A Novel Framework of Fuzzy Complex Numbers and its
Application to Compositional Modelling
Xin Fu and Qiang Shen
Abstract?Dealing with various inexact pieces of information
has become an intrinsically important issue in knowledge
based reasoning, because many problem domains involve im
precise, incomplete and uncertain information. Indeed, different
approaches exist for reasoning with inexact knowledge and
data. However, the common strategy they adopt is to integrate
various types of inexact information into a global measure. This
may destroy the underlying semantics associated with different
information components. This paper presents an innovative
notion of fuzzy complex numbers (FCNs), which extends real
complex numbers to representing twodimensional uncertainties
conjunctively without necessarily integrating them. This new
framework is applied to supporting Compositional Modelling
(CM). In particular, calculus of FCNs over arithmetic and
propositional relations is developed to entail scenario model
synthesis from model fragments, and modulus of FCNs is
introduced to constrain the scenario descriptions. The utility
and usefulness of this work are illustrated by means of an
example for constructing possible scenario descriptions from
given evidence in the crime investigation domain.
I. INTRODUCTION
Solving many realworld problems requires the use of
inexact information captured in a variety of forms. Concep
tually, such information may be classified into the following
four types, which are of particular interest to the present
work: 1) Vagueness: It occurs when the boundary of a piece
of information can not be determined precisely (e.g. Bob is
tall). 2) Uncertainty: It depicts the reliability or confidential
weight of a given piece of information, usually represented
by a numerical value (e.g. The suspect overpowers the
victim, with certainty degree 0.7). 3) Both vagueness and
uncertainty: That is, types 1 and 2 coexist (e.g. The amount
of collected fiber is a lot, with certainty degree 0.7). 4)
Both vagueness and uncertainty with the uncertainty also
expressed in vague terms: Instead of using numerical values,
the uncertainty is described by a fuzzy number or linguistic
term (e.g. The amount of collected fiber is a lot, with
certainty degree very likely).
To address such information, much work has been devel
oped, especially to support reasoning with inexact knowledge
and data. Although the application domains and problem
solving approaches may be rather different, they all aim to in
tegrate the underlying distinct bits of inexact information into
a global measure. Whilst different kinds of information may
be effectively integrated in these approaches, the underlying
semantics associated with different information components
may be destroyed. Thus, it is of great interest and potentially
beneficial to establish a new mechanism which will maintain
Xin Fu (email: xxf06@aber.ac.uk) and Qiang Shen (email:
qqs@aber.ac.uk) are with the Department of Computer Science,
Aberystwyth University, Aberystwyth, UK.
the associated semantics when perform reasoning without
necessarily using just one global scale.
Inspired by this observation, this paper proposes a novel
framework of fuzzy complex numbers (FCNs) that will entail
effective and efficient representation of aforementioned types
of inexact information conjunctively and explicitly. Note that
the term FCNs is not new; the concept of conventional
complex numbers has been extended with fuzzy set theory for
the purpose of providing richer representations. In particular,
a fuzzy complex number has been defined in [1] as a type
1 fuzzy set, mapping from the complex plane to [0,1].The
basic arithmetic operations including addition, substraction,
multiplication and division have been developed for such
fuzzy complex numbers, using the extension principle. Fur
ther, the differentiation and integration of this type of fuzzy
complex number is proposed in [2], [3], with more advanced
followon work given in [14], [15]. Also, another notion that
connects real complex numbers to fuzzy sets is introduced
in [11], where a new type of set, named complex fuzzy sets,
is suggested to allow the membership value of a standard
fuzzy set to be represented using a classical complex number.
Basic set theoretic operations are defined on complex fuzzy
sets, including complement, union and intersection. However,
as indicated in [11], it can be a difficult task to acquire
intuition for the concept of complexvalued membership.
This limits significantly the potential application of such
fuzzy sets. Nevertheless, work has continued along this
theme of research. This is evident in that a framework of
fuzzy reasoning systems that incorporate complex fuzzy sets
in propositional logic has been proposed in [4].
Briefly, existing work on socalled fuzzy complex numbers
or complex fuzzy sets is framed by either giving conventional
complex numbers a realvalued membership or assigning a
fuzzy set element with a complex number as its membership
value. These are rather different from what is proposed in
this paper, where both the real and imaginary values of an
FCN are in general, themselves fuzzy numbers, with each
having an embedded semantic meaning. In particular, the
paper introduces for the first time the mathematical definition
and operations of such FCNs. This work is applied to support
Compositional Modelling (CM) in an effort to deal with
inexact knowledge and data. CM has been developed to
synthesize and store plausible scenarios in many problem
domains with promising results [5], [9], [10], [12]. In this
paper, the calculus of FCNs over propositional and arithmetic
relations is introduced, thereby entailing scenario model gen
eration from model fragments. Also, it defines the modulus of
FCNs, which is applied to constrain the scenario descriptions.
The rest of this paper is organized as follows. Section II
introduces the background that sets the scene for the present
work. Section III proposes the innovative notion of FCNs,
which extends realvalued complex numbers to representing
twodimensional uncertainties. The proposed new type of
FCNs is integrated in the process of model composition,
with a propagation algorithm provided, in Section IV. This
is followed by an illustrative example in Section V, showing
the utility and usefulness of the proposed approach to support
equivocal death investigation. Finally, Section VI concludes
the paper and points out future research.
II. OUTLINE OF COMPOSITIONAL MODELLING
One of the most significant advantages of using CM is
its ability to automatically construct many variations of a
given type of scenario from a relatively small knowledge
base. In existing work, it is traditionally assumed that the
generic and reusable model fragments within the knowledge
base can all be expressed by precise and crisp information.
However, significant challenges arise for problems where
the degree of inexactness of available knowledge and data
can vary greatly. This work extends the existing CM work
to allow the generation of scenarios which is capable of
representing, storing and supporting inference about inexact
information, by the use of the propopsed notion of FCNs.
For completeness, the basic concepts of CM are briefly
introduced below.
The knowledge base (KB) in a compositional modeller
consists of a number of generic and reusable model frag
ments, each of which represents a set of relationships be
tween domain objects and their states for a certain type of
partial scenario. A model fragment (MF) has two parts that
encode domain knowledge: 1) The relations between domain
elements, often represented in a form that is similar to the
style of conventional production rules, but involving much
more general contents; and 2) a set of rule instances that
represent how certainly it is that the corresponding relation
ships hold. More formally, an MF is a tuple ?X,Y,H?, as
generally described below:
If X
Assuming H
Relations {X, H ? Y}
Distribution Y
{A
1
j
1
,...,A
n
j
n
,C
1
i
1
,...,C
q
i
q
? B
1
j
n+1
,...,B
m
j
n+m
: p}
In this definition, X = {x
1
,...,x
n
} is a set of antecedent
predicates. The If statement describes the required condi
tions for a partial scenario to become applicable; H =
{h
1
,...,h
q
} is a (possibly empty) set of assumptions, re
ferring to those pieces of information which are unknown or
cannot be inferred from others, but they may be presumed
to hold for the sake of performing hypothetical reason
ing; Y = {y
1
,...,y
m
} is a set of consequent predicates,
which describe the consequences when the conditions and
assumptions hold, including pieces of new knowledge or
relations which are derived from the hypothetical reasoning;
p indicates the certainty degree of its corresponding outcome
if the MF is instantiated; and finally within the Distribution
statement, the left hand side of the ?implication? sign in
each propositional instance is a combination of value pairs,
involving the values of antecedent and assumption variables,
and the right hand side indicates the corresponding possible
outcome if the MF is instantiated.
Note that, in CM, it is not required that every possible
combination of antecedent and assumption values has to be
assigned a certainty degree. This is because the number of
combinations will increase exponentially with the number of
variables. In acquiring domain knowledge, it is unlikely to
obtain all such details but the most significant components.
In the present work, the default certainty degree for those
unassigned combinations is set to 0.
For simplicity, in this work, the assumptions are treated
as part of the antecedent due to their logical equivalence.
Hence, a rule instance in an MF is of the form:
IF x
1
is A
1
j
1
?????x
n
is A
n
j
n
, THEN y
1
is B
1
j
n+1
?????y
m
is B
m
j
n+m
,
(1)
where the operator ? in the consequence is restricted to
logical conjunction in this paper, but ? in the antecedence
denotes either a conjunctive or disjunctive operator. Further,
this rule can be decomposed to multiple simpler rules each
involving a single consequence, and can equivalently be
written as:
IF x
1
is A
1
j
1
?????x
n
is A
n
j
n
,THENy
1
is B
1
j
n+1
,
???
IF x
1
is A
1
j
1
?????x
n
is A
n
j
n
,THENy
m
is B
m
j
n+m
.
Therefore, in the reminder of this work, only the rules with
single consequence will be considered.
In order to depict vague information in MFs, fuzzy rep
resentation has been introduced to CM in [6]. In particular,
the values of fuzzy variables can be represented by fuzzy
sets and certainty degrees can be represented in linguistic
terms or fuzzy numbers. Interested readers can refer to [6]
for further details.
III. FUZZY COMPLEX NUMBERS
A. Definition of FCNs
Inherit from the real complex numbers, an FCN, ?z,is
defined in the form of:
?z =?a + i
?
b, (2)
where both ?a and
?
b are fuzzy numbers with membership
functions ?
?a
(x) and ?
?
b
(x), regarding a given domain vari
able x. ?a is the real part of ?z while
?
b represents the imaginary
part, i.e. Re(?z)=?a and Im(?z)=
?
b.
An FCN can be visually shown as in Fig. 1. Importantly,
in general, for a given ?z, both Re(?z) and Im(?z) are fuzzy.
If
?
b does not exist, ?z degenerates to a fuzzy number. Further,
if
?
b does not exist and ?a iteself degenerates to a real number,
then ?z degenerates to a real number.
~
a
~
b
+ b
~
i
~
a
Im
Re
=z
~
Fig. 1. A fuzzy complex number
B. Operations on FCNs
The basic operations on FCNs form a straightforward
extension of those on real complex numbers. Let tildewidez
1
=?a+i
?
b
and tildewidez
2
=?c+i
?
d be two FCNs, where ?a,
?
b, ?c and
?
d are fuzzy
numbers with membership functions ?
?a
(x), ?
?
b
(x), ?
?c
(x) and
?
?
d
(x), respectively. The basic arithmetic operations on tildewidez
1
and tildewidez
2
are defined as follows:
Addition
tildewidez
1
+tildewidez
2
=(?a +?c)+i(
?
b +
?
d),
(3)
where ?a+?c and
?
b+
?
d are newly derived fuzzy numbers with
the following membership functions:
?
?a+?c
(y)=
logicalortext
y=x
1
+x
2
(?
?a
(x
1
) ? ?
?c
(x
2
)),
?
?
b+
?
d
(y)=
logicalortext
y=x
1
+x
2
(?
?
b
(x
1
) ? ?
?
d
(x
2
)).
(4)
Subtraction
tildewidez
1
?tildewidez
2
=(?a ? ?c)+i(
?
b ?
?
d),
(5)
where ?a??c and
?
b?
?
d are newly derived fuzzy numbers with
the following membership functions:
?
?a??c
(y)=
logicalortext
y=x
1
?x
2
(?
?a
(x
1
) ? ?
?c
(x
2
)),
?
?
b?
?
d
(y)=
logicalortext
y=x
1
?x
2
(?
?
b
(x
1
) ? ?
?
d
(x
2
)).
(6)
Multiplication
tildewidez
1
?tildewidez
2
=(?a?c ?
?
b
?
d)+i(
?
b?c +?a
?
d), (7)
where ?a?c?
?
b
?
d and
?
b?c+?a
?
d are newly derived fuzzy numbers
with the following membership functions:
?
?a?c?
?
b
?
d
(y)=
logicalortext
y=x
1
x
2
?x
3
x
4
(?
?a
(x
1
) ? ?
?c
(x
2
) ? ?
?
b
(x
3
) ? ?
?
d
(x
4
)),
?
?
b?c+?a
?
d
(y)=
logicalortext
y=x
1
x
2
+x
3
x
4
(?
?
b
(x
1
) ? ?
?c
(x
2
) ? ?
?a
(x
3
) ? ?
?
d
(x
4
)).
(8)
Division
tildewidez
1
tildewidez
2
=
parenleftBigg
?a?c +
?
b
?
d
?c
2
+
?
d
2
parenrightBigg
+ i
parenleftBigg
?
b?c ? ?a
?
d
?c
2
+
?
d
2
parenrightBigg
. (9)
For notational simplicity, let
?
t
1
=
?a?c+
?
b
?
d
?c
2
+
?
d
2
and
?
t
2
=
?
b?c??a
?
d
?c
2
+
?
d
2
,
where
?
t
1
and
?
t
2
are newly derived fuzzy numbers with the
following membership functions:
?
?
t
1
(y)=
logicalortext
y=
x
1
x
3
+x
2
x
4
x
2
3
+x
2
4
,x
2
3
+x
2
4
negationslash=0
(?
?a
(x
1
)? ?
?
b
(x
2
)? ?
?c
(x
3
)? ?
?
d
(x
4
)),
?
?
t
2
(y)=
logicalortext
y=
x
2
x
3
?x
1
x
4
x
2
3
+x
2
4
,x
2
3
+x
2
4
negationslash=0
(?
?a
(x
1
)? ?
?
b
(x
2
)? ?
?c
(x
3
)? ?
?
d
(x
4
)).
(10)
Modulus Given ?z
1
=?a + i
?
b, the modulus of ?z
1
is defined:
?z
1
 =
radicalBig
?a
2
+
?
b
2
. (11)
It is obvious that ?z
1
 is a newly derived fuzzy number
with the following membership function:
?
?z
1

(y)=
logicalordisplay
y=
?
x
2
1
+x
2
2
(?
?a
(x
1
) ? ?
?
b
(x
2
)). (12)
IV. PROPAGATION OF FCNSDURINGMODEL
COMPOSITION
When dynamically composing instantiated MFs into plau
sible scenarios, the vague and uncertain information (which
can be concisely represented in terms of FCNs) needs to be
propagated from individual fragments to their related ones.
This section describes how inexact information captured in
FCNs may be combined and propagated through relations.
Note that whilst CM is herein used to illustrate the combi
national propagation of FCNs, the underlying mechanism is
general and can be readily adapted to suit other problems.
A. Propagation
In CM, each proposition which reflects a node in the space
of plausible scenario descriptions (called emerging scean
rio space hereafter) has an FCN attached. Also, each rule
instance in any generic MF has an FCN attached, denoted
by ?z
r
, indicating the certainty degree of the corresponding
causal proposition. The propagation of FCNs during model
composition is a process of combining the FCNs given to the
variables (antecedents or consequent) with the FCN attached
to an instantiated rule instance (?z
r
).
For simplicity, the following rule instance in an MF:
IF x
1
is A
1
j
1
?????x
n
is A
n
j
n
,THENy is B
j
n+1
(?z
r
)
is rewritten as:
IF p
1
?????p
n
,THENc (?z
r
),
where p
1
,...,p
n
are the antecedent propositions, c is the
consequent proposition and ?z
r
is the FCN attached to the
rule. Again, ? in the antecedence may be interpreted as either
a conjunctive or disjunctive operator.
Proposition 1: If ? is a conjunctive operator, the aggre
gated FCN of the antecedent is:
?z
antecedent
= min(Re(?z
p
1
),...,Re(?z
p
n
))
+imin(Im(?z
p
1
),...,Im(?z
p
n
)).
(13)
Proposition 2: If ? is a disjunctive operator, the aggre
gated FCN of the antecedent is:
?z
antecedent
= max(Re(?z
p
1
),...,Re(?z
p
n
))
+imax(Im(?z
p
1
),...,Im(?z
p
n
)).
(14)
Since the rule instance in an MF only has the certainty
degree attached, i.e. ?z
r
= Re(?z
r
), the newly derived FCN
attached to the consequence is computed as:
?z
new
=?z
antecedent
? ?z
r
= Re(?z
antecedent
) ? ?z
r
+ iIm(?z
antecedent
) ? ?z
r
,
(15)
where Re(?z
antecedent
), Im(?z
antecedent
) and ?z
r
are repre
sented by different fuzzy numbers.
During model composition, given a set of collected ev
idence and a predefined KB, plausible scenarios can be
generated by joint use of two conventional inference tech
niques, namely backward chaining and forward chaining.
In forwardchaining, the values of the antecedent variables
are known, the value of the consequent variable is derived
by using the compositional rule of inference. However, due
to the lack of inverse operators over fuzzy sets, given the
value of the consequent variable, no general method exists
to derive the exact values of antecedent variables by direct
computation. This is especially the case when there are
more than one antecedent variable involved. To address this
problem, in this work, the fuzzy rule instances contained
within MFs are not used to derive the unknown values of
the antecedent variables. Instead, such causal constraints over
the antecedent and consequent variables are employed to
check for consistency amongst their respective values. Both
forward and backward chaining processes are explained in
detail below. Due to the nature of evidencedriven scenario
generation, backward chaining is described first.
1) Backward chaining: This involves the abduction of
domain variables and their states which might have led to
the available evidence. Plausible causes can be identified by
instantiating the conditions and assumptions of those MFs
whose consequent matches the given evidence. However, for
efficiency, only those possible causes whose consequences
matching with the given evidence have resulted in the great
est FCN modulus are created in the emerging scenario space.
As indicated earlier, during backward chaining, where ?z
c
and ?z
r
are given, no general method exists to derive the
exact value of ?z
antecedent
in a closed form. Hence, specific
fuzzy quantity spaces are built to approximately represent the
possible values of the real and imagery parts of ?z
antecedent
,
assuming that they can only take values from such predefined
quantity spaces. For computational simplicity, in this work,
it is further assumed that only fuzzy values from a common
quantity space, Q
FN
= {V
FN
1
,...,V
FN
j
,...,V
FN
n
},are
possibly taken by ?z
c
and ?z
r
.
Algorithm 1 is constructed by ensuring the generality
that each element in Q
FN
may be the possible value of
?z
antecedent
. Thus, all such values are checked using (15)
as the constraint. The value that best satisfies this constraint
when given ?z
c
, is selected to represent ?z
antecedent
. Note that
the Re(?z
c
) and Im(?z
c
) are checked against the constraint
separately.
Algorithm 1 BackwardPropagation(?z
c
, ?z
r
)
1: Re(?z
antecedent
) ??
2: Im(?z
antecedent
) ??
3: S
Re
=0
4: S
Im
=0
5: for j =1:n do
6: if S(Re(?z
c
),V
FN
j
? ?z
r
) >S
Re
then
7: S
Re
= S(Re(?z
c
),V
FN
j
? ?z
r
)
8: Re(?z
antecedent
)=V
FN
j
9: end if
10: if S(Im(?z
c
),V
FN
j
? ?z
r
) >S
Im
then
11: S
Im
= S(Im(?z
c
),V
FN
j
? ?z
r
)
12: Im(?z
antecedent
)=V
FN
j
13: end if
14: end for
15: return ?z
antecedent
Once ?z
antecedent
is obtained, the next step is to derive
the individual FCNs of each antecedent variable. Take the
conjunctive operator for example, the following constraints
are introduced:
min(Re(?z
p
1
),...,Re(?z
p
n
)) = Re(?z
antecedent
),
min(Im(?z
p
1
),...,Im(?z
p
n
)) = Im(?z
antecedent
).
(16)
These constraints simply state that, for any
(V
FN
m
,...,V
FN
k
) ? Q
FN
?????Q
FN
, where Q
FN
is
the fuzzy quantity space [13] for both real and imaginary
parts of FCN. If min(V
FN
m
,...,V
FN
k
)=Re(?z
antecedent
),
then V
FN
m
,...,V
FN
k
are possible solutions for each real
part of ?z
p
1
,...,?z
p
n
, respectively. Similarly, the possible
solutions of the imaginary parts can be obtained. Note that,
if ? is interpreted as a disjunctive operator, all is needed is
to change the min operator in (16) to max.
2) Forward chaining: This procedure applies logical de
duction to all such the MFs in the KB whose conditions
match the existing nodes in the emerging scenario space.
Therefore, the corresponding FCN propagation algorithm is
straightforward, as described in Algorithm 2. In particular,
?z
antecedent
and ?z
r
are known, the ?z
new
is computed by
applying (15), where ?z
new
is the calculated FCN of the
consequent variable. Given the fixed Q
FN
in this work, once
?z
new
is computed, it is used to match against the elements
of Q
FN
, the element receives the highest fuzzy matching
degree with ?z
new
is returned to represent ?z
c
.
Algorithm 2 ForwardPropagation(?z
p
1
,...,?z
p
n
, ?z
r
)
1: Calculate ?z
antecedent
2: ?z
new
=?z
antecedent
? ?z
r
3: S
Re
=0
4: S
Im
=0
5: for j =1:n do
6: if S(V
FN
j
,Re(?z
new
)) >S
Re
then
7: S
Re
= S(V
FN
j
,Re(?z
new
))
8: Re(?z
c
)=V
FN
j
9: end if
10: if S(V
FN
j
,Im(?z
new
)) >S
Im
then
11: S
Im
= S(V
FN
j
,Im(?z
new
))
12: Im(?z
c
)=V
FN
j
13: end if
14: end for
15: return ?z
c
B. FCN updating
In the emerging scenario space, if an existing node resulted
from instantiating another MF such that two instantiated
MFs share a common variable value, then the existing FCNs
associated with this node needs to be updated for consistency.
Suppose that for a certain variable to take value A, and
that the following two FCNs are obtained through different
inference procedures:
?z
A
= V
FN
i
+ iV
FN
j
,
?z
prime
A
= V
FN
p
+ iV
FN
q
,
where V
FN
i
,V
FN
j
,V
FN
p
and V
FN
q
denote different ele
ments in the predefined Q
FN
respectively. The work here
uses the modulus of an FCN to evaluate the overall infor
mation content of its associated variable value. If ?z
A
 and
?z
prime
A
 are the same (whilst ?z
A
negationslash=?z
prime
A
), then both ?z
A
and ?z
prime
A
will be kept as possible FCNs. However, if ?z
A
 negationslash= ?z
prime
A
, then
the newly updated FCN of this variable taking value A is
obtained by:
?z
primeprime
A
= min(V
FN
i
,V
FN
p
)+imin(V
FN
j
,V
FN
q
). (17)
This has an intuitive appeal because if (17) holds, then
both ?z
A
and ?z
prime
A
will also hold.
V. A PPLICATION TO CRIME INVESTIGATION
A. The crime case
This illustrative example is extracted and adapted from a
realistic case described in [7]. Consider the following case
description: The victim was discovered lying on this back
across his bed with a semiautomatic Remington .308 model
742 rifle between his legs. There were two bullet holes in
the roof of the victim?s trailer, suggesting that two shots had
been fired upward towards the ceiling. Also, the window near
the bed has been shattered. The preliminary observations
suggested that the victim had shot himself in the head with
the weapon and this case was classified as a suicide at the
beginning. However, according to the expertise, it is very
unlikely for a weapon to discharge twice during the suicide
process due to a reflex action, it has never been reported in
any forensic journal. Hence, another explanation comes up,
this is possibly a homicidal case.
B. Knowledge representation
In order to model the above case, four fuzzy variables are
extracted from the description as defined in Table II. Suppose
that the KB contains the following two MFs:
If {x
1
}
Relations {x
1
? x
2
}
Distribution {
r
1
: low powered ? low (Veryhigh)
r
2
: low powered ? medium (Medium)
r
3
: low powered ? aloud (Verysmall)
r
4
: medium ? low (Small)
r
5
: medium ? medium (Veryhigh)
r
6
: medium ? aloud (Small)
r
7
: high powered ? low (Verysmall)
r
8
: high powered ? medium (Medium)
r
9
: high powered ? aloud (Veryhigh) } (MF
1
)
If {x
2
,x
3
}
Relations {x
2
,x
3
? x
4
}
Distribution {
r
1
: aloud, near ? high (Veryhigh)
r
2
: aloud, medium ? high(High)
r
3
: aloud, far ? high (Small)
r
4
: medium, near ? high (High)
r
5
: medium, medium ? high (Medium)
r
6
: medium, far ? high (Small)
r
7
: low, near ? high (Medium)
r
8
: low, medium ? high (Small)
r
9
: low, far ? high (Verysmall) } (MF
2
)
In this work for simplicity, the fuzzy numbers used to
form FCNs (namely, both certainty and fuzzy matching
degrees) are defined in the following fuzzy quantity space,
as illustrated in Fig. 2:
Q
FN
= {Very small,Small,Medium,High,V ery high}.
C. Generation of plausible scenario space
Suppose that after examination of the death scene, two
pieces of evidence are collected:
e
1
: The degree of window shattered is quite high, with
certainty degree Very high.
e
2
: The Remington .308 model 742 rifle was found, with
certainty degree Very high.
Initial creation of an emerging scenario space is done
by matching a given piece of evidence, say e
1
, against the
KB. Assume that the matching degree of e
1
and ?x
4
is
high?inMF
2
is High. Hence, the first node, ?x
4
is high?,
is added to the emerging scenario space with the FCN:
Very high + iHigh. Given this and the instantiated MF
2
,
backward chaining is performed from x
4
. This leads to x
2
and x
3
being added to the emerging scenario space. By
applying Algorithm 1 to MF
2
, the results of Table I are
obtained.
In order to avoid generating unnecessary explanations, the
modeller produces only the current best or the most plausible
explanations in the first instance. That is, the approach will
not create alternative but less plausible explanation unless
the current best has to be discarded. With regard to the use
of the modulus of a derived FCN for plausibility evaluation,
r
2
and r
4
are established to outperform the rest. This means
that either ?the sound volume is aloud and the distance is
medium? or ?the sound volume is medium and the distance
is near? is the most plausible situation to have caused the
quite high degree of window shattered.
1
?
X
00.25 0.5 0.75 1
Very_small Very_highSmall Medium High
Fig. 2. Fuzzy quantity space for both certainty and fuzzy matching degrees
TABLE I
RESULTS OF BACKWARD CHAINING DUE TO e
1
Rule index Re(?z
antecedent
) Im(?z
antecedent
)
r
1
Very high High
r
2
Very high Very high
r
3
N/A N/A
r
4
Very high Very high
r
5
N/A Very high
r
6
N/A N/A
r
7
N/A Very high
r
8
N/A N/A
r
9
N/A N/A
TABLE II
FUZZY VARIABLES
Var Interpretation Domain
x
1
Power level of
the weapon
{low powered, medium, high powered}
x
2
Sound volume of
the weapon
{low, medium, aloud}
x
3
Distance between
the discharge
position and the
window
{near, medium, far}
x
4
Degree of win
dow shattered
{very low, low, medium, quite high, high}
According to Table I, ?z
antecedent
r
2
= Very high +
iV ery high, it follows from (16) that
Re(?z
antecedent
r
2
)=min(Re(?z
x
2
:aloud
),Re(?z
x
3
:medium
)),
Im(?z
antecedent
r
2
)=min(Im(?z
x
2
:aloud
),Im(?z
x
3
:medium
)).
It is obvious that
Re(?z
antecedent
r
2
)=Veryhigh ??
braceleftBigg
Re(?z
x
2
:aloud
)=Veryhigh
Re(?z
x
3
:medium
)=Veryhigh
Im(?z
antecedent
r
2
)=Veryhigh ??
braceleftBigg
Im(?z
x
2
:aloud
)=Veryhigh
Im(?z
x
3
:medium
)=Veryhigh
Therefore, the FCNs associated with x
2
: aloud and x
3
:
medium can be respectively written as:
?z
x
2
:aloud
= Veryhigh + iV ery high,
?z
x
3
:medium
= Veryhigh + iV ery high.
Similarly, the FCNs attached to r
4
can be derived. After
that, the newly created instances of plausible causes are
recursively used in the same manner as if each of them was
the original piece of evidence. As a result, MF
1
is then
instantiated, x
1
is added and the FCNs associated with the
states of x
1
can be obtained in the same way as above. This
phase of backward chaining (with given e
1
) leads to what is
shown in Fig. 3.
VH + i VHmedium VH + i VHmedium
VH + i VHnear
high VH + iH
high_powered VH + i VH
1
aloud VH + i VH
2
VH + i VHmedium
3
12
3
4
x
x
x
x
x
x
x
Fig. 3. Emerging scenario space after backward chaining initiated by e
1
Given e
2
and the same KB, no backward chaining is
needed as the information Remington .308 model 742 rifle
only matches the antecedent of MF
1
as a high powered
weapon with the FCN : Very high+iHigh. This piece of
evidence is then used to revise the emerging scenario space of
Fig. 3 by forward chaining. In particular, the FCN associated
with the node ?x
1
is high powered? is updated by using con
straint (17). This leads to the result that the FCN associated
with x
1
being high powered becomes Very high + iHigh.
As the variable value for x
1
is high powered, the strategy
of bestfirst generation of the scenario space will ensure
the removal of the lower branch of Fig. 3. The forward
chaining procedure is much easier to complete as it involves
straightforward calculations only. The results are listed in
Table III.
TABLE III
RESULTS OF FORWARD CHAINING PROPAGATION DUE TO e
2
Rule index Re(?z
c
) Im(?z
c
)
r
7
Very small Very small
r
8
Medium Medium
r
9
Very high High
According to the modulus of the FCN obtained for each
matched propositional instance, the one associated with r
9
in MF
1
outperforms those associated with the rest. The
consequence of r
9
in MF
1
,?x
2
is aloud?, is therefore firstly
instantiated. Because ?x
2
is aloud? already exists in the
emerging space, it is only needed to update the FCN as
sociated with it, which is done in the same manner as above.
The revised scenario space is depicted in Fig. 4. For this
simple example, no further instantiations and propagations of
variable states are possible. Hence, Fig. 4 is the final outcome
of the entire CM process.
In summary, the resulting scenario space offers the best
explanation possible for the collected pieces of evidence.
According to the initial case description, the bed is near
the window, hence, the victim lying on the bed did not shot
himself, because the distance between the discharge position
and the window is establish to be medium in the resultant
scenario description. Therefore, this case is suggested being
a homicide rather than a suicide.
x
1
aloud VH + iHhigh_powered
VH + iVHmedium
VH + iHhigh
x
2
x
3
x
4
VH + iH
Fig. 4. Emerging scenario space after forward chaining
VI. CONCLUSIONS
This paper has presented a novel framework of fuzzy
complex numbers, which is capable of representing and
propagating different kinds of inexact knowledge and data in
a unified manner. This ability is demonstrated by exploiting
the framework to support the task of compositional mod
elling. Additionally, from the CM point of view, the proposed
method not only provides a more concise and flexible knowl
edge representation formalism, but also enhances the capa
bility of CM to handle a wide range of inexact information.
Finally, the utility and usefulness of the proposed framework
are illustrated by means of an application to the construction
of plausible descriptions from given evidence in the crime
investigation domain. Note that whilst crime investigation is
herein used to demonstrate the applicability of FCNs, the
proposed framework is general and can be readily adapted
to suit different problems such as representing knowledge for
multifinered robot hand manipulation in [8], [10].
Whilst the results are promising, the present notion of
FCNs is only capable of jointly representing two kinds of
inexactness. It remains as an important piece of further
research to extend this framework to arbitrary ntypes of
inexact information. Also, a more general constraint satisfac
tion mechanism suitable for constraining variables modified
by FCNs would help to improve the generality of this work.
In particular, this may avoid the need to prefix just one
common quantity space which the real and imaginary parts
of any FCN may take values from.
REFERENCES
[1] J. J. Buckley. Fuzzy complex numbers. Fuzzy Sets Syst., 33(3):333?
345, 1989.
[2] J. J. Buckley. Fuzzy complex analysis II: integration. Fuzzy Sets Syst.,
49(2):171?179, 1992.
[3] J. J. Buckley and Y. Qu. Fuzzy complex analysis I: differentiation.
Fuzzy Sets Syst., 41(3):269?284, 1991.
[4] S. Dick. Toward complex fuzzy logic. IEEE Transactions on Fuzzy
Systems, 13(3):405?414, 2005.
[5] B. Falkenhainer and K. Forbus. Compositional modelling: Finding the
right model for the job. Artificial Intelligence, 51:95?143, 1991.
[6] X. Fu, Q. Shen, and R. Zhao. Towards fuzzy compositional modelling.
In Proceedings of the 16th International Conference on Fuzzy Systems,
pages 1233?1238, 2007.
[7] V. J. Geberth. Practical homicide investigation. CRC Press, 6000
Broken Sound Parkway, NW, (Suite 300) Boca Raton, FL 33487, USA,
fourth edition, 2006.
[8] Z. Ju, H. Liu, X. Zhu, and Y. Xiong. Dynamic grasp recognition using
time clustering, gaussian mixture models and hidden markov models.
To appear Journal of Advanced Robotics, 2009.
[9] J. Keppens and Q. Shen. On compositional modelling. Knowledge
Engineering Reivew, 16(2):157?200, 2001.
[10] H. Liu. A fuzzy qualitative framework for connecting robot qualitative
and quantitative representations. IEEE Transactions on Fuzzy Systems,
16(6):1522?1530, 2008.
[11] D. Ramot, R. Milo, M. Friedman, and A. Kandel. Complex fuzzy sets.
IEEE Transactions on Fuzzy Systems, 10(2):171?186, 2002.
[12] Q. Shen, J. Keppens, C. Aitek, B. Schafer, and M. Lee. A scenario
driven decision support system for serious crime investigation. Law,
Probability and Risk, 5(2):87?117, 2006.
[13] Q. Shen and R. Leitch. Fuzzy qualitative simulation. IEEE Transac
tions on Systems, Man and Cybernetics, 23(4):1038?1061, 1993.
[14] C. Wu and J. Qiu. Some remarks for fuzzy complex analysis. Fuzzy
Sets Syst., 106(2):231?238, 1999.
[15] G.Q. Zhang. Fuzzy limit theory of fuzzy complex numbers. Fuzzy
Sets Syst., 46(2):227?235, 1992.