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or English or both. Motivated by these interpretations, several relations are de-
fined on NISs (e.g. [16,21,22,5]). We list a few of them below. Consider a NIS
S
:= (
U,A,
a∈A
V
a
,F
) and an attribute set
B
.
Ind
B
if and only if
F
(
x, a
)=
F
(
y,a
) for all
a
Indiscernibility.
(
x, y
)
∈
∈
B
.
Sim
B
if and only if
F
(
x, a
)
Similarity.
(
x, y
)
∈
∩
F
(
y,a
)
∅
for all
a
∈
B
.
=
In
B
if and only if
F
(
x, a
)
Inclusion.
(
x, y
)
∈
ↆ
F
(
y,a
) for all
a
∈
B
.
NSim
B
if and only if
Negative similarity.
(
x, y
)
∈
∼
F
(
x, a
)
∩∼
F
(
y,a
)
=
∅
for all
a
∈
B
,where
∼
is the complementation relative to
V
a
.
Com
B
if and only if
F
(
x, a
)=
Complementarity.
(
x, y
)
∈
∼
F
(
y,a
) for all
B
.
Weak indiscernibility.
(
x, y
)
a
∈
wInd
B
if and only if
F
(
x, a
)=
F
(
y,a
)for
∈
B
.
Weak similarity.
(
x, y
)
some
a
∈
wSim
B
if and only if
F
(
x, a
)
∈
∩
F
(
y,a
)
=
∅
for some
B
.
Weak inclusion.
(
x, y
)
a
∈
wIn
B
if and only if
F
(
x, a
)
∈
ↆ
F
(
y,a
)forsome
a
∈
B
.
wNSim
B
if and only if
Weak negative similarity.
(
x, y
)
∈
∼
F
(
x, a
)
∩∼
B
.
Weak complementarity.
(
x, y
)
F
(
y,a
)
=
∅
for some
a
∈
wCom
B
if and only if
F
(
x, a
)=
∈
∼
F
(
y,a
)
for some
a
∈
B
.
Each of the relations defined above gives rise to a generalized approximation
space, where the relation may not be an equivalence. Thus, one can approximate
any subset of the domain using the lower and upper approximations defined on
these generalized approximation spaces.
Search for a logic which can be used to reason about the approximations
of concepts remains an important area of research in rough set theory. For a
comprehensive survey on this direction of research, we refer to [6,2]. It is not
di
cult to observe that in a logic for information systems, one would like to
have the following two features. (i) The logic should be able to describe aspects
of information systems such as attribute, and attribute-values. (ii) It should also
be able to capture concept approximations induced by different sets of attributes.
In literature, one can find logics with the first feature (e.g. [14,15,21,1]) as well
as logics with the second feature (e.g. [13,16,14,15,19,24,7,8]). But proposal for
a sound and complete modal logic of NIS having both feature is not known
to us. In [9], a sound and complete logic for
deterministic
information systems
with both of the above features was proposed. But recall that in the case of
NISs, an object takes a set of attribute-values instead of just one as in the case
of deterministic information system. Moreover, unlike deterministic information
system, many relations other than indiscernibility are relevant and studied in
the context of NISs. Therefore, in this article, our aim is to present a sound and
complete modal logic with semantics directlybasedonNIS,havingbothofthe
above mentioned properties.
The remainder of this article is organized as follows. In Section 2, we in-
troduced the syntax and semantics of the logic LNIS of NISs. This logic can
capture the approximations relative to indiscernibilitry, similarity and inclusion
