![](bg6.png)
5.3 Decidability
The discussion from the previous points gives us
also means for creating a procedure for deciding
satisfiability of modules in a DKB.
The decidability result from Th. 1 combined with
Prop. 2 allows for immediate stating that DDL(F,
N
n
) is decidable for acyclic DKBs. However, we can
extend this result a bit by including the DKBs which
can contain cycles.
A basic idea behind such extention is simple: we
proceed iteratively with determining c(O
i
) for each
module, assuming that in first iteration c
1
(O
i
) =
M(O
i
) and then, in the next k-th iteration taking
c
k
(O
i
) = F
n
M
i
c
k ‒ 1
(O
i
). As Serafini and Tamilin show
in (2007), the fixpoint will finally be reached, which
can be detected by adapted procedure for checking
whether an ontology is a conservative extension of
another (Lutz, Walther and Wolter, show in (2007)
that this problem for ALC is decidable).
This leads us to the following conclusion:
Proposition 3. For a given DKB ä = ({O
i
}, {μ
ij
}),
i, j ∈ I, i ≠ j, an recursive procedure for converting
modules in the following way: c
1
(O
i
) = M(O
i
), c
k
(O
i
)
= F
n
M
i
c
k ‒ 1
(O
i
), repeated until c
k
(O
i
) is a conservative
extension of c
k ‒ 1
(O
i
) for all i ∈ I, is a terminating,
sound and complete procedure for deciding satisfia-
bility of modules for ALC and DDL(F, N
n
).
6 CONCLUSIONS
In this section we summarize the main observations
and contributions of the paper and relate them to
other studies.
From the point of view of DDL, the results
allows us to show some insight in the relation
between mapping and importing (Homola and
Serafini, 2010). Here we show how different kinds
of mappings relate to specific kinds of importing
(especially “putting under”). Further work will allow
us to include also E-Connection (Kutz, Lutz,
Wolter, and Zakharyaschev, 2004) and P-DL (Bao,
Voutsadakis, Slutzki, and Honavar, 2009), two other
major methods of modularization.
The other result is an alternative way of proving
decidability of DDL(F, N
n
) for ALC. Though at the
current stage of research it does not extend the
results already available in literature, it shows the
practical application of the results from Th. 1. The
further development might result in a set of
techniques for proving decidability for a wide range
of modularization methods.
From the perspective of s-module framework the
presented discussion provides interesting hints about
its further development. The s-module framework
cannot easily handle situations in which we want to
refer to a tuple of elements of a domain. Sec. 5.3
suggests it may be useful to extend the framework
by some kind of treatment for limits (i.e. the ability
to determine bounds for an arbitrary set of modules).
Finally, the paper presents some extensions to
the framework of s-modules: definition of s-module
space, restriction operator, and a slightly extended
result for decidability (cf. Sec. 3).
ACKNOWLEDGEMENTS
This work is partially supported by the Polish
National Centre for Research and Development
under Grant No. SP/I/1/77065/10 by the strategic
scientific research and experimental development
program: „Interdisciplinary System for Interactive
Scientific and Scientific-Technical Information”.
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