From: FLAIRS-02 Proceedings. Copyright © 2002, AAAI (www.aaai.org). All rights reserved.
From the Egg-Yolk to the Scrambled-Egg Theory
Hans W. Guesgen
Computer Science Department, University of Auckland
Private Bag 92019, Auckland, New Zealand
hans@cs.auckland.ac.nz
Abstract
The way we deal with space in many everyday situations is
on a qualitative basis, allowing for imprecision in spatial descriptions when we interact with each other. This is often
achieved by specifying the spatial relations between the objects or regions that we talk about. In artificial intelligence,
a variety of formalisms have been developed that deal with
space on the basis of relations between objects or regions that
objects might occupy. One of these formalisms is the RCC
theory, which is based on a primitive relation, called connectedness, and uses a set of topological relations, defined on
the basis of connectedness, to provide a framework to reason
about regions. This paper discusses an extension of the RCC
theory, which deals with vagueness in spatial representations.
The extension is based on the egg-yolk theory, but unlike the
egg-yolk theory uses fuzzy logic to express vagueness.
Motivation
There are many ways of dealing with spatial information,
but researchers have been arguing successfully that human
beings often do so in qualitative way. Therefore, computer systems should support such a form of reasoning
(see, e.g., (Hernández 1991)). In (Guesgen & Hertzberg
1993), for example, we introduce a form of spatial reasoning that extends Allen’s temporal logic (Allen 1983) to the
three dimensions of space by applying very simple methods for constructing higher-dimensional models and for reasoning about them, namely combination (i.e., building tuples of one-dimensional relations) and projection (i.e., extracting one-dimensional aspects from the tuples). There
are other approaches that proceed in more or less the same
way (Freksa 1990; Hernández 1991; Mukerjee & Joe 1990).
However, it seems that more recently the RCC theory (Randell, Cui, & Cohn 1992) has gained a particular interest in
the research community. This first-order theory is based on
a primitive relation, called connectedness, and uses eight
topological relations, defined on the basis of connectedness,
to provide a framework to reason about regions.
The original RCC theory has been designed to deal with
precisely defined regions, but later on has been extended to
cope with vagueness in spatial representations, in particular vague or indeterminate boundaries (Cohn & Gotts 1996).
­
Copyright c 2002, American Association for Artificial Intelligence (www.aaai.org). All rights reserved.
476
FLAIRS 2002
Point in the region
Vague point
Point outside the region
Figure 1: A vague region represented by two crisp regions
(the egg and the yolk).
The extension, called the egg-yolk theory, uses two crisp regions, the egg and the yolk, to characterize a vague region.
All points within the yolk are considered to be in the region,
whereas all points outside the egg are outside the region. The
white characterizes the points that may or may not belong to
the region (see Figure 1).
In this paper, we introduce an extension of the egg-yolk
theory, called the scrambled-egg theory. The idea is to utilize the neighborhood structure that is inherent in the RCC
theory and to define fuzzy sets for the relations between regions based on this neighborhood structure. Rather than
distinguishing between points that are definitely in the region (the yolk) and those that are possibly in the region (the
white), we mix the two sets of points into one region (the
scrambled egg).
Our approach is to some degree related to the one described in (Egenhofer & Al-Taha 1992), which uses the concept of 9-intersection instead of the RCC theory as basis.
Egenhofer and Al-Tahu do not use fuzzy logic, but they utilize a structure similar to the RCC neighborhood structure,
called the closest topological relationship graph, to deal with
indeterminate boundaries.
The paper is organized as follows. We start with a brief
review of the RCC theory and its extension to regions with
indeterminate boundaries. We then show a way of associating fuzzy sets with the relations in the RCC theory, which
can be viewed as an alternative for representing relations between regions with indeterminate boundaries. Finally, we
will sketch an algorithm to reason about these fuzzy sets.
The RCC Theory Revisited
The idea of using relations to reason about spatio-temporal
information dates back at least to the beginning of the eight-
Relation/Interpretation
DC(X,Y)
EC(X,Y)
X externally connected to Y
PO(X,Y)
Figure 2: The relations between two rectangles with respect
to the -axis and -axis, where denotes the Allen relation
.
ies, when Allen (1983) introduced an interval logic for reasoning about relations between time intervals. Although
Allen’s logic can be used to reason about three-dimensional
space (Guesgen 1989), it often leads to counterintuitive results, in particular if rectangular objects are not aligned to
the chosen axes (see Figure 2).
The RCC theory (Randell, Cui, & Cohn 1992) avoids this
problem by defining the relation between two regions based
on their topological properties and therefore independently
of any coordinate system. The basis of the RCC theory is
the connection relation, which is a reflexive and symmetric
relation, satisfying the following axioms:
1. For each region : X partially overlaps Y
EQ(X,Y)
X identical with Y
TPP(X,Y)
X tangential proper part of Y
TPPi(X,Y)
Y tangential proper part of X
NTPP(X,Y)
X nontangential proper part of Y
2. For each pair of regions , : From this relation, additional relations can be derived, which
include the eight jointly exhaustive and pairwise disjoint
relations:1
Reasoning about space is achieved in the RCC theory by
applying a composition table to pairs of relations, similar to
the composition table in Allen’s logic. Given the relation between the regions and , and the relation between
the regions and , the composition table determines the
relation between the regions and , i.e., .
with more than three reIn the case of a set of regions
gions, the composition table can be applied repeatedly to
three-element subsets of
until no more relations can be
updated, resulting in a set of relations that is locally consistent. This is basically the idea behind Allen’s algorithm.
X disconnected from Y
Æ
Imprecision in Spatial Relations
Reasoning about space often has to deal with some form of
imprecision. For example, when we talk about a region like
1
See Figure 3 for an interpretation and a graphical illustration
of the relations.
Illustration
NTPP(X,Y)
Y nontangential proper part of X
Figure 3: An illustration of the relations.
the city of Auckland, we usually do not know exactly where
the boundaries are for that regions. Nevertheless, we are
perfectly capable to reason about such a region. Or if we
hear on the radio that a cold front is moving in from Antarctica, we can estimate when this front “connects” to New
Zealand, although we might not be able to decide with certainty whether the cold front is still disconnected from (),
externally connected to (
), or already partially overlapping () New Zealand.
Lehmann and Cohn (1994) have introduced an extension
to the RCC theory, called the egg-yolk theory, which deals
with imprecision in spatial representations by using two
crisp regions to characterize an imprecise region. One of
these regions is called the yolk, the other one the egg. All
points within the yolk are considered to be in the region,
whereas all points outside the egg are outside the region.
FLAIRS 2002
477
The white (i.e., the egg without the yolk) characterizes the
points that may or may not belong to the region.
The egg-yolk theory uses a set of five base relations,
called , instead of the eight base relations in :
Given two imprecise regions and , the relations
are used to describe the relationship between (1) the egg of
and the egg of , (2) the yolk of and the yolk of , (3)
the egg of and the yolk of , and (4) the yolk of and
the egg of , resulting in 46 possible relationships between
and .
As Lehmann and Cohn point out, it is possible to use more
than two regions to describe an imprecise region. We follow
this idea here and combine it with an approach that we used
before to introduce imprecise reasoning into Allen’s logic.
The approach is based on the concept of conceptual neighborhoods, which was first introduced by Freksa (1992) for
Allen relations and later applied to the RCC theory (Cohn et
al. 1997; Cohn & Gotts 1996).
Conceptual Neighborhoods and Fuzzy Sets
Two relations on regions and are conceptual neighbors if the shape of or can be continuously deformed
such that one relation is transformed into the other relation
without passing through a third relation. Figure 4 shows the
conceptual neighbors for the relations.
The notion of conceptual neighbors can be used to introduce imprecision into reasoning about spatial relations
(Guesgen & Hertzberg 1996). For that purpose, we first
represent each relation by a characteristic function as
follows:
The function yields a value of 1 if and only if the argument
is equal to the relation denoted by the characteristic
function:
if else
The next step towards the introduction of imprecision is
to transform the relations into fuzzy sets. A fuzzy
of a domain is a set of ordered pairs, ,
set where is an element of the underlying domain and
is the membership function of . In other
words, instead of specifying whether an element belongs
to a subset of or not, we assign a grade of membership
to . The membership function replaces the characteristic
function of a classical subset of .
In the context of the relations, this means that each
relation is represented as a set of pairs, each pair consisting of an element of (which is the underlying domain) and the value of the characteristic function of the relation applied to that element. For example, if two regions
and are externally connected (i.e., ), we use
the characteristic function of the relation to convert this
statement into the following:
478
FLAIRS 2002
Instead of having two classes, one with the accepted relations where results in 1 and another with the discarded
relations where results in 0, we now assign acceptance
grades (or membership grades, to use the term from fuzzy
set theory) with the relations. If the relation is , we assign the membership grade 1; if the relation is a neighbor of
, we choose a membership grade with ; if
the relation is a neighbor of a neighbor of , we assign a
grade with ; and so on.
Since there is no general formula for determining the
membership grades , choosing the right grade
for each degree of neighborhood can be a problem. On the
other hand, there are experiments showing that fuzzy membership grades are quite robust, which means that it is not
necessary to have precise estimations of these grades (Bloch
2000). The explanation given for this observation is twofold:
first, fuzzy membership grades are used to describe imprecise information and therefore do not have to be precise, and
second, each individual fuzzy membership grade plays only
a minor role in the whole reasoning process, as it is usually combined with several other membership grades. If the
membership grades are combined using the min/max combination scheme, as it is the case in the rest of this paper, we
do not even need numeric values for the alphas. In this case,
reasoning can be performed on symbolic values, provided
that there is a total order on the alphas.
Non-atomic relations (i.e., disjunctions of relations) can be transformed into fuzzy relations by
using the same technique. A non-atomic relation is
given by a set of atomic relations, which is interpreted
in a disjunctive way. We therefore transform each atomic
relation in the set into a fuzzy relation and compute
the fuzzy union of the resulting sets. There are different
ways of computing the union of fuzzy sets. Here, we choose
the one introduced in (Zadeh 1965), which associates with
each element in the resulting fuzzy set the maximum of the
membership grades that the element has in the original fuzzy
sets.
Formally, a fuzzy relation can be defined by using a function that denotes the conceptional distance between the relation and a relation , i.e., results in 1 if
is a neighbor of , in 2 if is a neighbor of a neighbor of
, and so on:
can be defined recursively as follows:
1. If , then
2. Otherwise,
neighbor of Given a sequence of membership grades, , the function can be used to associate relations with membership grades, depending on some given
relation (see Figure 5 for an example). In particular,
we can define a membership function as follows:
Figure 4: The relations arranged in a graphs showing the conceptual neighbors.
«¾
«½
1
«½
«¿
«¾
«¿
«¾
Figure 5: The assignment of membership grades to the relations with as reference relation.
With this definition, the fuzzy relation of a relation
is given by the following:
We now extend the formulation of relations as characteristic functions to the composition of relations,
starting again with crisp relations and continuing with fuzzy
relations. In the crisp case, the composition table can be represented as a set of characteristic functions of the following
form:
½ Æ ¾ The function yields a value of 1 for arguments that are elements of the corresponding entry in the composition table;
otherwise, a value of 0:
½Æ ¾
if else
Æ
For example, if and , then the characteristic function of the relation is defined as follows:
Æ Æ
Æ
if else
Adopting the min/max combination scheme from fuzzy
set theory, we can now define the fuzzy composition of two fuzzy relations and as the following
fuzzy relation:
Æ Æ
where ½Æ ¾
½Æ ¾
½
¾
is given by the following:
½ ¾ Ê ½ ÆÊ ¾ ½
¾
The fuzzy composition of relations plays a central role
in a number of algorithms for reasoning about fuzzy FLAIRS 2002
479
relations. One of these algorithms is an Allen-type algorithm for computing local consistency in networks of fuzzy
relations. Input to this algorithm is a set of regions
and a set of (not necessarily atomic) fuzzy relations.
The aim of the algorithm is to transform the given relations
into a set of relations that are consistent with each other.
This is achieved through an iterative process that repeatedly
looks at three regions , , and , and their fuzzy relations
, , and , computes the composition of two of the relations, and compares the result with the
third relation:
Æ
Unlike Allen’s original algorithm, the fuzzy version of the
algorithm does not make a yes/no decision about whether a
relation is admissible or not, but computes a new membership grade for that relation. The new membership grade is
compared with the initial membership grade of the relation.
If the new grade is smaller than the initial grade, the membership grade of the relation is updated with the new grade.
Summary
This paper introduces an extension of the RCC theory, which
is based on work done by Bennett, Cohn, Gooday, Gotts,
and others at the University of Leeds. The main idea is to
associate fuzzy sets with the relations in the RCC theory,
utilizing the notion of conceptual neighborhoods.
Unlike the egg-yolk theory, our approach encodes vagueness in the relations between regions, rather than the regions
themselves. In other words, we do not distinguish between
egg white and yolk, but view each region as a “scrambled”
region that has a set of relations, each to a certain degree,
with another region.
References
Allen, J. 1983. Maintaining knowledge about temporal
intervals. Communications of the ACM 26:832–843.
Bloch, I. 2000. Spatial representation of spatial relationship knowledge. In Proc. KR-00, 247–258.
Cohn, A., and Gotts, N. 1996. The ‘Egg-Yolk’ representation of regions with indeterminate boundaries. In Burrough, P., and Frank, A., eds., Geographical Objects with
Undetermined Boundaries, GISDATA Series No. 2. London, England: Taylor and Francis. 171–187.
Cohn, A.; Bennett, B.; Gooday, J.; and Gotts, N. 1997.
Representing and reasoning with qualitative spatial relations about regions. In Stock, O., ed., Spatial and Temporal
Reasoning. Dordrecht, The Netherlands: Kluwer. 97–134.
Egenhofer, M., and Al-Taha, K. 1992. Reasoning about
gradual changes of topological relationships. In Frank, A.;
Campari, I.; and Formentini, U., eds., Theories and Methods of Spatio-Temporal Reasoning in Geographic Space,
Lecture Notes in Computer Science 639. Berlin, Germany:
Springer. 196–219.
Freksa, C. 1990. Qualitative spatial reasoning. In Proc.
Workshop RAUM, 21–36.
480
FLAIRS 2002
Freksa, C. 1992. Temporal reasoning based on semiintervals. Artificial Intelligence 54:199–227.
Guesgen, H., and Hertzberg, J. 1993. A constraint-based
approach to spatiotemporal reasoning. Applied Intelligence
(Special Issue on Applications of Temporal Models) 3:71–
90.
Guesgen, H., and Hertzberg, J. 1996. Spatial persistence.
Applied Intelligence (Special Issue on Spatial and Temporal Reasoning) 6:11–28.
Guesgen, H. 1989. Spatial reasoning based on Allen’s temporal logic. Technical Report TR-89-049, ICSI, Berkeley,
California.
Hernández, D. 1991. Relative representation of spatial knowledge: The 2-D case. In Mark, D., and Frank,
A., eds., Cognitive and Linguistic Aspects of Geographic
Space. Dordrecht, The Netherlands: Kluwer. 373–385.
Lehmann, F., and Cohn, A. 1994. The EGG/YOLK reliability hierarchy: Semantic data integration using sorts with
prototypes. In Proc. 3rd International Conference on Information and Knowledge Management (CIKM-94), 272–
279.
Mukerjee, A., and Joe, G. 1990. A qualitative model for
space. In Proc. AAAI-90, 721–727.
Randell, D.; Cui, Z.; and Cohn, A. 1992. A spatial logic
based on regions and connection. In Proc. KR-92, 165–
176.
Zadeh, L. 1965. Fuzzy sets. Information and Control
8:338–353.