=Paper=
{{Paper
|id=Vol-1698/CS&P2016_06_Polkowski_A-version-of-rough-mereology-suitable-for-rough-sets
|storemode=property
|title=A Version of Rough Mereology Suitable for Rough Sets
|pdfUrl=https://ceur-ws.org/Vol-1698/CS&P2016_06_Polkowski_A-version-of-rough-mereology-suitable-for-rough-sets.pdf
|volume=Vol-1698
|authors=Lech Polkowski
|dblpUrl=https://dblp.org/rec/conf/csp/Polkowski16
}}
==A Version of Rough Mereology Suitable for Rough Sets==
https://ceur-ws.org/Vol-1698/CS&P2016_06_Polkowski_A-version-of-rough-mereology-suitable-for-rough-sets.pdf
A version of rough mereology suitable for rough
sets
Lech T. Polkowski
Polish-Japanese Academy IT
Koszykowa str. 86, 02-008 Warszawa, Poland
email: lech.polkowski@pja.edu.pl;polkow@pjwstk.edu.pl
Abstract. We address in principle the notion of a boundary and we
propose a version of mereology better adapted to rough set theory than
the original version. We discuss the motivation and differences between
the original and proposed now versions of rough mereology and we show
that the Pawlak notion of a boundary in rough set theory is a particular
case of the more general notion of a boundary in the rough mereological
theory proposed in this work.
Keywords: rough set theory, rough mereology, truly rough inclusion, bound-
ary
1 Introduction: the idea of a boundary, in general and in
Pawlak’s theory
It is evident to all who study rough set idea that the most important notion is
the notion of a boundary and most important things that conform to that notion
are boundaries of concepts as they witness the uncertainty of the concept. The
notion of a boundary has been the subject of investigation by philosophers,
logicians, topologists from ancient times to now. The basic problem with the
notion of a boundary stems of course from the fact that our perception of the
world is continuous whereas the world has a discrete structure. Whence follow
the philosophical dilemmas like Bolzano’s paradox of the two touching balls A
and B. The question is where A ends and B begins? If q is the last point on A
which must exists by closedness of A, then if p is the first point on B and p is not q
then A and B do not touch and there is no boundary between them but we know
they touch as we are not able to push A or B any further. Similar are problems
by Leonardo of air and water: what is the boundary between water in the river
and air? Those and other similar questions have occupied philosophers since
times of antiquity. One needs not to reach for real physical phenomena in order
to find problems and difficulties with boundaries, e.g., time imagined continuity
have caused similar problems: what is the last moment an event occurs? See
Varzi [9] and Smith [8] for a discussion of philosophical aspects of the notion of
a boundary.
Mathematicians resolved the problem by postulating the completeness of
the real line which cannot be dissected into two open disjoint non–empty sets;
returning to philosophical aspects of the notion of a boundary, this implies that
when an event has the last moment p in which it happens, we are not able to
point to the first moment in which it does not happen. They also approached the
problem of a boundary from a local point of view with the idea of a neighborhood
and closeness: the boundary of a set is the set of points ‘infinitely close’ to the set
in the sense that each neighborhood of a boundary point must needs intersect
the set and its complement, see Engelking [1].
This point of view prevails in Pawlak’s idea of a boundary see Pawlak [3]
but the implementation of it proceeds in a distinct way. In the first place, one
needs the data about reality in the form of an information system i.e. a tuple
(U, A, val, V ) where U is a set of objects representing physical entities, processes,
moments of time etc., A is a set of attributes each of which maps objects in U
into the set of values V by means of a mapping val : U × A → V . The value
val(u, a) is often written down in a simpler form of a(u).
Objects are then coded as information sets of the form Inf (u) = {a(u) : a ∈
A}. The crux of the rough set approach is in identification of objects having the
same information sets: Ind(u, v) if and only if Inf (u) = Inf (v), where Ind(u, v)
is the indiscernibility relation . From that moment on, objects loose their real
names and become visible only by their information sets which allows for making
some of them indiscernible. The equivalence relation of indiscernibility partitions
the universe U into classes which are also regarded as primitive granules of
knowledge [u] or black boxes.
One addresses the problem of concepts understood as subsets of the universe
U and distinguishes among them certain ones as those which can be assembled
from granules Sby the union of sets operator i.e. a concept C ⊆ U is certain if and
only if C = {[u] : [u] ⊆ C}. Other concepts are not certain, commonly called
rough. A rough concept R then must have an object u such that the class [u] is
not contained in it but it does intersect the complement U \ R. Such objects in
R constitute the boundary of R, BdR, i.e. BdR = {u ∈ U : [u] ∩ (U \ R) 6= ∅ = 6
[u] ∩ R}. One may say that the boundary of R consists of objects which have
their copies both in R and in U \ R. This is clearly a topological approach as
the class [u] is the least neighborhood of u in the partition topology induced by
the indiscernibility relation Ind.
Let us point to advantages of this approach: first it is objective as the shape of
the boundary follows by the data without any intervention of subjective factors
which are so essential in e.g. fuzzy set theory see Zadeh [12]; next, it relies solely
on data without resorting to e.g. real numbers or other auxiliary external to data
factors.
2 Mereology and its extensions into domain of
uncertainty
Mereology due to Stanislaw Leśniewski, see Leśniewski [2], Sinisi [7], is based on
the notion of a part relation, part(x, y) (‘x is a part to y’) which satisfies over a
universe U conditions:
M1 For each x ∈ U it is not true that part(x, x);
M2 For each triple x, y, z of things in U if part(x, y) and part(y, z), then
part(x, z).
The notion of an element is defined as the relation el(x, y) which holds true
if part(x, y) or x = y. It follows that the relation of being an element is a partial
order on U and el(x, y) and el(y, x) are true simultaneously if and only if x = y.
Clearly, part(x, y) if and only if el(x, y) and x 6= y.
The last important notion is that of the class understood as the object which
represents a collective entity i.e. a property: for a property Ψ on U which is not
void, the class pf Ψ , ClsΨ , is the object such that:
C1 If Ψ (u) holds true, then el(u, ClsΨ );
C2 For each u, if el(u, ClsΨ ) then there are objects p, q such that el(p, u),
el(p, q), Ψ (q) hold true.
2.1 Our rough mereology as up to now
We proposed a scheme which extended mereology based on the part relation
to the version based on the ‘part to a degree’ relation, see Polkowski [6], writ-
ten down as the relation µ(x, y, r) (x is a part of y to a degree of at least of r)
on the universe U endowed with the mereological notion of the element el. The
assumptions abut µ reflected the basic true properties of the partial containment:
RM1 µ(x, x, 1) for each x ∈ U ;
RM2 µ(x, y, 1) if and only if el(x, y) holds true;
RM3 If µ(x, y, 1) and µ(z, x, r) hold true then µ(z, y, r) holds true;
RM4 If µ(x, y, r) and s < r hold true then µ(x, y, s) holds true.
This approach has required the a priori mereology on the universe U of which
the relation µ was a diffusion or fuzzification, cf. Varzi [10]. Assume that f :
[0, 1]2 → [0, 1] is a continuous in each coordinate and symmetric function such
that f (1, r) = r for each r ∈ [0, 1]. We will call any such f a pre–norm.
We say for a pre–norm f that relation µ is f –transitive if and only if from
true conditions µ(x, y, r) and µ(y, z, s) the true condition µ(x, z, f (r, s)) follows.
Proposition 1. If the relation µ on the universe U is f –transitive then the
relation π(x, y) which holds true if and only if µ(x, y, 1) and µ(y, x, 1) hold true
is an equivalence relation on U .
Indeed, π(x, x1) holds true by RM1; symmetry is evident by definition; tran-
sitivity follows by f −transitivity of µ.
An attempt to apply this version of µ in the rough set context is handicapped
by the fact that by RM2, the relation π should be the identity which does not
capture the full scope of rough set cases. We need something more flexible to
accomodate equivalence relations of indiscernibility.
2.2 A rough mereology for rough set theory
As stated below, we need a mereology which may account for rough set theoretic
contexts. We assume an information system (U, A, val, V ) as our playground.
We define a truly rough inclusion, µtr (x, y, r) as a relation which satisfies on
U the following assumptions:
TRM1 µtr (x, x, 1);
TRM2 There is a partition P on U such that µtr (x, y, 1) if and only if x and y
are in the same partition class [x]P ;
TRM3 If µtr (x, y, 1) and µtr (z, x, r) then µtr (z, y, r);
TRM4 If µtr (x, y, r) and s < r then µtr (x, y, s).
TRM5 The truly rough inclusion µtr (x, y, r) is f –transitive for some pre–norm f .
The predicate el(x, y) if µtr (x, y, 1) defines x as an element of y.
We sum up basic consequences of our assumptions.
Proposition 2. The following are true by conditions TRM1-TRM5.
1. µtr (x, y, 1) implies µtr (y, x, 1) i.e. µtr is symmetric.
2. The relation el(x, y) is symmetric and el(x, y) and el(y, x) imply that
[x]P = [y]P .
3. {y : µtr (y, x, 1)}=[x]P .
4. The relation µtr (x, y, 1) is transitive in the sense that µtr (z, x, 1) and
µtr (x, y, 1) imply µtr (z, y, 1).
3 Boundaries in rough mereology for rough sets
We use the language of predicates on the universe U in definitions of boundaries
by means of a truly rough inclusions µtr .
3.1 A general scheme for boundaries
For a truly rough inclusion µtr , and x ∈ U , r ∈ [0, 1], we define a new predicate
N (x, r)(z) if there exists an s ≥ r such that µ(z, x, s). N (x, r) is the neighborhood
granular predicate about x of radius r.
Consider a predicate Ψ on U having a non–empty meaning [Ψ ]. The com-
plement to Ψ is the predicate −Ψ such that −Ψ (x) if and only if not Ψ (x). We
define the upper extension of Ψ of radius r, denoted Ψr+ by letting Ψr+ (x) if there
exists z such that Ψ (z) and N (x, r)(z). Similarly, we define the lower restriction
of Ψ of radius r, denoted Ψr− by letting Ψr− (x) if and only if not (−Ψ )+ r (x).
Proposition 3. 1. Predicates Ψr+ and Ψr− are disjoint in the sense that there
is no z ∈ U such that Ψr+ (z) and Ψr− (z) hold true. 2. If Ψr+ (x) holds true then
Ψr+ (y) holds true for each y such that µtr (y, x, 1). 3. If Ψr− (x) holds true then
Ψr− (y) holds true for each y such that µtr (y, x, 1).
Proof. Claim 1 follows by definitions of the two predicates. For Claim 2, consider
x, y such that Ψr+ (x) and µtr (y, x, 1). There exists z such that Ψ (z), N (x, s)(z)
hold true with some s ≥ r so µtr (z, x, s) holds true. By symmetry of µtr , we have
µtr (x, y, 1) true and f –transitivity of µtr for an adequate pre–norm f implies that
µtr (z, y, f (1, s)) holds true i.e. µtr (z, y, s) holds true which means that N (y, r)(z)
holds true and finally Ψr+ (y) holds true. For Claim 3, assume that Ψr− (x) and
µtr (y, x, 1) hold true i.e.
¬∃z, s ≥ r.µtr (z, x, s) ∧ ¬Ψ (z) (1)
which is equvalent to
µtr (z, x, s) → Ψ (z). (2)
As µtr (y, x, 1) is equivalent to µtr (x, y, 1), we have by f –transitivity of µtr that
µtr (z, y, s) → Ψ (z), (3)
which is equivalent to the thesis Ψr− (y).
We will say that a predicate Ψ is el–saturated if and only if true formulas
Ψ (x) and el(y, x) imply that Ψ (y). A corollary to Claim 3 in Proposition 3 says
that for each r ∈ [0, 1], predicates Ψr+ and Ψr− are el–saturated.
A global and local definition of the boundary For a predicate Ψ , we define
the predicate boundary of Ψ with respect to a truly rough inclusion µtr , denoted
Bdµtr Ψ as follows:
Bdµtr Ψ ↔ (¬Ψ1+ ) ∧ (¬Ψ1− ). (4)
Arguing like in proof of Proposition 3, we prove the following
Proposition 4. 1. Bdµtr Ψ is el–saturated 2. For no z ∈ U , Bdµtr Ψ (z) ∧ Ψ1+ (z)
is true and for no z ∈ U , Bdµtr Ψ (z) ∧ Ψ1− (z) is true.
Proposition 5. For each x ∈ U , Bdµtr Ψ (x) holds true if and only if there exist
z, y ∈ U such that Ψ (z), −Ψ (y), µtr (z, x, 1), µtr (y, x, 1).
A predicate Open is defined on predicates on U and a predicate Φ on U is
open, Open(Φ) in symbols if and only if it is el–saturated.
Corollary 1. Open(Ψr+ ) and Open(Ψr− ) hold true for each r ∈ [0, 1].
Open(Bdµtr Ψ ) holds true.
Proposition 6. For a finite collectionWof predicates {Ψ1 , Ψ2 , . . . , Ψk } if Open(Ψi )
holds true for each i ≤ k, then Open( i Ψi ) holds true.
A predicate Closed holds true for a predicate Ψ if and only if Open(−Ψ )
holds true.
Corollary 2. Closed(Ψr+ ) and Closed(Ψr− ) hold true for each r ∈ [0, 1].
Closed(Bdµtr Ψ ) holds true.
For the mereotopological notion of boundary see also Polkowski and Semeniuk–
Polkowska [4], [5] and Varzi [11].
3.2 The Pawlak notion of a boundary is a special case of truly
rough mereological notion of a boundary
We return to an information system (U, A, val, V ). We derive a truly rough inclu-
sion from any Archimedean t–norm. There exist two non–isomorphic Archimedean
t–norms:
– the Lukasiewicz t–norm L(x, y) = max{0, x + y − 1};
– the product t–norm P (x, y) = x · y.
Both these t–norms admit a Hilbert–style representation
t(x, y) = g(f (x) + f (y)),
where f : [0, 1] → [0, 1] is a continuous decreasing function with f (0) = 1, and
g : [0, 1] → [0, 1] is the inverse to f . In case of the t–norm L, f (x) = 1 − x and
g(y) = 1 − y. We let for an Archimedean t–norm t:
card(Dis(x, y))
µt (x, y, r) if and only if g( ) ≥ r, (5)
card(A)
where Dis(x, y) = {a ∈ A : a(x) 6= a(y)}.
In particular, as for the Lukasiewicz t–norm we have g(y) = 1 − y, the
Lukasiewicz truly rough inclusion can be defined as
card(Ind(x, y))
µL
tr (x, y, r) if and only if ≥ r, (6)
card)A)
where Ind(x, y) = A \ Dis(x, y). In particular, µL is L–transitive.
The predicate of element el(x, y) holds true if and only if µL tr (x, y, 1) holds
true if and only if Ind(x, y) i.e. x, y are indiscernible. Hence, a predicate is el–
saturated if and only if its meaning is the union of a family of indiscernibility
classes and rough mereological notions of Ψ1+ and Ψ1− become, respectively, the
notions of the upper and the lower approximations of the meaning of Ψ and the
meaning of the boundary predicate BdµL Ψ is the boundary of the meaning of
Ψ.
4 Conclusions
We have proposed a new version of rough mereology suitable for rough set theory
and we show that the rough set theory is a particular case of an abstract rough
mereotopological theory.
5 Acknowledgement
The author would like to dedicate this work to the memory of Professor Zdzislaw
Pawlak (1926-2006).
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