350 research outputs found
The Complexity of Reasoning with Cardinality Restrictions and Nominals in Expressive Description Logics
We study the complexity of the combination of the Description Logics ALCQ and
ALCQI with a terminological formalism based on cardinality restrictions on
concepts. These combinations can naturally be embedded into C^2, the two
variable fragment of predicate logic with counting quantifiers, which yields
decidability in NExpTime. We show that this approach leads to an optimal
solution for ALCQI, as ALCQI with cardinality restrictions has the same
complexity as C^2 (NExpTime-complete). In contrast, we show that for ALCQ, the
problem can be solved in ExpTime. This result is obtained by a reduction of
reasoning with cardinality restrictions to reasoning with the (in general
weaker) terminological formalism of general axioms for ALCQ extended with
nominals. Using the same reduction, we show that, for the extension of ALCQI
with nominals, reasoning with general axioms is a NExpTime-complete problem.
Finally, we sharpen this result and show that pure concept satisfiability for
ALCQI with nominals is NExpTime-complete. Without nominals, this problem is
known to be PSpace-complete
Tractable approximate deduction for OWL
Acknowledgements This work has been partially supported by the European project Marrying Ontologies and Software Technologies (EU ICT2008-216691), the European project Knowledge Driven Data Exploitation (EU FP7/IAPP2011-286348), the UK EPSRC project WhatIf (EP/J014354/1). The authors thank Prof. Ian Horrocks and Dr. Giorgos Stoilos for their helpful discussion on role subsumptions. The authors thank Rafael S. Gonçalves et al. for providing their hotspots ontologies. The authors also thank BoC-group for providing their ADOxx Metamodelling ontologies.Peer reviewedPostprin
Complexity Results and Practical Algorithms for Logics in Knowledge Representation
Description Logics (DLs) are used in knowledge-based systems to represent and
reason about terminological knowledge of the application domain in a
semantically well-defined manner. In this thesis, we establish a number of
novel complexity results and give practical algorithms for expressive DLs that
provide different forms of counting quantifiers.
We show that, in many cases, adding local counting in the form of qualifying
number restrictions to DLs does not increase the complexity of the inference
problems, even if binary coding of numbers in the input is assumed. On the
other hand, we show that adding different forms of global counting restrictions
to a logic may increase the complexity of the inference problems dramatically.
We provide exact complexity results and a practical, tableau based algorithm
for the DL SHIQ, which forms the basis of the highly optimized DL system iFaCT.
Finally, we describe a tableau algorithm for the clique guarded fragment
(CGF), which we hope will serve as the basis for an efficient implementation of
a CGF reasoner.Comment: Ph.D. Thesi
Description Logics as Ontology Languages for the Semantic Web
The vision of a Semantic Web has recently drawn considerable attention, both from academia and industry. Description logics are often named as one of the tools that can support the Semantic Web and thus help to make this vision reality. In this paper, we describe what description logics are and what they can do for the Semantic Web. Descriptions logics are very useful for defining, integrating, and maintaining ontologies, which provide the Semantic Web with a common understanding of the basic semantic concepts used to annotate Web pages. We also argue that, without the last decade of basic research in this area, description logics could not play such an important rˆole in this domain
A Tree Logic with Graded Paths and Nominals
Regular tree grammars and regular path expressions constitute core constructs
widely used in programming languages and type systems. Nevertheless, there has
been little research so far on reasoning frameworks for path expressions where
node cardinality constraints occur along a path in a tree. We present a logic
capable of expressing deep counting along paths which may include arbitrary
recursive forward and backward navigation. The counting extensions can be seen
as a generalization of graded modalities that count immediate successor nodes.
While the combination of graded modalities, nominals, and inverse modalities
yields undecidable logics over graphs, we show that these features can be
combined in a tree logic decidable in exponential time
The Complexity of Enriched Mu-Calculi
The fully enriched μ-calculus is the extension of the propositional
μ-calculus with inverse programs, graded modalities, and nominals. While
satisfiability in several expressive fragments of the fully enriched
μ-calculus is known to be decidable and ExpTime-complete, it has recently
been proved that the full calculus is undecidable. In this paper, we study the
fragments of the fully enriched μ-calculus that are obtained by dropping at
least one of the additional constructs. We show that, in all fragments obtained
in this way, satisfiability is decidable and ExpTime-complete. Thus, we
identify a family of decidable logics that are maximal (and incomparable) in
expressive power. Our results are obtained by introducing two new automata
models, showing that their emptiness problems are ExpTime-complete, and then
reducing satisfiability in the relevant logics to these problems. The automata
models we introduce are two-way graded alternating parity automata over
infinite trees (2GAPTs) and fully enriched automata (FEAs) over infinite
forests. The former are a common generalization of two incomparable automata
models from the literature. The latter extend alternating automata in a similar
way as the fully enriched μ-calculus extends the standard μ-calculus.Comment: A preliminary version of this paper appears in the Proceedings of the
33rd International Colloquium on Automata, Languages and Programming (ICALP),
2006. This paper has been selected for a special issue in LMC
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