Table Of ContentPure Rough Mereology and Counting
A. Mani
Department of Pure Mathematics
University of Calcutta
9/1B, Jatin Bagchi Road
Kolkata(Calcutta)-700029, India
Email: [email protected]
Homepage: http://www.logicamani.in
Abstract—The study of mereology (parts and wholes) in the (GOS) are introduced as a variant of the framework in [4],
7
context of formal approaches to vagueness can be approached thenatureofparthoodoverGOSisinvestigated,theontology
1
in a number of ways. In the context of rough sets, mereological
of meaning associated explored and counting strategies are
0
conceptswithaset-theoreticorvaluationbasedontologyacquire
2 complexanddiversebehavior.Inthisresearchageneralroughset developed for drawing inferences about the nature of data at
n framework called granular operator spaces is extended and the hand. The counting strategies refer antichains (derived from
a nature of parthood in it is explored from a minimally intrusive parthood) of rough objects, but can be extended to other
J point of view. This is used to develop counting strategies that parthoodsinanaturalway.Thesecanbeusedforconstructing
help in classifying the framework. The developed methodologies
8 models and also confirming whether approximations found in
would be useful for drawing involved conclusions about the
2 practice can possibly be explained from a rough set ontol-
nature of data (and validity of assumptions about it) from
antichains derived from context. The problem addressed is also ogy/view.
]
I about whether counting procedures help in confirming that the
A approximations involved in formation of data are indeed rough A. Background
. approximations? An Information System I, is a relational system of the
s
c Keywords: Mereology, Parthood, General Granular Operator form I = (cid:104)S, A, {V : a∈A}, {f : a∈A}(cid:105) with S, A and
a a
[ Spaces,RoughMereology,RoughMembershipFunctions,AI,Rough V being respectively sets of Objects, Attributes and Values
a
1 Objects, Granulation, Contamination Problem 1 respectively.Informationsystemsgeneratevarioustypesofre-
v I. INTRODUCTION lationalorrelatorspaceswhichinturnrelatetoapproximations
1 of different types and form a substantial part of the problems
0 Mereology is the study of parts and wholes and has been
encountered in general RSTs.
3 studied from philosophical, logical, algebraic, topological and
In classical RST, equivalence relations of the form R are
8 applied perspectives. Almost every major philosophical ap-
0 derivedbytheconditionx, y ∈ SandB ⊆ A,let(x, y) ∈ R
proach has its approach to the basic questions related to
. if and only if (∀a ∈ B)ν(a, x) = ν(a, y). (cid:104)S, R(cid:105) is then
1 parts and wholes in the context of knowledge, knowledge
0 representation and world view. an approximation space. On the power set ℘(S), lower and
7 upper approximations of a subset A ∈ ℘(S) operators, (apart
Rough set theory is a formal approach to vagueness and
1 from the usual Boolean operations), are defined as per: Al =
knowledgethatinvolvesawidearrayoflogico-algebraic,com-
v: putational,mathematicalandappliedphilosophicaltechniques. (cid:83)[x]⊆A[x],Au =(cid:83)[x]∩A(cid:54)=∅[x],with[x]beingtheequivalence
i class generated by x ∈ S. If A,B ∈ ℘(S), then A is said to
X The study of mereology in the context of rough sets can
be roughly included in B, (A(cid:118)B) if and only if Al ⊆Bl&
be approached in at least two essentially different ways. In
ar the approach aimed at reducing contamination by the present Au ⊆ Bu. A is roughly equal to B (A ≈ B) if and only if
A(cid:118)B and B(cid:118)A (the classes of ≈ are rough objects).
author [1], [2], the primary motivation is to avoid intrusion
In rough sets, the objects of interest may be all types of
intothedatabywayofadditionalassumptionsaboutthedata.
objects,onlyroughobjectsofspecifictypeorroughandexact
In the rough membership function based approach [3], the
objectsofsometypesandcorrespondingtothesethedomains
strategy is to base definitions of parthood on the potential
of interest would be the classical domain or rough domain or
values of the function.
hybrid versions thereof respectively [1]. Boolean algebra with
Oneofthemostgeneralset-theoreticframeworksforgranu-
approximation operators forms a classical rough semantics.
larroughsetshasbeendevelopedbythepresentauthorin[4].
This fails to deal with the behavior of rough objects alone.
There it is shown by her that the granular framework is ideal
The scenario remains true even when R in the approximation
for handling antichains formed by mutually distinct objects
space is replaced by arbitrary binary relations. In general,
and related models. Other general set theoretic frameworks
℘(S) can be replaced by a set with a parthood relation and
avoidgranulationandimposemorerestrictionsonapproxima-
some approximation operators defined on it as in [1]. The
tion.Inthepresentpaper,generalizedgranularoperatorspaces
associated semantic domain is the classical semantic domain
1IEEE-Xplore:WIECON-ECE’2017 for general RST. The domain of discourse associated with
roughlyequivalentsetsinclassicaldomainisaroughsemantic Definition 2. By a roughly consistent object will be meant
domain. Hybrid semantic domains, have also been used in the a set of subsets of S of the form H = {A;(∀B ∈ H)Al =
literature (see [1]). Bl,Au = Bu}. The set of all roughly consistent objects
The contamination problem is the problem of reducing is partially ordered by the inclusion relation. Relative this
confusion among concepts from one semantic domain in maximalroughlyconsistentobjectswillbereferredtoasrough
another during the process of construction of semantics. The objects.Bydefiniteroughobjects,willbemeantroughobjects
useofnumericfunctionslikeroughmembershipandinclusion of the form H that satisfy
functions based on cardinalities of subsets is one source of
contamination.Therationalecanalsobeseeninthedefinition (∀A∈H)All =Al&Auu =Au. (3)
ofoperationslike(cid:116)inthedefinitionofpre-roughalgebra(for
example)thatseektodefineinteractionbetweenroughobjects Otherconceptsofroughobjectswillalsobeusedinthispaper.
but use classical concepts that do not have any interpretation
Proposition 1. When S is a granular operator space, (cid:98) is a
in the rough semantic domain. Details can be found in [2],
boundedpartialorderonS|≈.Moregenerallyitisabounded
[5].
quasi order.
II. GENERALGRANULAROPERATORSPACES In quasi or partially ordered sets, sets of mutually incom-
Definition 1. A General Granular Operator Space (GOS) S parable elements are called antichains. Some of the basic
is a structure of the form S = (cid:104)S,G,l,u,P(cid:105) with S being a properties may be found in [7], [8]. Antichains of rough
set, G an admissible granulation(defined below) over S, l,u objects have been used by the present author for forming
being operators : ℘(S) (cid:55)−→ ℘(S) and P being a definable algebraic models in [4]. In the paper the developed semantics
binarygeneralizedtransitivepredicate(forparthood)on℘(S) is applicable for a large class of operator based rough sets
satisfyingthesameconditionsasinDef.??exceptforthoseon including specific cases of RYS [1] and other less general
admissible granulations (Generalized transitivity can be any approacheslike[9].In[9],negationlikeoperatorsareassumed
proper nontrivial generalization of parthood (see [6]). P is ingeneralandthesearenotdefinableoperationsrelativeorder
proper parthood (defined via Pab iff Pab&¬Pba) and t is related operations/relation.
a term operation formed from set operations):
A. Parthood in GOS
(∀x∃y1,...yr ∈G)t(y1, y2,... yr)=xl It is necessary to clarify the nature of parthood even
and(∀x)(∃y , ... y ∈G)t(y , y ,... y )=xu, in set-theoretic structures like granular operator spaces. The
1 r 1 2 r
(Weak RA, WRA) restriction of the parthood relation to the case when the first
(∀y∈G)(∀x∈℘(S))(Pyx −→ Pyxl), argumentisagranuleisparticularlyimportant.Thetheoretical
assumption that objects are determined by their parts, and
(Lower Stability, LS)
specifically by granules, may not reasonable when knowledge
(∀x, y∈G)(∃z∈℘(S))Pxz, &Pyz&zl =zu =z,
of the context is evolving. This is because in the situation:
(Full Underlap, FU)
• granulation can be confounded by partial nature of infor-
In the granular operator space of [4], P =⊆, P =⊂ only mation and noise,
in that definition), P is proper parthood (defined via Pab • knowledge of all possible granulations may not be pos-
iff Pab&¬Pba) and t is a term operation formed from set sible and the chosen set of granules may not be optimal
operations. for handling partial information, and
On℘(S),iftheparthoodrelationPisdefinedviaaformula • the process has strong connections with apriori under-
standing of the objects in question.
Φ as per
Pab if and only if Φ(a,b), (1)
then the Φ-rough equality would be defined via a ≈
Φ
b if and only if Pab&Pba. In a granular operator space,
P=(cid:60) is defined by Fig. 1: Which Classes and What Parts of Flowers?( FreeFoto)
a(cid:60)b if and only if al ⊆bl&au ⊆bu. (2)
Parthood can be defined in various ways in GOS. Not
The rough equality relation on ℘(S) is defined via a ≈ that they are always definable in terms of other predi-
b if and only if a (cid:60) b&b (cid:60) a. Regarding the quotient cates/operations,butthatmanyaresodefinable-itispossible
S| ≈ as a subset of ℘(S), the order (cid:98) will be defined to do rough sets with parthood as a primitive relation. In
as per α (cid:98) β if and only if Φ(α,β) Here, Φ(α,β) is an picking lavender in a garden as in Fig 1, agents restrict
abbreviation for (∀a ∈ α,b ∈ β)Φ(a,b). (cid:98) will be referred themselves to classes approximating lavenders for example.
to as the basic rough order. Rough inclusion (Sec I-A) is a simpler example of parthood.
The following are more direct possibilities:
Assign f(x )=1 =s0(1 )
1 1 1
Pab←→al ⊆bl (Very Cautious) If f(x )=sr(1 )&(x ,x )∈R,
i j i i+1
Pab←→al ⊆bu (Cautious)
then assign f(x =1
i+1 j+1
Pab←→al ⊆bu\bl (Lateral) If f(x )=sr(1 )&(∀k<i+1)(x ,x )∈/ R,
i j k i+1
Pab←→au ⊆bu (Possibilist) then let f(x )=sr+1(1 )
i+1 j
Pab←→au ⊆bl (Ultra Cautious)
For this section, it will be assumed that S is a granular
Pab←→au ⊆bu\bl (Lateral+) operator space, S ⊆ ℘(S), R ⊂ S is the set of rough objects
Pab←→au\al ⊆bu\bl (Bilateral) (somesense),C⊆Sisthesetofcrispobjectsandthereexists
Pab←→au\al ⊆bl (Lateral++) a map ϕ : R (cid:55)−→ C2 satisfying (∀x ∈ R)(∃a,b ∈ C)ϕ(x) =
(a,b)&a ⊂ b, #(S) = n < ∞, #(C) = k, R∩C = ∅ and
Pab←→(∀g∈G)(g∈x−→g∈y) (G-Simple)
#(R)=n−k<n.
Proposition 2. Even though it is not required that ϕ(x) =
All of these except for lateral++ are transitive concepts of (a,b)&xl =a&xu =b, R mustberepresentable byafinite
parthoods that make sense in contexts as per availability and subset K⊆C2\C.
nature of information. Very cautious parthood makes sense
in contexts in which cost of misclassification is high or the Primitive Counting on Antichains(PCA)
association between properties and objects is confounded by The following conveys the basic idea of primitive counting
the lack of clarity in the possible set of properties. G-Simple on antichains (better algorithms may be possible). Every
is a version that refers granules alone and avoids references parthood P can be associated with P-antichain generated by
to approximations. the relation R defined by Rab if and only if ¬Pab&¬Pba.
Theorem 1. Lateral++ parthood is not a transitive and • Assign x1 to category C1 and set f(x1)=11.
reflexive relation but is a strictly confluent relation (as it • If¬Rx1x2 thenassignx2 tocategoryC1 andsetf(x2)=
satisfies Pab&Pac −→ (∃e)Pbe&Pce). In all other cases, 21 else assign x2 to a new category C2 and set f(x2)=
1 .
in the above, P is reflexive and transitive. Antisymmetry need 2
not hold in general. • If f(x2) = 21, ¬Rx1x3 and ¬Rx2x3 then assign x3 to
category C and set f(x ) = 3 . If f(x ) = 2 and
1 3 1 2 1
Proof: If P is a bilateral relation as defined above, then (Rx x or Rx x ) then assign x to category C and
1 3 2 3 3 3
Paabecauseau\al =au\al.IfPabandPbc,thenau\al ⊆ set f(x ) = 1 . If f(x ) = 1 and ¬Rx x then assign
3 3 2 2 1 3
bu\bl and bu\bl ⊆ cu\cl so au ⊆ al = cu\cl that is x to category C and set f(x )=2 . If f(x )=1 and
3 1 3 1 2 2
Pac. Similarly other cases can be verified. Counterexamples (Rx x and ¬Rx x then assign x to category C and
1 3 2 3 3 2
for lateral++ parthood are easy to construct. set f(x )=2
3 2
• Proceed Recursively under the following conditions:
• No two distinct elements a, b of a category Ci satisfy
Rab
III. COUNTINGFORSEMANTICS
• For distinct i,j, (∀a∈Ci)(∃b∈Cj)Rab
Theorem 2. In the above context all of the following hold:
Ifitispossibletocount(inageneralizedsense)acollection
of objects in accordance with rules dependent on the relative • Objects of category C1 form a maximal antichain.
types of objects then these may be useful for discovering new • The objects in category Ci would be enumerated by
inferences about the collection - this idea has been used in seque(cid:80)nces of the form (Qi = Card(Ci)) 1i,2i,...,Qi
[1] in the context of the contamination reduction approach and Qi =n.
for actually deducing algebraic semantics for classical rough • Objectsofeachcategoryformsanantichainandeachxi
sets. The History based primitive counting (HPC) method [1] belongs to exactly one of the categories.
is a way of counting discernible and indiscernible objects in
Proof: Most of the proof follows from the construction.
which full memory of objects being counted is retained over
All objects of the category C are mutually compared and
1
temporalprogression.Thisdoesnotleadtoantichains,butthe
objects not in C are R-related to at least one object in C .
1 1
modification proposed in this section permits as much.
So C is a maximal antichain.
1
The collection being counted is taken to be
{x , x , ..., x , ..., } for simplicity and R is a general History Aware Primitive Counting on Antichains (HPCA)
1 2 k
indiscernibility over it. The relation with all preceding steps InthisvariationofPCA,objectswillbepermittedtobelong
of counting is taken into account and when R is symmetric, to multiple categories - with the aspect being reflected in
the following is the HPC algorithm: the end result of the counting process. This allows for a
enumerationforamaximalantichaindecomposition.Thesteps of A being numbered as for C , that is in the form
3 3
for this method of counting are as follows: 1 ,2 ,...,Q .
3 3 3
• Assign x1 to category C1 and set f(x1)=11. • Proceed till stopping criteria is satisfied.
• If¬Rx1x2 thenassignx2 tocategoryC1 andsetf(x2)= Theorem4. Letthesetofmaximalantichainsobtainedbythe
2 else assign x to a new category C and set f(x )=
1 2 2 2 FHCA method be A. A need not be associated with a HPCA
T .
2 coherent order <.
• If f(x2) = 21, ¬Rx1x3 and ¬Rx2x3 then assign x3 to
category C and set f(x ) = 3 . If f(x ) = 2 and
1 3 1 2 1
A. Applications
(Rx x or Rx x ) then assign x to category C and
1 3 2 3 3 3
set f(x3)=T3. The novel application scenarios to which the developed
• For j > 2, if f(xj) = k1 for k ≤ j, ¬Rxxj+1 for all methodology apply qualify as parts of the inverse problem
x∈C1 (C1 at this step) then assign xj+1 to category C1 [1]. Based on some understanding of properties and their
and set f(xj+1) = k+11. If f(xj) = k1 and (Rxxj+1 connection with objects (this need not be clearly understood
for at least an x ∈ C1 at this stage) then assign xj+1 to beforehand), suppose an agent specifies approximations and
category Cj−k+1 and set f(xj+1)=Tj+1. indiscernibilitiesofobjects.Intheusualwayofapplyingrough
• Continue till all objects have been covered for the con- sets,approximationsarediscoveredfromconcreteinformation
structionofC1.TheonlyfunctionofTj isinlocatingthe systems/tables. But here the origin can be very dense.
start point for the next step. With the approximations and indiscernibilities of objects,
• StoppingCondition:∪Ci =S(atthisstage)andfori(cid:54)=j it would be certainly possible to count the objects in various
Ci (cid:42)Cj or all start points are exhausted. orders in the senses mentioned in this section. Using this it
• Find the argument for f for which j in Tj is a minimum wouldbepossibletopredictwhethertheinformationhasbeen
• Assign xj to the category C2 and set f(xj)=12. derived from a rough perspective or not. Algebraic semantics
• Proceed as for x1, but over the set may also be derivable if admissible. More details relating to
S(j) ={x ,x ,...,x ,x ,x ,...,x } (4) this would be part of a forthcoming paper.
j j+1 n 1 2 j−1
If rough sets are interpreted from the perspective of knowl-
to derive all the elements of C2. Continue till stopping edge (see [10], [11] and references therein), definite objects
condition is satisfied. represent exact concepts, while others are not exact. Non
Theorem 3. HPCA may possibly yield a decomposition of S definiteobjectsmaybeapproximatedbylowerdefiniteobjects
into maximal antichains C ,C ,...,C or just a collection that correspond to the part of knowledge that is definitely
1 2 q
of maximal antichains C ,C ,...,C satisfying contained in the object. For this to be valid it is neces-
1 2 q
sary that for the object A in question, All = Al and
(cid:91)
Ci ⊆S. (5) Alu = Al hold. Similarly when A is approximated by its
upperapproximationAu,thelattercorrespondstothepossible
Proof:TheorderstructureinitiallyassumedonS,restricts
concepts that may correspond to A. For this interpretation
the maximal antichains that may be found by the HPCA
to be valid, it is again necessary that Auu = Au hold.
algorithm.
In general, the approximations Al,Au may not correspond
It is possible that the order in which the discernible ob-
to exact knowledge and many generalized concepts of exact
jects are found restricts the discernibility of objects found
knowledge may be definable/relevant. In fact, GOS permits
subsequently. So many maximal antichains are bound to be
encoding knowledge that can be successively refined on the
excluded. Counter examples are easy to construct.
possible knowledge aspect for every choice of parthood. The
The above proof motivates the following definition:
present paper also motivates the problem of characterization
Definition 3. A total order < on S will be said to be HPCA of the correspondence between the general counts and rough
coherentifandonlyiftheHPCAproceduregeneratesasetof ontologyandthestructuresanalogoustoantichainsassociated
maximal antichains {Ci :i=1,2,...,q} such that ∪Ci =S. withgeneralizedtransitiveparthoods.Specifically,thealgebras
corresponding to FHCA and HPCA are of interest.
Full History Based Counting on Antichains (FHCA)
In this method the steps shall be the same as for HPCA if
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