A NEW APPROXIMATE REASONING BASED ON SPMF

Dae-Young Choi

Dept. of MIS, YUHAN College, Koean-Dong, Sosa-Ku, Puchon City

Kyungki-Do, South Korea

Keywords: Approximate reasoning, Linear time complexity.

Abstract: A new approximate reasoning based on standardized parametric membership functions (SPMF) is proposed.

It provides an efficient mechanism for approximate reasoning within linear time complexity.

1 INTRODUCTION

Approximate reasoning is generally expressed in the

form of the following syllogism :

Rule : IF X is A THEN Y is B

Observation : X is A′ (1)

Conclusion : Y is B′

where X and Y are linguistic variables, and A, B and

A′ are fuzzy subsets defined in the universe of

discourse U, V and U, respectively. Generally, in

order to obtain the deduction of conclusions from

observations and rules in a knowledge base, there

are three alternative ways of doing this in fuzzy sets

framework: truth value restriction (Zadeh, 1975),

compositional rule of inference (CRI) (Zadeh, 1973)

and an approximate analogical reasoning schema

(AARS) (Turksen and Zhong, 1988). The CRI has

been more widely accepted and applied in

development studies. The CRI is essentially based

on matrix operation. The effect of such matrix

operations on membership function value

propagation is not conceptually clear. Some of its

undesirable consequences were pointed out in

(Mizumoto, 1985, Turksen and Zhong, 1988). It has

major flaws that it does not satisfy the modus ponens.

That is, when A′=A, the deduced B′ is obtained as

follows : μ

B′

= (1+μ

B

)/2 ≠ μ

B

. This inference result

indicates that the CRI does not satisfy the modus

ponens, i.e., (A∧(A→B))→B, which is quite

reasonable demand in the approximate reasoning. In

addition, ‘indetermination’ part of the consequence

occurs because of the incompatibility between the

membership functions of A in the premise of rule

and A′ from observation. This incompatibility

happens when the insignificant part (i.e., the zero

membership range) of A includes that of A′ (Chang

et al., 1991). In the meantime, the AARS uses the

term similarity to express the semantics of inference.

Here, the similarity of two fuzzy sets is expressed by

the following equation: SM=(1+DM)

-1

where

SM∈[0,1]. That is, the similarity measure (SM) is

obtained by using the distance measure (DM).

However, it did not define clearly how to obtain the

DM. In addition, it did not define clearly the

modification function (MF) that plays an important

role in the approximate reasoning. To handle these

problems, a new approximate reasoning based on

SPMF is proposed.

2 SPMF

Let A be a fuzzy set for a linguistic term and be a

subset of the universal set X, then, for x∈X, a

triangular-type membership function can be

represented by using 3 points μ

A

(x

L

, x

M

, x

H

), where

x

L

<x

M

<x

H

,

and if the result of this membership

function is normalized to [0, 1] then μ

A

(x

L

, x

M

, x

H

) =

0 for every x∈[-∞, x

L

]∪[x

H

, ∞] and μ

A

(x

L

, x

M

, x

H

) = 1

at x

M

. A trapezoidal-type can be represented by using

4 points μ

A

(x

L

, x

I

1

, x

I

2

, x

H

), where x

L

<x

I

1

<x

I

2

<x

H

,

and if

the result of this membership function is normalized

to [0, 1] then μ

A

(x

L

, x

I

1

, x

I

2

, x

H

) = 0 for every x∈[-∞,

x

L

]∪[x

H

, ∞] and μ

A

(x

L

, x

I

1

, x

I

2

, x

H

) = 1 at [x

I

1

, x

I

2

]. A

more comprehensive study of SPMF can be found in

(Chang et al., 1991).

383

Choi D. (2008).

A NEW APPROXIMATE REASONING BASED ON SPMF.

In Proceedings of the Tenth International Conference on Enterprise Information Systems - AIDSS, pages 383-386

DOI: 10.5220/0001669703830386

Copyright

c

SciTePress

3 THE PROPOSED METHOD

A new approximate reasoning based on SPMF

makes the DM to compare two fuzzy sets in the

pattern matching phase, and then the DM is used

to construct the MF. The MF will adjust the right

side of the rule in the consequent deducing phase.

We first consider a simple rule case where only

one observation A′ and one simple rule as in Eq.

(1).

3.1 Distance Measure (DM)

Based on their behavioral experiment (Zwick et al.,

1987), they recommended the five good DM

between fuzzy subset A and B of a universe of

discourse U. We note that the five good DM

concentrate their attention on a single value rather

than performing some sort of averaging or

integration. We know that the reduction of

complicated membership functions to a single ‘slice’

may be the intuitively natural way for human beings

to combine and process fuzzy concepts. Moreover,

we know that the DM between two fuzzy subsets can

be efficiently represented by a limited number of

features. From these ideas, we define the DM based

on the structure of the SPMF.

(1) Triangular-type Membership Functions

If the antecedent A of a rule is represented by A =

(x

L

,

x

M

, x

H

) and an observation A′ is represented by A′

= (x

L

′,

x

M

′, x

H

′), then each DM is obtained regarding

its corresponding 3 points, respectively.

DM

L

= x

L

′

- x

L

DM

M

= x

M

′-x

M

(2)

DM

H

= x

H

′

- x

H

(2) Trapezoidal-type Membership Functions

If the antecedent A of a rule is represented by A =

(x

L

,

x

I

1

,

x

I

2

, x

H

) and an observation A′ is represented

by A′ = (x

L

′,

x

I

1

′,

x

I

2

′, x

H

′), then each DM is obtained

regarding its corresponding 4 points, respectively.

DM

L

= x

L

′

- x

L

DM

I

1

= x

I

1

′ - x

I

1

(3)

DM

I

2

= x

I

2

′

- x

I

2

DM

H

= x

H

′

- x

H

3.2 Pattern Matching

The pattern matching is achieved through the use of

the least distance measure (LDM) between the

observed fact and the antecedent of a rule.

(1) Triangular-type Membership Functions

LDM

p

= min {DM

L

+DM

M

+DM

H

} for all rules in a

rule base. (4)

(2) Trapezoidal-type Membership Functions

LDM

I

= min {DM

L

+ DM

I

1

+ DM

I

2

+ DM

H

} for

all rules in a rule base. (5)

Thus, the rule with LDM

p

or LDM

I

is selected in a

rule base.

3.3 Modification Functions (MF)

In the proposed method, a rule R

i

: A

i

→ B

i

is to be

fired with each MF that modifies the consequent B

i

of the rule R

i

. We construct each MF based on its

corresponding DM in Eqs. (2), (3). When deducing a

consequent, the MF enables us to bypass the matrix

operations of CRI. Each MF is achieved by using the

following formulas :

(1) Triangular-type Membership Functions

Let the maximum support interval (MSI) of two

fuzzy subsets (i.e., A, A′) represented by using the

triangular-type membership functions be [x

LL

, x

MH

],

where x

LL

is derived from min {x

L

, x

L

′} and x

MH

is

derived from max {x

H

, x

H

′}, and let the distance of

MSI of two fuzzy subsets (DMSI) be |x

MH

-x

LL

|, then

each MF is obtained regarding its corresponding 3

points, respectively.

MF

L

= (1+ (DM

L

/DMSI))

MF

M

= (1+ (DM

M

/DMSI)) (6)

MF

H

= (1+ (DM

H

/DMSI)) where each DM

is derived from Eq. (2).

(2) Trapezoidal-type Membership Functions

Let the MSI of two fuzzy subsets (i.e., A, A′)

represented by using the trapezoidal-type

membership functions be [x

LL

, x

MH

], where x

LL

is

derived from min {x

L

, x

L

′}, and x

MH

is derived from

max {x

H

, x

H

′}, and let the DMSI of two fuzzy subsets

be |x

MH

-x

LL

|, then each MF is obtained regarding its

corresponding 4 points, respectively.

MF

L

= (1+ (DM

L

/DMSI))

MF

I

1

= (1+ (DM

I

1

/DMSI)) (7)

MF

I

2

= (1+ (DM

I

2

/DMSI))

MF

H

= (1+ (DM

H

/DMSI)) where each DM is

derived from Eq. (3).

We can determine the overall MF(OMF) by

averaging all MF in Eqs. (6) or (7), respectively.

OMF = avg{all MF(A, A′)} where the all MF(A, A′)

ICEIS 2008 - International Conference on Enterprise Information Systems

384

are derived from Eqs.(6) or (7), respectively. (8)

In Eq. (8), we consider a simple rule case where only

one observation A′ and one simple rule in the form

‘IF X is A THEN Y is B’. The construction of MF is

subjective in (Turksen and Zhong, 1988). To handle

this problem we suggest the efficient OMF based on

the structure of the SPMF.

3.4 Deducing a Consequent

It is assumed that we consider a simple rule case

where only one observation A′ and one simple rule

in the form ‘IF X is A THEN Y is B’. Let B be a

fuzzy subset of the linguistic variable ‘Y’ and be

represented by the SPMF then, for

∀

y∈Y, the

linguistic value B can be represented by (y

L

,

y

M

, y

H

)

or (y

L

,

y

I

1

,

y

I

2

, y

H

) in the triangular-type and

trapezoidal-type membership functions, respectively.

In the proposed method, we construct the deduced

consequent B′ by applying the OMF to B.

(1) Triangular-type Membership Functions

y

L

′ = OMF

× y

L

y

M

′ = OMF × y

M

(9)

y

H

′ = OMF

× y

H

where the OMF is

derived from Eq. (8).

(2) Trapezoidal-type Membership Functions

y

L

′ = OMF × y

L

y

I

1

′ = OMF × y

I

1

(10)

y

I

2

′ = OMF × y

I

2

y

H

′ = OMF × y

H

where the OMF is

derived from Eq. (8).

The OMF obtained in the pattern matching phase is

applied to the points such as y

L

, y

H

, etc, in the

consequent deducing phase as in Eqs. (9), (10).

Definition 1. According to Eqs. (2), (3), (6)-(8), in

case of a positive dependency (e.g., ‘good → big’,

see Example 1) between A and B in a rule, the

directionality of modification in the consequent

deducing phase is determined.

Case 1 : If OMF < 1, then the left shift with OMF

occurs regarding all points such as y

L

, y

H

, etc.

Case 2 : If OMF = 1, then no shift occurs. As a

special case, for a pair (A, A′), if all DM in Eqs. (2)

or (3) is zero, then the exact matching occurs

between the observed fact A′ and the antecedent A

of a rule.

Case 3 : If OMF > 1, then the right shift with OMF

occurs regarding all points such as y

L

, y

H

, etc.

On the contrary, in case of a negative dependency

(e.g., ‘high weight → low speed’) between A and B

in a rule, the directionality of modification in the

consequent deducing phase is determined reversely.

We note that when the special case of Case 2 of the

Definition 1 occurs (i.e., A = A′), the reasoning

result of the proposed method becomes B′ = B. This

is one of the advantages of the proposed method

over CRI. In other words, the proposed method

satisfies the modus ponens but the CRI does not

satisfy the modus ponens.

Example 1. We consider a simple rule case where

only one observation A′ and one simple rule in the

form ‘IF X is A THEN Y is B’. It is assumed that the

selected rule is ‘IF economic conditions were good

THEN the earning was big’, and one observation is

‘economic conditions are good′’. We assume that the

stockholder defines fuzzy subsets regarding the

goodness of the linguistic variable economic

conditions in the interval [0, 100] by using the

trapezoidal-type.

μ

1

80 85 88 90 92 95 96 100 X

Figure 1: An example of fuzzy subsets regarding the

goodness of economic conditions (X).

In Figure 1, the antecedent A of the selected rule is

assumed to be ‘good’, whereas the observation A′ is

assumed to be ‘good′’. In this case, each MF is

computed by using Eq. (7).

MF

L

= (1+ ((88-80)/(100-80)) = (1 + (8/20)) = 1.4.

MF

I

1

= (1+ ((92-85)/(100-80)) = (1 + (7/20)) = 1.35.

MF

I

2

= (1+ ((96-90)/(100-80)) = (1 + (6/20)) = 1.3.

MF

H

= (1+ ((100-95)/(100-80)) = (1 + (5/20)) = 1.25.

Thus, we obtain the OMF = (1.4+1.35+1.3+1.25)/4 =

1.33 by using Eq. (8). In the meantime, we assume

that the stockholder defines the fuzzy subset ‘big

earning’ as in Figure 2. Using Eq. (10), we construct

the deduced consequent B′ by applying the OMF to

B as in Figure 2.

y

L

′ = OMF

× y

L

= 1.33 × $10 = $13.3.

y

I

1

′ = OMF

× y

I

1

= 1.33 × $12 = $15.96.

y

I

2

′ = OMF

× y

I

2

= 1.33 × $13 = $17.29.

y

H

′ = OMF

× y

H

= 1.33 × $15 = $19.95.

Thus, we obtain the deduced consequent B′, i.e., the

Good

Goo

d

′

A NEW APPROXIMATE REASONING BASED ON SPMF

385

deduced earning = (y

L

′, y

I

1

′, y

I

2

′, y

H

′) = (13.3, 15.96,

17.29, 19.95).

μ

1

10 12 13 13.3 15 15.96 17.29 19.95 Y

Figure 2: An example of fuzzy subsets.

Now, we consider the composite rules with ‘OR’ and

‘AND’ connectives.

(1) ‘OR’ Composition

Given a rule with the following format : [A

i

1

OR A

i

2

OR …OR A

i

K

] → B

i

, it can be decomposed into

simple rules as A

i

1

→ B

i

, A

i

2

→ B

i

, … ,

A

i

K

→ B

i

,

and can be treated as individual simple rules,

respectively (Turksen and Zhong, 1988).

(2) ‘AND’ Composition

Given a rule with the following format : [A

i

1

AND

A

i

2

AND … AND A

i

K

] → B

i

, we can determine the

overall MF(OMF

i

) based on Eqs. (6) or (7) by

averaging MF

ij

regarding all corresponding pairs of

(A

ij,

A

ij

′), where i denotes the i

th

rule and j = 1,2,…,k.

In this case, Eq. (8) is changed into as follows :

OMF

i

=avg {avg MF

ij

(A

ij,

A

ij

′)} where each MF

ij

(A

ij,

A

ij

′) is derived from Eq.(6) or (7), respectively, and a

group of observations has the same form [A

i

1

′ AND

A

i

2

′

AND… AND A

i

K

′]. (11)

Example 2. We consider the i

th

rule with ‘AND’

connectives in the form ‘IF X

1

is A

i

1

AND X

2

is A

i2

AND X

3

is A

i3

THEN Y is B’. For simplicity, let A

i

1

=(1,2,3), A

i

2

=(3,4,5,6), A

i

3

=(6,7,8), and A

i

1

′=(2,3,4),

A

i

2

′=(4,5,6,7), A

i

3

′=(7,8,9), respectively, (i.e., k = 3)

then the OMF

i

is obtained by using Eqs.(6),(7), (11).

OMF

i

= (

∑

=

3

1j

avg (MF

ij

))/3

={[(1+((2-1)/(4-1)))+(1+((3-2)/(4-1)))+(1+((4-3)/(4-

1)))]/3+[(1+((4-3)/(7-3)))+(1+((5-4)/(7-3)))+(1+((6-

5)/(7-3)))+(1+((7-6)/(7-3)))]/4+[(1+((7-6)/(9-

6)))+(1+((8-7)/(9-6)))+(1+((9-8)/(9-6)))]/3}/3

= {(1+(1/3))+(1+(1/4))+(1+(1/3))}/3 = 1.3.

The OMF

i

(i.e., 1.3) will be used in the consequent

deducing phase as follows :

μ A

i1

A

i1

′A

i2

A

i2

′A

i3

A

i3

′ μ B

i

B

i

′

1 1

1 2 3 4 5 6 7 8 9

X

1 1.3 2 2.6 3 3.9 Y

Figure 3: An example of ‘AND’ connectives.

4 COMPARISONS

Some comparisons are in Table 1.

Table 1: Comparisons.

Attributes Existing methods Proposed method

Mem. fn. Ad-hoc SPMF

Method Genreally, CRI Based on SPMF

Formula Complex Simple

Operation Generally, matrix Linear

5 CONCLUSIONS

The proposed method provides an efficient

mechanism for approximate reasoning within linear

time complexity.

REFERENCES

Chang, T. C., Hasegawa, K., Ibbs, C. W., 1991. The

Effects of Membership Function on Fuzzy Reasoning,

Fuzzy Sets and Systems 44, 169-186

Mizumoto, M., 1985. Extended Fuzzy Reasoning, In :

Gupta et al. (Eds.), Approximate Reasoning in Expert

Systems (North-Holland) 71-85

Turksen, I. B., Zhong, Z., 1988. An Approximate

Analogical Reasoning Approach Based on Similarity

Measures, IEEE trans. on SMC 18(6), 1049-1056

Zadeh, L.A., 1973. Outline of a New Approach to the

Analysis of Complex Systems and Decision Processes,

IEEE Trans. on SMC 3, 28-44

Zadeh, L.A.,

1975. The Concept of a Linguistic Variables

and Its Application to Approximate Reasoning I, II, III,

Part I: Inf. Sci. 8, 199-249, Part II: Inf. Sci. 8, 301-357,

Part III: Inf. Sci. 9, 43-80.

Zwick, R. et al. 1987, Measures of Similarity among

Fuzzy Concepts : A Comparative Analysis, Int. J. of

Approximate Reasoning 1, 221-242

Big earning (B)

Deduced

earning

(B′)

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