@InProceedings{SchroderPattinson08c,
  author = {Lutz Schr{\"o}der and Dirk Pattinson},
  title = {How Many Toes Do I Have? Parthood and Number Restrictions in Description Logics},
  year = {2008},
  editor = {Gerhard Brewka and Jer{\^o}me Lang},
  booktitle = {Principles of Knowledge Representation and Reasoning (KR 2008)},
  publisher = {AAAI Press; Menlo Park, CA},
  pages = {307-218},
  keywords = {Coalgebra modal logic description logic number restriction parthood trees},
  comment = { <a href = "http://www8.informatik.uni-erlangen.de/~schroeder/papers/TreeDL.pdf"> [preprint] </a>},
  abstract = {The modelling of parthood relations in description logics via
transitive roles often leads to undecidability when combined with
number restrictions and role hierarchies.  Here, we introduce the
description logic PHQ that explicitly supports reasoning about
parthood in the presence of qualified number restrictions. Our main
results are completeness and decidability in NEXPTIME. Conceptually, we argue that PHQ provides a better semantic fit for many
applications: more often than not, parthoods occurring e.g. in
biomedical ontologies are expected to be tree-like. In such cases,
PHQ supports stronger inferences than standard description
logics.  Technically this is achieved by explicitly excluding the
merging of descendants, which, at the same time, eliminates the
prime source of undecidability.  We work in the general setting of
coalgebraic modal logic, a generic semantic framework for
not-necessarily-normal modal logics. This added generality allows
the re-use of many of our results for other logics of sometimes
quite different flavour.
},
}