On Possible Applications of Rough Mereology to
Handling Granularity in Ontological Knowledge
Pavel Klinov and Lawrence J. Mazlack
Applied Computational Intelligence Laboratory
University of Cincinnati
Cincinnati, Ohio 45221
Overview
...specifying and analyzing of a complex structure
where the inference engine must take into account the
uncertain character of knowledge about objects and
complex structures (Polkowski & Skowron 1996).
Representing and reasoning about knowledge is critical in
Artificial Intelligence. There is a distinction between factual
and ontological knowledge. Factual knowledge represents a
set of facts about individual objects that are known or believed whereas ontological (background) knowledge represents implicit concepts and relationships that are assumed
to exist in the world. Ontological knowledge is often represented as a hierarchy of concepts because splitting things of
the real world into categories and sub-categories is a natural
way of human thinking. One example of conceptual hierarchies in AI is ontologies that are widely used in such areas
as Natural Language Processing, Semantic Web, etc.
Representation of both types of knowledge becomes difficult when the knowledge is imprecise. One example of
imprecision is granularity i.e. inability to distinguish between the individual objects. In this case, knowledge cannot
be represented precisely but can be approximated with respect to the granularity of the domain. Approximation of
factual knowledge has been extensively researched and often employs Rough Set Theory (Pawlak 1982) for dealing
with indiscernibility of objects. Similar approach has been
applied to ontologies to approximate concepts in the hierarchy (Doherty et al. 2003). The open problem is the approximations of hierarchical relationships (such as ”is-a”,
”part-of”) between concepts. This paper addresses this issue using Rough Mereology (Polkowski & Skowron 1996)
complemented with Interval Analysis (Moore 1966).
The principal contribution is to provide rough mereological functions that can be used for representation and reasoning with formal ontologies in approximation spaces. Specifically, approximate concept membership and approximate
concept subsumption functions will be provided. It can be
demonstrated that the interval based functions are free of the
shortcomings of the previously suggested definitions (Cao,
Sui, & Zhang 2003) (Klinov & Mazlack 2007).
The primitive notion of Rough Mereology is a function
called rough inclusion that defines the extent to which one
possibly complex object is a part of another. Rough inclusion functions can be used in knowledge representation
(Cao, Sui, & Zhang 2003). For example, the following problems that often occur in ontology engineering may be approached by means of rough inclusion functions:
• Whether the object, say, x belongs to the concept A? The
problem can be reduced to computing the rough inclusion
of x’s equivalence class into the set that models A (membership function).
• Whether the concept A ”is-a” concept B (whether A is
subsumed by B)? The problem can be reduced to computing the rough inclusion of set A into set B (subsumption
function).
Both questions are subject to a degree in imprecise domains. Granularity, as a special type of imprecision, can be
modeled by approximation spaces. In this case, it is of interest to compute lower and upper approximations of membership and subsumption degrees. This can be done using
interval rough inclusion functions.
Degree of Membership
It is natural to model imprecise ontological concepts as
rough sets in the case of granular knowledge. Then the approximate object membership can be computed using the
standard Pawlak’s rough membership function μ (Pawlak &
Skowron 1994). The problem with the rough membership
in reasoning with ontologies is the computation of membership degrees to intersections and unions of concepts. This is
not possible in the general case (Pawlak & Skowron 1994).
However, it has been shown that μA∩B and μA∪B must fall
into certain intervals (Yao 2004). This property suggests use
of closed intervals instead of real numbers for the representation of rough membership degrees. Operations on intervals
can be adopted as defined in Interval Analysis.
Let ν denote the interval domain. Interval membership
function μ : 2|U | × 2|U | → ν can be defined as an extension
Interval Rough Mereology
Rough Mereology is a recently proposed paradigm for approximate reasoning based on Rough Set Theory (Polkowski
1996). It was developed for:
c 2007, Association for the Advancement of Artificial
Copyright Intelligence (www.aaai.org). All rights reserved.
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of the standard rough membership (Pawlak 1982):
μA (x) = μ([x]R , A) = [
[x]R ∩ A [x]R ∩ A
,
]
[x]R
[x]R
unified definition of the arithmetic operations of intervals
(Moore 1966). This is not an accident. Rough sets can be
viewed as interval structures. Analogously to how real numbers fall into imprecision intervals, crisp sets fall into intervals induced by approximation spaces.
It is straightforward to verify that the function 8 is free
of the described above shortcomings and obeys mandatory
rough inclusion properties (Polkowski & Skowron 1996).
(1)
Then approximations of memberships to union or intersection of concepts can be computed as follows:
μA∩B (x)
=
μA∪B (x)
[max(0, μA (x) + μB (x) − 1),
min(μA (x), μB (x))]
=
[max(μA (x), μB (x)),
min(1, μA (x) + μB (x))]
(2)
Summary and Future Directions
The paper proposes rough mereological approach to approximating concepts and hierarchical relationships for knowledge representation. The approach can be effectively used in
systems that employ Rough Set Theory to handle the granularity of knowledge. Hierarchical relationships are approximated using the subsumption function that computes the degree of inclusion of one rough concept into another. This
is useful in ontology engineering where concept hierarchy
occupies a central place. Formal ontologies for imprecise
domains can be approximated using the proposed mereological approach and be used by for approximate reasoning.
There is a number of future research directions. In particular, the approach may be generalized to work with similarity (tolerance) spaces. This is useful when similarity better
characterizes relationships among objects rather than indiscernibility. Another future direction is to use rough fuzzy
hybridization to handle different types of imprecision, for
example, granularity and vagueness. Interval approach is
still expected to be beneficial because intervals can naturally
represent lower and upper approximations of fuzzy membership values.
(3)
Degree of Subsumption
Another common problem in ontological knowledge representation is determining whether or not a concept is more
specific/generic than another concept. The problem can be
reduced to measuring the extent to which one possibly imprecise set is a part of another. It can be done by providing a mereological subsumption function. Two rough inclusion functions for computing subsumption have been recently proposed (Cao, Sui, & Zhang 2003):
|B(X) ∩ B(Y )|
(4)
|B(X) ∪ B(Y )|
1
|X ∩ Y | |X ∩ Y |
(5)
ϑ(X, Y ) = ×
+
2
|X ∪ Y | |X ∪ Y |
where B(X) and B(Y) are boundary regions.
Unfortunately these functions have some undesirable
properties (Klinov & Mazlack 2007). For example, the function 4 returns counterintuitive result, namely 0, in the case
when: X ⊆ Y . The function 5 is unable to distinguish
between the case when X ⊆ (Y \Y ) and when both pairs
(X, Y ) and (X, Y ) have 50% overlap. This may lead to a
significant loss of information when reasoning in an imprecise environment.
Analogously to the membership, the subsumption function can be interval based: η : 2|U | × 2|U | → ν. Then given
some inclusion function for crisp sets ηc , η must have the
following properties (η(X, Y ) = [α, β]):
ρ(X, Y ) =
0 ≤ α ≤ minX⊆A⊆X;Y ⊆B⊆Y (ηc (A, B))
(6)
maxX⊆A⊆X;Y ⊆B⊆Y (ηc (A, B)) ≤ β ≤ 1
(7)
References
Cao, C.; Sui, Y.; and Zhang, Z. 2003. Rough mereology in
knowledge representation. In International Conference On
Rough Sets, Fuzzy Sets, Data Mining and Granular Computing, 329–333.
Doherty, P.; Grabowski, M.; W., L.; and Szalas, A. 2003.
Towards a framework for approximate ontologies. Fundamenta Informaticae 57(2–4):147–165.
Klinov, P., and Mazlack, L. J. 2007. Interval rough mereology for approximating hierarchical knowledge. In Joint
Rough Set Symposium. To appear.
Moore, R. E. 1966. Interval Analysis. Englewood Cliffs,
NJ: Prentice-Hall.
Pawlak, Z., and Skowron, A. 1994. Rough membership
functions. In Advances in the Dempster-Shafer theory of
evidence. New York, NY, USA: John Wiley & Sons, Inc.
251–271.
Pawlak, Z. 1982. Rough sets. International Journal of
Computers and Information Science 11:341–356.
Polkowski, L., and Skowron, A. 1996. Rough mereology:
a new paradigm for approximate reasoning. International
Journal of Approximate Reasoning 15:333–365.
Yao, Y. Y. 2004. Semantics of fuzzy sets in rough set
theory. In Transactions on Rough Sets II. Springer Berlin
Heidelberg. 297–318.
The properties can be given the following interpretation
if viewing rough sets X and Y as collections of all sets enclosed between their lower and upper approximations:
If any imprecision were eliminated, the degree of inclusion of X into Y computed according to ηc cannot drop
below α and exceed β.
According to this interpretation, η can be computed as
follows:
[α, β] = [min(V1 , V2 , V3 , V4 ), max(V1 , V2 , V3 , V4 )] (8)
where: V1 = ηc (X, Y ), V2 = ηc (X, Y ), V3 = ηc (X, Y ),
V4 = ηc (X, Y ). Note that this definition is similar to the
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