Quantification in Defeasible DLs matters again

                  Maximilian Pensel? and Anni-Yasmin Turhan

     Institute for Theoretical Computer Science, Technische Universität Dresden,
                       first-name.last-name @tu-dresden.de

    Defeasible DLs have nonmonotonic semantics and are a prominent approach
for reasoning w.r.t. inconsistency or default assumptions. A defeasible knowl-
edge base (DKB) contains defeasible concept inclusions (DCIs) in the so-called
DBox. DCIs capture concept inclusions that typically hold and could be ignored
during reasoning if they contradict more specific DCIs or the TBox. The more
DCIs are consistent with an instance of a concept, the more typical it is consid-
ered. Rational reasoning as characterised by the well-known KLM postulates for
propositional logic has previously been lifted to ALC by Casini et al. in 2010.
The existing computation algorithms for entailments use a reduction to classi-
cal reasoning (by so-called materialisation) for rational entailments and for the
stronger relevant entailments. Materialisation transforms each DCI C @   ∼ D into a
concept ¬C t D which can be used for classical subsumption tests in conjunction
with the (potential) subsumee to include the information from the DCIs. Obvi-
ously, such a conjunction does not propagate defeasible information to concepts
nested in existential or value restrictions. Thus un-defeated DCIs are ignored,
expected inferences are lost, and all role successors are of uniform (a-)typicality.
—a well-known problem to the Defeasible DL community.
    We introduced the first approach for rational reasoning in EL⊥ mending these
shortcomings by extending (classical) canonical models to typicality models. In-
tuitively, the domain of a typicality model contains multiple copies of the domain
of an EL⊥ canonical model. Each such copy satisfies a different (sub-)set of DCIs
from the DKB. The copies allow to tailor the typicality of role successors required
by, say ∃r.C, as follows: for (d, e) ∈ rI and e ∈ C I element d can be related
to a more typical representative of C than e. Typicality models are computed
by first performing classical reasoning w.r.t. a TBox and then successively “up-
grading” typicality of role successors, i.e., picking more typical, yet consistent
representatives of the (successor) concept. We showed that subsumption under
rational closure is achieved, if the considered sets of DCIs are a particular se-
quence of decreasing subsets of the DBox. We have generalised our approach to
the stronger relevant closure by using the complete lattice of subsets of the DBox
for generating copies of the canonical domain. We show for rational and rele-
vant closure that typicality models yield more entailments than the respective
materialisation-based approach.
Our results for rational closure are published at the 14th International Confer-
ence on Logic Programming and Nonmonotonic Reasoning (LPNMR 2017) and
the results for relevant closure are published at the 4th International Workshop
on Defeasible and Ampliative Reasoning (DARe 2017).
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    Supported by DFG in the Research Training Group QuantLA (GRK 1763).