectives.