5 research outputs found
On Bisimulations for Description Logics
We study bisimulations for useful description logics. The simplest among the
considered logics is (a variant of PDL). The others
extend that logic with inverse roles, nominals, quantified number restrictions,
the universal role, and/or the concept constructor for expressing the local
reflexivity of a role. They also allow role axioms. We give results about
invariance of concepts, TBoxes and ABoxes, preservation of RBoxes and knowledge
bases, and the Hennessy-Milner property w.r.t. bisimulations in the considered
description logics. Using the invariance results we compare the expressiveness
of the considered description logics w.r.t. concepts, TBoxes and ABoxes. Our
results about separating the expressiveness of description logics are naturally
extended to the case when instead of we have any sublogic
of that extends . We also provide results
on the largest auto-bisimulations and quotient interpretations w.r.t. such
equivalence relations. Such results are useful for minimizing interpretations
and concept learning in description logics. To deal with minimizing
interpretations for the case when the considered logic allows quantified number
restrictions and/or the constructor for the local reflexivity of a role, we
introduce a new notion called QS-interpretation, which is needed for obtaining
expected results. By adapting Hopcroft's automaton minimization algorithm and
the Paige-Tarjan algorithm, we give efficient algorithms for computing the
partition corresponding to the largest auto-bisimulation of a finite
interpretation.Comment: 42 page
Computing Crisp Bisimulations for Fuzzy Structures
Fuzzy structures such as fuzzy automata, fuzzy transition systems, weighted
social networks and fuzzy interpretations in fuzzy description logics have been
widely studied. For such structures, bisimulation is a natural notion for
characterizing indiscernibility between states or individuals. There are two
kinds of bisimulations for fuzzy structures: crisp bisimulations and fuzzy
bisimulations. While the latter fits to the fuzzy paradigm, the former has also
attracted attention due to the application of crisp equivalence relations, for
example, in minimizing structures. Bisimulations can be formulated for fuzzy
labeled graphs and then adapted to other fuzzy structures. In this article, we
present an efficient algorithm for computing the partition corresponding to the
largest crisp bisimulation of a given finite fuzzy labeled graph. Its
complexity is of order , where , and are
the number of vertices, the number of nonzero edges and the number of different
fuzzy degrees of edges of the input graph, respectively. We also study a
similar problem for the setting with counting successors, which corresponds to
the case with qualified number restrictions in description logics and graded
modalities in modal logics. In particular, we provide an efficient algorithm
with the complexity for the considered problem in
that setting