A Method for Resolving the Consistency Problem Between
Rule_Based and Quantitative Knowledge using Fuzzy Simulation
Gyooseok Kim, Paul A. Fili i' 1:
Computer & Information Science and Engineering Department
University of Florida
Abstract
Given a physical system, there are experts who have knowledge about how this sys
tem operates. In some cases, there exists quantitative knowledge in the form of deep
models for the identical system. One of the main issues dealing with these different
types of knowledge is 'I. v. does one address the difference between the two model
types, each of which represents a different level of knowledge about the system?" We
have devised a method that starts with 1) the expert's knowledge about the system,
and 2) a quantitative model that can represent all or some of the behavior of the sys
tem. I In, method then ;,i.ljl the knowledge in either the rulebased system or the
quantitative system to achieve some degree of .., 'I. .. between the two representa
tions. I I!i, il!j checking and resolving the inconsistencies, we provide a way to obtain
better models in general about systems by exploiting knowledge at all levels, whether
qualitative or quantitative.
1 Introduction
Given a physical system, knowledge about the system is often obtained from experts in
the form of rules. Although the rulebased model is occasionally associative or shallow in
nature, this model can easily capture human heuristic and problem solving knowledge in an
efficient way [2, 11, 22]. In some cases, there exists a quantitative model which represents
all or part of the behaviors of the physical system. This model provides deeper and more
theoretical knowledge when expert system developers want to find solutions for technical
problems [11, 7, 22]. Assuming the above two different model types for the identical system,
some important questions can arise: how much do the models differ? and how can one resolve
the inconsistencies?. Answering to these types of questions is not easy since there is a big
gap between expert rules and quantitative models as shown in Fig 1.
One way of handling the inconsistencies between the expert's level of qualitative knowl
edge and the lower level of deep knowledge is to form a knowledge acquisition cycle as in Fig
2 [11]. Approaches for creating model bases are discussed within the context of computer
simulation [9]. For example, the model base represents compiled knowledge about many
domains such as the mathematical queuing model for waiting line problems. If a match
is found, then shallow rules are generated by means of qualitative or quantitative simula
tion based on this deep model. Since the number of the shallow rules resulting from the
deduction process is usually too big for study and validation against the original expert's
rules, induction methods can be employed to obtain a more comprehensive and general set
of rules [11].
For domains in which the expert rules contain many linguistic terminologies whose bound
aries are not exact, we need a way to encode this vagueness to permit the use of computer
Expert Rules
IF knob is ON during short time
THEN temperature of water will be cool
IF knob is ON during long time and then OFF during moderate time
THEN temperature of water will be tepid
IF knob is ON during moderate time
THEN height of water will decrease slightly
GAP
Quantitative Model
COLD: T= = =0, H=
HEATING: T= k1 (100T), IW= 0, Hf = 0
COOLING: T= k2(uT), HW= 0, Hf= k3
BOILING: T= 100, IL= k4,Hf=k5
Figure 1: A gap between expert rules and quantitative models
Model base search
Figure 2: A knowledge acquisition cycle
Resolving
inconsistency
(by quantitative
comparison and
interactive user control)
Figure 3: A knowledge acquisition cycle using fuzzy simulation
simulation. For such cases, we can use either qualitative or quantitative simulation with fuzzy
set concepts [1, 18, 8] for the deduction process. However, a well known problem in using
the qualitative methods is the possibly generating of spurious behaviors of the system during
the reasoning process [12, 11]. ':.i, recover, in order to get a compressed and generalized set
of fuzzy rules, additional methods such as fuzzy induction or fuzzy system identification
methods [15, 24, 17, 23] should be adopted. Consequently, forming the above knowledge
acquisition cycle to employ fuzzy set concepts requires a series of difficult tasks.
Our research is concerned with devising a method for resolving the inconsistency between
two different levels of knowledge in efficient and systematic manner. To achieve this goal, we
have partially developed a method as shown in Fig 3. This method starts with the expert's
rules about the system and a quantitative deep model that can represent all or some of the
behavior of the system. This method then adjusts the knowledge in either the rulebased
system or the quantitative system to achieve some degree of consistency between the two
representations. Here, a fuzzy simulation approach [7, 6, 8] is used for directly encoding
uncertainty arising from human linguistic vagueness into simulation components as well as
for utilizing quantitative models for the deduction process. Since this method uses a linguistic
mapping process to map simulation inputs and outputs into fuzzy linguistic values that were
also used by experts, direct comparison is possible without an additional induction step.
Ti, method consists of two phases: 1) consistency checking phase, and 2) resolving
phase. In the consistency checking phase, experts provide various levels of estimates (i.e.,
central points, intervals, approximate fuzzy sets) for a fuzzy set and then, through fuzzy
simulation and incremental optimization over the error surface, fuzzy set boundary vertices
are created to fill in the expert's knowledge. Currently, an implementation has been made
for this first phase where the estimates are presented in the form of central points. For
quantitative comparison between the two levels of knowledge, quantitative measures have
been formulated to gauge the sources and the degree of inconsistency. Thl final products
of this stage are rules derived from quantitative models, approximate fuzzy membership
functions for those rules, and the amount of inconsistency against the expert's rules. If
the amount of inconsistency exceeds a reasonable range, the resolving phase is necessary.
In this phase, human intervention is present: either expert rules (including the definitions
of fuzzy numbers) or simulation model components are modified to reduce the amount of
inconsistency. Even at this point, the quantitative measures mentioned above help them
identify and revise the most inconsistent component rapidly and analyze the effectiveness of
that modification, thereby allowing the two different levels of knowledge to gradually reach
a consensus with high resolution. Tli, knowledge acquisition cycle presented here forms a
more potentially organized framework that resolves the inconsistency between two knowledge
sources in an efficient and systematic manner.
Tli, primary contribution of this research is that through checking consistency and re
solving inconsistency, we provide a way to obtain better models in general about systems by
exploiting knowledge at all levels, whether qualitative or quantitative. We provide benefits
to expert systems from simulation and benefits to simulation modeling from expert knowl
edge [7, 6, 8, 13, 16, 3, 20, 21]. In particular, when expert system researchers are studying the
acquisition of deep knowledge from an expert or validating the expert's knowledge against
quantitatively compiled knowledge, the first type of benefits can be obtained from simula
tion models [7, 6, 8, 13, 16]. Tli, advantage from the reverse direction is also obtained when
simulation model validations are performed during the simulation modeling process with the
aid of the expert knowledge [13, 3, 20, 21].
Searching for the appropriate fuzzy membership functions that adequately capture the
meaning of the linguistic terms employed in a particular application belongs to the gen
eral problem area of knowledge acquisition within the underlying framework of fuzzy set
theory [10]. Tliil using the proposed method, we deliver a secondary contribution from
automatically generating the approximate forms of fuzzy membership functions (see Fig 3)
with which expert rules and quantitative models match maximally in spite of the expert's un
certainty about the exact definitions of the fuzzy values in his linguistic rules. 'i, ni'eover, we
interact with the user of the knowledge acquisition tool (to be constructed) to permit man
ual control over fuzzy set boundaries. Once we obtain those approximate definitions, fuzzy
simulations for all different combinations of the fuzzy values, defined by these definitions,
give us more detailed rules as a hypothesis of the expert's knowledge.
Fuzzy set theory, which is relevant to this paper and its relation to computer simulation,
is discussed in section 2. TI ii i in sections 3 and 4, we discuss research accomplished to date
and preliminary results with an example. Fini.lly, section 5 discusses our future research.
2 Background
2.1 Fuzzy set theory
Tli, theory of fuzzy sets can be found in [25, 26, 10, 5]. Fuzzy sets may be viewed as an
attempt to deal with a type of imprecision which arises when the boundaries of classes are not
sharply defined. A fuzzy set A of a universe of discourse X is characterized by a membership
function /tA : X + [0, 1] which associates with each element x of X a number t A(X) in the
interval [0, 1] which represents the grade of membership of x in A.
Definition 2.1: A fuzzy set A of the universe of discourse X is convex if and only if for
all x, X2 in X, 1A(Arx + (1 A)X2) > ln(tA(), t(2)),
where A C [0, 1].
Definition 2.2: A fuzzy set A of the universe of discourse X is called a normal fuzzy set
if 3xi E X, A(Xi) = 1.
Definition 2.3: A fuzzy number is a fuzzy subset in the universe of discourse X that is
both convex and normal.
To simplify the representation of fuzzy sets, a finite fuzzy subset, A, of X is expressed as
A= Y A(X1)/X1 + [A(X2)IX2 + ... + pA(Xn)/X., or A = E=l1 A(Xi)Xi,
where + sign denotes the union rather than an arithmetic sum.
If the fuzzy subset, A, is not finite, A may be represented in the form A = f pA(x)/x
in which the integral sign stands for the union of the fuzzy singletons ptA(x)/x.
Definition 2.4: Tih complement of A is denoted by A and is defined by
A= ( (X))/
Tih operation of complementation corresponds to negation.
Definition 2.5: Th, union of fuzzy sets A and B is denoted by A U B and is defined by
AUB= (R J A B(x) V pB(x))/x. (2)
where V is the maximum operator.
Definition 2.6: TIl, intersection of fuzzy set A and B is denoted by A n B and is defined
by
A n B = (A(x) A/ B(x))/x. (3)
where A is the minimum operator.
Let A and B represent two fuzzy numbers and let denote any of the four basic arith
metic operations. Ti,11 we define fuzzy set, A*B on R, where R is a set of all real numbers, as
tA*B () = ,"" (A A(x) A BY((Y)), (4)
for all z E R. ThliL for example, if A, B C R are two fuzzy numbers with respective
membership functions pA(x) and pB(y), then the four basic arithmetic operations, i.e., ad
dition, subtraction, multiplication and division, give for each x, y, z C R the following results:
tA+B(Z) = """ ,+(Y (iA(x) A B (y)). (5)
AB(Z)= '""' y (JA(x) A Y ()). (6)
AxB(Z) ='"' xy(A(x) A YB(y)). (7)
1 ,AB(Z) =",, ,y(p A(x) A tB (y)). (8)
2.2 Fuzzy set theory in computer simulation
Probability based methods are useful when most of the uncertainty can be effectively de
scribed through the use of large data sets and their associated moments. However, experts
often do not think in probability values, but in terms such as much, usually, always and
sometimes. In domains where estimation or measurement of probabilities is not amenable,
fuzzy set theory offers an alternative [14]. Here, we can use any type of fuzzy number, such
as an intervalvalued fuzzy number, a triangular fuzzy number, a trapezoidal fuzzy number
or a general discrete (or continuous) fuzzy number depending on the degree of uncertainty.
Owing to the extension principle [27] in the fuzzy set theory, nonfuzzy mathematical struc
tures can be made fuzzy. Here is a sample of how this relates to simulation. We can make
fuzzy: [8, 26] 1) a state variable value including initial conditions, 2) parameter values, 3)
inputs and outputs, 4) model structures, and 5) algorithmic structures.
To simulate mathematical models using the fuzzy set concept, three kinds of fuzzy simu
lation approaches have been reported: Qualitative Simulators (i.e., Qua.Si [1]), Fuzzy Qual
itative Simulation (i.e., Fusim [18]), and three methods (Monte Carlo, Uncorrelated Uncer
tainty, and Correlated Uncertainty) of fuzzy simulation introduced by Fihli" i, [8]. While
the first two kinds of fuzzy simulation are useful when there is not enough information to
simulate quantitatively, the third kind takes linguistic information from the expert and per
forms computer simulation quantitatively on continuous and discrete event models. Rules
or FSA (Finiti State Automata) can be extracted from these quantitative models through
linguistic mappings, and these results can be validated directly against the expert domain
knowledge. Th, fuzzy simulation method we present is an extension version of the correlated
uncertainty method [7, 6, 8]. Th, correlated uncertainty method assumes that all errors or
uncertainties over time are correlated. In many circumstances, this is the case, since un
certainties specified by a heuristic or a belief are often correlated in that humans are often
consistent in their beliefs. For such a process, every vertex in the fuzzy number is issued
independently to the simulation function and the outputs of the simulation are mapped into
the most closely matched fuzzy linguistic value by using a distance metric.
3 A Proposed Method
In this section, we propose a method for resolving the inconsistencies between the expert's
rules and the quantitative models. As we discussed, our method consists of two phases:
consistency checking and resolving inconsistency. While the first phase is done through an
automatic process, the second phase is performed semiautomatically. An algorithm has
been developed for the first phase. Before exploring the algorithm, we must first introduce
the input of the algorithm and two important usages of fuzzy simulation that we've devised.
3.1 Format of expert rules as input of proposed method
In what follows, we assume that the format of expert rules is one of the following two types.
T11 input of the proposed method is a collection of the expert's rules below, with conclusions
from the same fuzzy variable:
IF (X is A1) THEN (T is B); CF; CLA,; CLB
IF (Xi is A1) OP (X2 is A2) OP,..., OP (Xn is A,)
THEN (T is B); CF; CLA,; CLA2; ...; CLA,; CLB
where
Xi, i = 1, 2,..., n, and T are fuzzy variables that take real numbers from some universal set
Table 1: Notation
Notation Usage
1 [Fpremise Membership Function of fuzzy value in rule premise.
SfFconselq Membership Function of fuzzy value in rule consequence.
RULEsimplex Expert's simplex rule.
RULEcompound Expert's compound rule.
CFexpert Confidence Factor presented by an expert.
CF uzzy Confidence Factor calculated using fuzzy simulation.
X, Y respectively, Ai, i = 1, 2, ...,n, and B are fuzzy values on X, Y respectively, CF is
a confidence factor in the rule consequence given that the premise conditions are satisfied,
OP is a fuzzy logic (or or and) or fuzzy arithmetic (+, , x or + ) operator, and CLA,,
i = 1, 2,..., n, and CLB are expert's confidence levels on the fuzzy values in each rule.
Tih two types of rules above will be called complex rules and compound rules respectively.
Tih value of CL can be a central point estimate, an interval estimate, an approximate
fuzzy number or a complete fuzzy number depending on the expert's confidence level on the
linguistic term he used. In this paper, we restrict our discussion within a situation where
the values of the CL are central point estimates.
3.2 Two fuzzy simulation methods
In what follows, the notation in Table 1 will be used for simplicity. In the proposed
method, the fuzzy simulation approach has two important roles: 1) calculation of CFfuzzy
of RULEcompound, and 2) estimation of .\Fconseq. We discuss these two roles of the fuzzy
simulation in the following two sections.
3.2.1 Calculation of CFfuzzy of RULEcompound
Since the uncertainty arising from the human reasoning process is easily represented by a
rule associated with CFexpert, we introduced a way for emulating such a process by showing
how fuzzy simulation can derive the confidence factors from quantitative models. By doing
this, we benefit from the comparison of the two rules in terms of their CF values. However,
since the CFexper, involves a subjective opinion, there is no theoretical formulation to derive
the CFfuzzy whose value is exactly the same as the CFexpet. Our solution is to define an
equation in such a way that its results agree with human intuition as much as possible. We
used a 'jl.'.l average method to create such an intuition.
Let us define the CFfuzzy using the .,, 'jll'../ average method. Given a RULEcompound, let
its two ./[Fpremises be A and B, where A and B are fuzzy subsets of a universe discourse X,
and its .[Fconseq be C, where C is a fuzzy subset of a universe discourse Y. Tili we define
the CFfuzzy by the following equation:
CFfUY EJ=I(! (xj) x cy ) (9)
EjTlfuzzy I ) (9)
A B C
1.0 1.0 
x y
5 15 20 30 35 45 0 5 10
Figure 4: Definitions of fuzzy numbers A, B and C
where 0 denotes a fuzzy logic or arithmetic operator,
xj,j = 1, 2,..., n, denote real values on the fuzzy set resulted from the operation of A 0 B,
yj,j = 1, 2,..., n, denote real values on Y obtained from fuzzy simulation using xj.
Equation (9) can be divided into the following three steps for simplifying its calculation:
1) perform the fuzzy logic/arithmetic operation, 2) simulate using the fuzzy set obtained
from the above step, and 3) calculate CFf,,,, using the weighted average method. For
example, given a RULEompound, IF X1 is A OR X2 is B THEN T is C, with definitions of A,
B and C as shown in Fig 4, CFfuzzy for the RULEcompound can be calculated by performing
the following steps:
1. Perform the fuzzy OR operation for A and B. For each element x in X, the degree of
membership of A OR B, YA OR B(x), is obtained by (2). Fig 5(a) shows the result of
the operation.
2. Perform the fuzzy simulation on the fuzzy set of Fig 5(a). Ti, result is shown at Fig
5(b).
3. Calculate CFf,,,,y using the weighted average method.
(0.3 x 0.5) + (0.7 x 0.8) + (1.0 x 0.9) + (0.7 x 0.2)
CFuzzY 0.3 + 0.7 + 1.0 + 0.7 + 1.0 + 0.7 + 0.3
=0.37
Tli, validity of calculating CFfuzzy in this way can be easily shown as in Fig 6. CFfuzzy,
using (9), is 1.0 and 0.0 for Fig 6(a) and for Fig 6(b) respectively. T1l, results exactly match
our intuition. When the CF falls into some range between the two extreme cases above,
we can intuitively say that each member in A supports the conclusion B with a higher
confidence, the greater CF we get. Using (9), we also get the results which support such an
intuition.
3.2.2 Estimation of .~Fconseq
In the previous section, we used a fuzzy simulation to derive the CFfuzzy when all definitions
of the linguistic terms in a rule are already known. Conversely, without knowing the definition
of the linguistic term, particularly the definition of the linguistic term in the consequence of
the rule, we can use the fuzzy simulation to estimate its approximate range.
I A ORB
5 20 25 30 45
5 20 25 30 45
A ORB
I I I I I
5 10 15 20 25 30 35 40 45
(a) The result of operation A OR B (b) The result of fuzzy simulation on fuzzy set A ORB
Figure 5: Calculation of CFfuzzy as an example
A k B k A LB
1.0 1.0 
0.5  0.5 
Sx yI I I
Fuz y I I I I I I I I I
SSimulationL_ ~~~ Simulation
(a) All members of A support the conclusion B (b) None of the members of A support
the conclusion B
Figure 6: All members or none of members support the conclusion
1 23 6 9 14
Let's assume B is a symmetric triangular fuzzy number whose members are real numbers
y. Knowing its center point c and the width w of B, the degree of membership of any real
number, yl, ; _,..., y, can be obtained from the equation,
2 x y c
SB(Yi) =1 (10)
where i =1,2,..., m.
Let's assume another fuzzy number A whose members are real numbers x. Given an
expert rule, IF X is A THEN T is B, with its CFexpet, performing a fuzzy simulation on A
and applying the weighted average method to B yields
C = EiU(A(Xi) X YB(Yi))
C/uzzy = (11)
where ;I is a result of the fuzzy simulation on xi.
However, consider a situation where a fuzzy simulation is executed on A, but the width
of B is unknown. Letting the CFfzzy in (11) be equal to the CFxpr, of the rule above, and
substituting the righthand side of (10) for the pB(Yi) in (11), we get
CF E? A(Aiu ) X (1 ())
C e EA() = (12)
= Ei'p I l/1 (Xi)
From this equation, we can obtain the following equation to estimate the unknown width w
of B.
2xE 1A(XAi)X 'I CI)
W = (13)
E=l A(Xi) (CFexpert X E i'(1)) (1i
Equation (13) has an important meaning: if we know an expert rule, its CFexpert, its
I[Fpremise, and the center point of the fuzzy number in the consequence of the rule, then we
can estimate the range of the fuzzy number with an aid of fuzzy simulation.
For example, with the rule, IF X is A THEN T is B; CFexpe, = 0.5; CLA = 0.03; CLB
= 27.0, and unknown width w of B, suppose that the result of a fuzzy simulation is shown
in Fig 7(a). By applying (13), a symmetric triangular membership function for B can be
obtained as shown in Fig 7(b).
One constraint of applying (13) is that the CFexper should not be equal to 1.0 (i.e., less
than 1.0). Otherwise, the value of the denominator in (13) would be zero. Even though such
a case is currently a limitation of the equation, a preliminary approach has been developed.
3.3 Consistency algorithm
An algorithm introduced here has been made for handling the first phase (i.e., checking
consistency phase) of the proposed method. Once the expert's rules for a physical system
have been presented, and a relevant quantitative model has been found during the model
base search, we can apply the algorithm presented here for checking consistency between
the two models of knowledge. Ti, algorithm generates the approximate definitions of fuzzy
linguistic values by increasing the ranges of fuzzy sets from their initial minimal width to fill
in the expert knowledge. For such a process, two confidence factors, CFexper, and CFfuzzy,
SA B B
1.0  1.0  1.0 
S A y y
0.03 26.027.0 29.0 32.0 23.0 27.0 31.0
Fuzzy 8.0
Simulation
(a) Fuzzy simulation with unknown width of B (b) Symmetric fuzzy membership function of B
Figure 7: Estimation of unknown width of B using fuzzy simulation
are used to calculate local and global inconsistencies. TI, , serve as the quantitative close
ness measures between the two different levels of knowledge. Two different methods of fuzzy
simulation, discussed in section 3.3.1 and 3.3.2., are involved in this process. When the
algorithm reaches a point where tuning membership functions does not improve the amount
of closeness any further, the algorithm stops and returns the membership functions that
have been tuned so far as an approximate set with which two levels of knowledge match
maximally. T11,11 is, the proposed algorithm uses a gradientbased optimization technique
to find the proper sets of fuzzy definitions. If the closeness is out of a reasonable range,
human intervention is required for resolving the inconsistencies: either the expert rules or
the simulation components which show the inconsistency can be reevaluated, or the defini
tions of the linguistic values generated by the algorithm can be changed interactively. Tih
algorithm presented here is also useful for this resolving phase, since the comparison results
are quantitatively calculated and visualized in response to the human interaction. When the
goodness of fit reaches a reasonable point, another fuzzy simulation with different values of
fuzzy variables creates a more detailed level of rules than the level of the expert's rules.
For this algorithm, we employ an iterative improvement method. This algorithm consists
of the following three basic steps:
1. Hypothesize membership functions in RULEsimplexs.
2. Apply hypothetical membership functions to RULEompounds.
3. Improve hypothetical membership functions using RULEcompounds.
Fig 8 shows detailed substeps in each basic step. We explore them in the following three
subsections.
3.3.1 Step 1: hypothesize membership functions in RULEsimplexs
Tih purpose of this step is to hypothesize each .lFpemisen in RULEsimplex and to obtain its
corresponding hypothetical .FFconseq using (13).
In the first substep, two cases should be handled differently. T,11.i is, when the algorithm
initially starts, we construct an initial hypothetical .~Fpremise so that its range is 2Ad with
Step
1 : Hypothesize membership functions in RULEimplexs
Hypothesize MFpremises in RULEsimplexs
Fuzzy simulation for each RULEsimplex
Obtain MFconseqs in RULEsimplexsusing equation (13)
2: Apply hypothetical membership functions to RULEcompounds
Fuzzy Logic/Arithmetic Operations for MFpremises in RULEcompounds
I
Fuzzy Simulations for RULEcompounds
Calculation of CFf zy for each RULEcompoundusing equation (9)
Calculation of LIs and Glfor RULEcompounds
I1
    04
3 : Improve hypothetical membership functions using RULEcomounds
Pick a RULEcompound which has the largest LI
Find all MFpremises causing the largest LI
Find a proper subset leading to the smallest GI
Compare this smallest GI to PI
GI < PI
GI> PI
Final approximate fuzzy membership functions
Figure 8: Three basic steps and their substeps of algorithm
12
Step
Step
the center point, where Ad is a optimal resolution size for simulation execution. Ad can be
determined by experts or simulationists. For other case, this substep modifies .[Fpem,,i, by
increasing its range by Ad on either sides. After executing the last substep, we obtain a
hypothetical pair of .\[Fpremise and .IFcons.eq for each RULEsimplex which satisfies CFfu zzy
CFrexpert.
3.3.2 Step 2 : apply hypothetical membership functions to RULEcompounds
Tli, obtained [\Fpremises and .\Fconseqs from the previous step are consistent only for the
RULEsimplexs in a sense that CFfuzzy, ; CFexpert for each RULEsimplex. Our claim is that if
those membership functions are really consistent, then this also should be the case with the
all RULEcompounds. Tli,, the purpose of this step is to apply these hypothetical membership
functions to the RULEcompounds to check their validities.
For each RULEcompound, we define its local inconsistency, LI, as
LI = ICFfuzy CFexpert (14)
Ti, 1, using the LI, we define the global inconsistency for all RULEcompounds, GI, as
GI= EL1, (15)
i=1
where m = total number of RULEcompounds. Searching for the largest LI enables us to identify
the most inconsistent RULEcompound between two different knowledge sources. :., recover the
GI calculated in this way allows us to measure the total amount of inconsistency.
3.3.3 Step 3 : improve hypothetical membership functions using RULEcompounds
Tli, purpose of this step is to reduce the GI by picking up a RULEcompound which has the
largest LI and modifying a proper subset of the .fFpremises among all subsets of .fFpremises
which caused that LI. We can find the proper subset by searching for a combination of
the .\[Fpremises which leads the GI to the smallest value among all combinations of the
.\[Fpremises which caused the largest LI. Notice that we should not regard the .\lFpremises
that can reduce the largest LI into the smallest amount as the proper subset. T11i reason is
that if any such .I[Fpremise is used also for other rules, then the modification of this definition
could make other LIs in those rules worse than before, possibly causing increased GI as a
whole. For this reason we introduce the GI instead of the LI as a performance index (PI).
Tli i; Fre, we need to find a subset of ._[Fpremises which improves the GI by the greatest
amount by executing step 1 and step 2 for each subset of ./[Fpremises. When we eventually
reach the smallest GI after incrementally reducing the inconsistencies, we can regard the
hypothetical set of the .fFpremises and the .\[Fconseqs as the final approximate fuzzy set with
which the expert's rulebased model matches maximally the quantitative simulation model.
IOFF
I=ON
Cold I H I=ON
I=ON I \ IOFFFF y ^ ""
1 IOFF Hw=0 \
1F Cooling I=ON Hw O
\ M3 Inderflow ,
M6 Hw=
T=
IOFF
I=ON or IOFF
Figure 9: Six state automaton controller for boiling water
4 Example: Boiling Water Model
To illustrate the application of the proposed method, we have chosen boiling water [9] as a
simple example. All steps in Fig 8 has been implemented using C programming language,
and a graphical user interface using Tk/Tcl has been built for illustrating the result of the
each step in the algorithm. Consider a pot of boiling water on a stovetop electric element.
Initially, the pot is filled to some predetermined level with water. A small amount of detergent
is added to simulate the forming activity that occurs naturally when boiling certain foods.
This system has one input the temperature knob. Tli, knob is considered to be in one of
two states:on or of.
Tli, output consists of T (the temperature of the water over time), H, (height of the
water) and Hf (height of the form). Tli, behavior of this model can be represented by six
states of FSA as in Fig 9. Tli, low level continuous models for .11, ..., \1,. in that figure are
defined as shown below by combining : 'ton's law with the capacitance law [9].
1. (.li) COLD: T = a, H, = 0, Hf= 0.
2. (.11) HEATING: T = k(100 T), H, = 0, Hf = 0.
3. (\1 ) COOLING: i = k2(o T), H,, =0, H = I .
4. (.1,) BOILING: T = 100, H, = k4, Hf = k5.
5. (.1.) OVERFLOW: same as BOILING with constraint Hf = Ht.
where ki, i = 1,...,5 are rate constants and a is the ambient temperature of the water.
Among those three types of output (i.e., T, H,, and Hf), let's assume that we are particu
larly interested in the T (temperature of water) when we turn the knob on and off over some
time period. We used a total of sixteen rules (eight RULEsimplex and eight RULEompo,,nds)
and their center point estimates as shown in Fig 10(a) as the expert's rules to describe the
water temperatures depending on the on and off position of the knob over time. Using these
expert rules as inputs and applying the proposed method to the quantitative simulation
model [9] in Fig 9, we obtained sixteen rules and an approximate set of fuzzy membership
PREMISES CONSEQUENCES CF
expert
KNOB ON KNOB OFF TEMPERATURE
veryveryshort cold 0 8
veryshort cool 0 5
short tepid 0 7
slightly moderate warm 0 8
moderate hot 0 6
slightly long veryhot 0 3
long veryhot 0 6
verylong veryhot 0 8
slightly moderate very veryshort warm 0 8
slightly moderate long tepid 0 9
moderate short hot 0 5
short very long cool 0 4
verylong very veryshort veryhot 0 8
moderate very long tepid 0 6
veryshort very long cool 0 7
veryveryshort very long cold 0 9
CENTER POINT ESTIMATE( CL)
veryveryshort 0 0 cold 20 0
veryshort 0 03 cool 27 0
short 0 06 tepid 37 0
slightly moderate 0 09 warm 44 0
moderate 0 13 hot 520
slightly long 0 17 very hot 70 0
long 02
very long 0 23
20 27
(a) Expert's rules and center point estimates
37 44 52
(b) Rules extracted from fuzzy simulations
and final approximate fuzzy membership functions
Figure 10: TIi, inputs presented by expert and the outputs generated by the proposed
method
PREMISES CONSEQUENCES CF
^ fu z zy
KNOB ON KNOB OFF TEMPERATURE
veryveryshort cold 0 8
veryshort cool 0 5
short tepid 0 7
slightlymoderate warm 0 8
moderate hot 0 6
slightly long very hot 0 3
long very hot 0 6
verylong very hot 0 8
slightlymoderate veryveryshort warm 0 766667
slightlymoderate long tepid 0 892000
moderate short hot 0553164
short verylong cool 0410000
verylong veryveryshort very hot 0 840909
moderate very_long tepid 0 684073
veryshort verylong cool 0 700000
veryveryshort verylong cold 0 900000
70 temp
Consistent Rule Malaer between Highlevel Rules and Lowlevel Simulation Models
Editing Rules Wth Simple Rules... Trends of GI(Global Inconsistency)
Fuzzy Simulation
Construct Conseq. MFs
With Complex Rules... 2.0
Fuzzy Simulation
Optimal MFs I Calculate CF Values 1.0
lear All H
Exit Search for smallest G Number of iteration
Membership Functions Expert Rules
Knob ON Knob OFF inclusion CF
l sot short m t moderate lY long oen 'veryery short zer cold 0.8
very_short zero cool 0.5
short zem tepid 0.7
slightly moderate zero warm 0.8
moderate zero hot 0.6
slightly_long zem very_hot 0.3
S long zero very _hot O.6
o 0.03 0.06 0.09 0.13 0.17 0.20 3 ng m v t0.
verylong zero very_hot 0.8
Sme ghly moder very very s o wann
slightly_moderate long tepid 0.9
cold cool tepid warm hot very_hot modrat rt hot 05
short verylong cool 0.4
very_long very_very_short very_hot 0.8
moderate verylong tepid 0.6
very_short verylong cool 0.7
very_veryshort verylong cold 0.9
2 27 37 44 52
(Temperature)
Mconseqs
premises
Window B
Window C
RULE
I UEsimplexs
RULEcompounds Window A
Figure 11: Initial GUI when expert's rules are first processed
functions for those rules as shown in Fig 10(b). In the remained part of this section, we will
show how to obtain such results by illustrating all steps depicted in Fig 8.
Fig 11 shows the initial GUI when the expert rules are first processed. Tii, GUI consists
of the following three windows:
Window A for displaying the trends of GI over the number of iterations of algorithm,
Window B for displaying the membership functions made so far, where the upper and
the lower parts represent Fpem,,ises and .\/F:onseq, respectively,
Window C for displaying expert rules, where the upper and the lower parts represent
RULEsimplexs and RULEcompounds respectively.
4.1 Algorithm Execution
1. Hypothesize .\IFprmises in RULEsimpexs
Tii, hypothetical .\Fp,,rmiss with the resolution size of 0.01 is initially made, and
corresponding .\[Fconseqs are obtained as shown in Fig 12 by using equation (13).
Consistent Rule Maker between Highlevel Rules and Lowlevel Simulation Models
Editing Rules ith Simple Rules... Trends of GI(Global Inconsistency)
Fuzzy Simulation
GI
Construct Conseq. MFs
With Complex Rules... 2.O
Fuzzy Simulation
Optimal MFs Calculate CF Values 1.0
Clear All I
Exit Search for smallest umber of iteration
Membership Functions Expert Rules
Knob ON Knob OFF Conclusion CF
hemot gAfa e moderate slI' long oen very very short zeo cll 0.
veryshort zero cool 0.5
short zero tepid 0.7
slightly_moderate zero warm 0.8
moderate zero hot 0.6
slightlylong zero veryhot 0.3
0.0 0.03 0.06 0.09 0.13 0.17 0.20 ngv t
(ime) slightly_moderate very_veryshort warm 0.8
S slightly moderatee long tepid 0.9
cold cool tepid warm hot very_hot moderate short hot 0.5
short very_long cool 0.4
very_long veryvery_short very_hot 0.8
moderate very_long tepid 0.6
very_short very_long cool 0.7
very_veryshort verylong cold 0.9
21 27 7 44 5 a
(Temperature)
a A IrJ
Obtaining MF seqs
Figure 12: Obtain .\[F,,, sq, in RULEsimplexs
17
2. Apply hypothetical membership functions to RULEompound
CFfuzzy for each RULEompound obtained by applying equation (9) and GI are shown
at Window C in Fig 13. LI are implicitly calculated.
Consistent Rule Maker between Highlevel Rules and Lowlevel Simulation Models
Editing Rules WI ith Simple Rules... Trends of GI(Global Inconsistency)
Fuzzy Simulation 2.022727
I 1
Construct Conseq. MFs
With Complex Rules... 2.0
Fuzzy Simulation I
Optimal MFs Calculate CF Values 1.0
Clear Al I
Exit earch for smallest GI 1 Number of iteration
Membership Functions Expert Rules
Knob ON Knob OFF Conclusion CF
sho sor short m ehe moderate l long in veryveryshort zero cold 0.
very_short zero cool 0.5
short zero tepid 0.7
slightlymoderate zero warm 0.8
moderate zem hot 0.6
slightlylong zer veryhot 0.3
long zero very hot 0.6
0.10 0.03 0.06 0.09 0.13 0.17 0.20 0.3 vryong zmeryhot .
very long rem very hot 0.0
(Tfme) 0 Q= ... r
slightlymoderate long tepid 0.9 0.000000
cold cool tepid warm hot very hot
short verylong cool 0.4 0.000000
very_long veryvery_short very_hot 0.8 0.872727
moderate very _ong tepid .6 0.000000
very_short veryong cool 0.7 .750000
very_veryshort very Ion
20 27 37 4 52 0 Global Inconsistency: 2.822727
S(TeZpe 37 rat
(TemperaturB)
Possible catalysts for the largest LI
Rule leading to the largest LI
CFzyfor each RULEompound GI
tizzy compound G
Figure 13: Calculations of CFf,,,u, LIs and GI
3. Improve hypothetical membership functions using RULEcompounds
Ti second rule of RULEcompound in Fig 13 was chosen since it has the largest LI, where
LI = 0.9. Ti, current GI, 2.822727 is saved as an uptodate PI. Three [Fpremises
(i.e., ./,/*l0, _, '. ,". long and short) are identified as the possible catalysts for the
largest LI as shown at Window B in Fig 13. Thli all subsets of these catalysts
are {. l.',i/.,_ .'if }, {long}, {short}, {..il. tl,_,,.fJ, ..,' long}, {.fl.', l, ,.1.,'
short}, {long, short} and {..//, I,/.; '.1 1, ', ,/ long, short}. For each subset, step 1 and
step 2 are executed. Among the seven subsets above, a proper subset was identified
as {..f llt, /..' 1. 1, f.l, long, short} because when we modify these IFpr emises, we get
the smallest GI, 1.487727, as shown in Fig 14. Since this GI, 1.487727, is smaller than
PI, 2.822727, the stop condition is not met. Thli, the algorithm repeats step 3. After
ten iterations of step 3, we finally met the stop condition as shown in Fig 15. At the
10th iteration, we got the smallest GI, 0.262242, when the moderate ~Fpremise was
modified. However, since this GI is not smaller than PI (i.e., GI at the 9th itera
tion), 1 *_'" i',I the \IFpremises and .\Fconseqs shown in Fig 15 are final approximate
membership functions which lead to minimal PI, 01 '"'I '11
Consistent Rule Maker between Highlevel Rules and Lowlevel Simulation Models
Editing Rules With Simple Rules... Trends of GI(Global Inconsistency)
Fuzzy Simulation 2.:22727
SI 1.487777
Construct Conseq. MFs I
With Complex Rules... 2.0 \
Fuzzy Simulation The smallest GI, 1.487727
Optimal MFs Calculate CF Values 1.0
Clear All I
Exit Seah for smallest G 2 Number of iteration
Membership Functions Expert Rules
Knob ON Knob OFF Conclusion CF
shot shomt short moe moderate Sn long on veryry_very_short zero cold 0.8
very_short zero cool 0.5
short zer tepad 0.7
slightly moderate zero warm 0.6
moderate zero hot 0.6
slightly_long zero very_hot 0.3
long zero very hot 0.6
3 006 0.09 0.13 v017 ery long zer very hot 0.
(lime) slighty_moderate very_very_short warm 0.0
slightly moderate long tepid 0.9
cold cool tepid warm hot very_hot moderate short hot 0.5
short verylong cool 0.4
very_long very_very_short very_hot 0.0
moderate very_long tepid 0.6
very_short very_long cool 0.7
veryvery_short very_long cold 0.9
D 27 37 44 52 J
(Temperature)
The most proper subset leading to the smallest GI
Figure 14: Thi smallest GI and its proper subset leading to the GI
4.2 Analysis of output
T11, final rules generated at the 9th iteration are shown in the Fig 10(b). Notice that, in that
figure, each CF,,xrt is closely equal to the corresponding CFf,,,. .", ieover, PI turns out
to be 0.229479, which can be regarded as fairly consistent. T11 i fi. e, the expert's rulebased
model and the quantitative simulation model for this particular boiling water problem can
be considered consistent without processing an additional resolving inconsistency phase. If
the closeness is out of a predetermined range, human interaction is required for resolving the
inconsistencies.
By executing another fuzzy simulation for all different combinations of the fuzzy values
defined by the above membership functions, we got more detailed rules (8 x 8 = 64 rules) as
a hypothesis of the expert's knowledge. Table 2 shows a part of such knowledge.
5 Discussion and Future Research
Ti1, proposed algorithm is an iterative improvement algorithm employing the gradient descent
method, because it executes a loop that continually moves in the direction of decreasing GI.
It keeps track of only the current states, and does not look ahead beyond the immediate
neighbors of that state. Its solution may be a local minima as shown in Fig 16. This local
minima problem can be cured if we choose all paths whose GIs are better than PI as shown
Final approximate fuzzy membership functions for MFpreses and MFconse
premises conseqs
Figure 15: Satisfying stop condition and final membership functions
Table 2: A part of detailed rules extracted from fuzzy simulation
KNOB_ON KNOB_OFF TEMP. CFfuzy
moderate very_very_short hot 0.611507
moderate very_short hot 0.603797
moderate short hot 0.553164
moderate slightlymoderate warm 0.403333
moderate moderate warm 0.6 , 'i
moderate slightly Jong warm 0.552000
moderate long tepid 0.5 11,~I11
moderate very_long tepid 0.684073
Imtial
{A} {B} {A, B}
K 7K^
060 666
LEGEND
Path investigated
SPath actually chosen
O Local Minima
S Global Minima
Figure 16: Local minima
in Fig 17, instead of choosing the path which has the best GI. Clearly, this solution costs
more in terms of simulation time and memory than before, but we can better avoid the
local minima problem. Alternatively, we can take a middle position between these extreme
strategies. For example, when the problem space is too large to adopt the latter strategy,
we can choose two or three best paths at every iteration.
Having developed the consistency checking phase with only central point estimates for
handling the uncertainty arising from linguistic vagueness, we now would like to focus on
considering the following issues:
extending the method to handle the various forms of uncertainty representation.
developing a phase for resolving inconsistency when the amount of inconsistency ex
ceeds a reasonable range.
applying the method to more application oriented and realistic rulebased examples.
Ti, following sections discuss the above issues in detail.
5.1 Handling various forms of uncertainty representation
We showed that the proposed algorithm can deal with central point estimates. However, in
order to handle the various levels of uncertainties arising from linguistic vagueness, we will
extend the method to cover the other cases as well. Specifically, we will enhance our method
to cover the situations in which experts present various forms of uncertainty about their
linguistic terms in the following ways:
* central point estimates
LEGEND
S~Path actually chosen
A A, B} Global Mnlmma
Figure 17: Global minima
y r s In a t t w
without any kind of estimatesB} and other rules ay have different f{r}s of uncertaintyB
07 27 36 2 02 09 36 09 29
{A} {B} {A, B {A} {B} {A, B) {A} {B} {A B} {A} } {A B}
39 36 38 09 06 28y 29 28 35 29 28 35
Figure 17: Global minima
interval estimates
approximate fuzzy membership functions such as triangular or trapezoid fuzzy numbers
fuzzy membership functions aith their complete definitions
Fig 18 shows the four forms of the uncertainty and the wmay e discretize such uncertainties
into afuzzy real space In addition to these forms, we are interested in exploring the following
aspects:
Expert's knowledge may be filled with large gaps. Some rules may have linguistic values
without any kind of estimates, and other rules may have different forms of uncertainty
mentioned above.
Other forms of uncertainty such as probability can be introduced into the overall
process.
5.2 Developing a phase for resolving inconsistency
Tih algorithm presented in section 3.3 has been implemented for handling the consistency
checking phase. When the amount of the inconsistency exceeds a reasonable range, human
intervention is required for resolving the inconsistency. Fi i expert rules (including the
definitions of the fuzzy membership functions and the confidence factors of the rules) or
simulation model components should be reevaluated or modified. For such a process, we are
going to consider the following aspects:
type form of uncertainty explanation Discretization into fuzzy space
confidence
A
central point When experts present 1.0
1 estimate the center point c
of A
c
confidence
I, A
When experts present 1.0  
the interval [a, b]of A
with a full confidence
Interval a b
estimate
confidence
When experts present 1.0
the possible range [d, e] A
of A
d e
confidence
When experts present A
both the center point c 1.0  
and the possible range
[d, e] of A
Approximate d c e
fuzzy number
confidence
when experts present \ A
both the interval [a, b] 1.0   
of Awith a full confidence
and its possible range
[d,e] d a b e
confidence
A
1.0  
Complete When experts present 1.0
fuzzy number the complete definition
of A
5
Figure 18: Four forms of uncertainty about the linguistic terms
Ti, implementation of the resolving inconsistency phase needs to be interactive so that
experts or users can dynamically change their rules and fuzzy set endpoints. Every
time this modification happens, the consistency checking phase should be reinvoked
with some visual aids so that the effect of the modification can be easily recognized.
This reflects a theoretical modification of the proposed algorithm as well as an imple
mentation change of the code, since the overall processes now involve humans in the
loop during the checking consistency and the resolving inconsistency.
Thl corresponding quantitative model itself may be considered a source of the incon
sistency. Some parameters or even the structure of the model needs to be adjusted.
TI i, 1 i, fe, some guidelines or facilities for aiding such processes need to be devised.
5.3 Application
Til, boiling water example discussed in section 4 is a simple domain that we selected for the
purpose of illustration. 'i r, e applicationoriented and realistic rulebased examples are being
studied for good assessment and future development of the proposed method. We are partic
ularly interested in its application within lOOSE ('.i [ il i nidel Object Oriented Simulation
Environment) [19] which is now under construction at the University of Florida. '.lOOSE
is an enabling environment for modeling and simulation based on OOP' (Object Oriented
Physical :'.deling). OOP'.I extends objectoriented program design with visualization and
a definition of system modeling that reinforces the relation of ii I, to "pi, ii.
By incorporating the proposed method into '.OOSE, we can obtain two types of ben
efits: one from validating the expert's rules against simulation models, and another from
validating the simulation models against the expert's knowledge. To make this point more
understandable, we consider the general modeling process depicted in Fig 19 [20, 21, 4].
Tih conceptual model represents the mathematical, logical or verbal representation of the
problem entity developed for a particular study, and the computerized model represents the
conceptual model implemented on a computer. Thl general purpose of the conceptual model
validation depicted in this figure is to validate the underlying assumptions and theories. '. 1, re
specifically, the process is concerned with whether this specific model's representation of the
problem entity being modeled and its structure, logic and mathematical and causal relation
ships are reasonable for the intended use of the model [20]. One of the primary validation
techniques used for this evaluation is face validation [21]. Face validation involves having
domain experts evaluate the conceptual model to determine if they believe it is correct and
reasonable for its purpose. This usually means examing the flowchart or graphical model, or
the set of model equations.
Tih counterpart of the above modeling process in :.IOOSE can be depicted in Fig 20.
iOOSE does not yet employ any form of validation or verification techniques. '.IOOSE
supports many different types of models including FSM (Fiiii; State '., I, 1ii ii), FI I (Func
tional Block '.i del) and EQN (EQ ii1 iin :.N; Constraint model) for the conceptual modeling
process. ThI ii, by translating the conceptual model into C++ code, it constructs the com
puterized model. Here, the proposed method can come to play. By having the expert's
knowledge in the form of rules, and through consistency checking against the results ob
tained from implementing FSM, Fi'.I and EQN, we can form an verification arc between
Problem
Entity
Operational / Analysis Conceptual
Validity Experimentation and Model
/Data Modeling Validity
// Validity
Computer Programming ,
and Implementation
Computerized
Modelri
Verification
Figure 19: '.i, dealing process and its related validation/verification
Problem
Entity
1 1
Experimentation
Experimentation
Computerized
Model
(C++)
Conceptual
and
Physical Modeling
Model Translation
Conceptual Model
(FSM, FBM, EQN)
Figure 20: '.i,1deling process in '.iOOSE
/ Conceptual
/ and
/ Physical Modeling
Experimentation
/
Conceptual Model
SExpert's rules
Computerized Model Translation
Model (FSM, FBM, EQN)
(C++)
Consistency Checking
Figure 21: Consistency checking in .iOOSE
the conceptual model and computerized model as shown in Fig 21. Since we skipped the
conceptual model validation, any inconsistency found can be due to an inadequate concep
tual model of '.iOOSE or the expert's rules, or an improperly programmed or implemented
conceptual model on the computer.
6 Conclusion
Tli, motivation for this work lies with the problem of resolving the difference between qual
itative and quantitative forms of knowledge about physical systems. Til, fuzzy simulation
method introduced here bridges the gaps between the two different levels of knowledge.
We showed how two different extreme levels of knowledge can be directly compared and
maintained in a systematic manner. Since the uncertainty arising from the human reasoning
process is easily represented by rules associated with confidence factors, we devised a way for
emulating such processes by showing how fuzzy simulation can derive the confidence factors
from quantitative models. For handling another form of uncertainty arising from linguistic
vagueness, we assumed that central point estimates were presented by experts. Although
this form of estimation is a very limited form of uncertainty representation, we assert that
the presented method serves as a stepping stone for developing a more robust method which
can capture the other forms of expert's confidence levels in the future. By devising a method
of integrated qualitative and quantitative dynamical system knowledge refinement, we hope
to provide a way to obtain better models in general about physical systems by exploiting
knowledge at all levels, whether qualitative or quantitative.
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