LATTICE-VALUED LOGIC AND NEURAL NETWORKS

Yunfeng Liu and P.K.C. Wang Department of Electrical Engineering

University of California Los Angeles, CA 90095-1594

Abstract

Lattice-valued logic is a generalized logic whose definition function is set-valued. It can be applied to switching systems by defining a lattice-valued switch (LVS) whose parallel connection "U" and cascade connection " n" are generalized operators which may correspond respectively to the "max" (V) and "min" (A) operators in fuzzy logic, or "+" and " e " operators in neural networks. Here, it is shown that both fuzzy logic based systems (FLS) and neural network based systems (NNS) can have a unified representation in the form of LVS. A fuzzy Boolean switching system which has the advantages of both FLS and NNS is introduced along with a method for its design.

1. Introduction

Fuzzy-logic based systems (FLS) are capable of processing fuzzy information derived from expert systems [l], while neural network based systems (NNS) are only capable of processing data from sen- sors [2]. They have been used in diverse areas such as control systems [3],[4], pattern recognition, and signal processing [5]. Recently, attempts have been made in integrating FLS and NNS [6],[7].

In this paper, we show, with the aid of exam- ples, that FLS and NNS can be integrated in the framework of lattice-valued logic (LVL) which is a generalized logic whose definition functions are set- valued [8]. Based on LVL, we introduce a latticed- valued suitch(LVS). The parallel connection "U" and cascade connection "n" of LVS are generalized oper- ators which may correspond respectively to " m a " (V) and "min" (A) operators in fuzzy logic, or "+" and "." operators in a neural network. Thus, both FLS and NNS have a unified mathematical represen- tation in the form of LVS. To facilitate the treat- ment of integrated FLS and NNS, we also introduce a fuzzy Boolean switching system (FBSS) which has the advantages of both FLS and NNS. Namely, its operations are simple, and it has capability of pro- cessing fuzzy information as in FLS. Moreover, it has the ability to learn as in NNS, and its internal struc- ture is easy to understand and update as in FLS. The operations of FBSS are based on fuzzy Boolean algebra [9]. A design method for FBSS is introduced.

Its basic steps are indicated in the block diagram shown in Fig.l.1.

Boolean Neural Network Learner

I Data Base I Yl i x

- 1 i I

_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ - - - - - - - - - - - I r

I_ ( x , Y )

1 -valued Blocks I I B'=(X', Y') I I I I

I I Simplify I ( I I

I I I I I

I

I I I

I I I I I

Y ! I

I

I

I c i

Boolean Fumy Logic Worker

Fig. 1.1 Fuzzy Boolean switching system.

2. Lattice-Valued Switch We begin with a few definitions. Let X,,i =

1,. . . , n , be nonempty sets; X = X 1 x .. . x X,; I

the longitudinal interval , where a, b E R are the

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endpoints o f I. Let 8, a subset of P ( I ) = {Ii : I* E I , i E I}, be a lattice, where Z is a suitable index set.

Definition 2.1 A set-valued function (def ini t ion func t ion of A) x + ~ A ( z ) on X into 8 is called a n-dimensional lattice-valued switch (LVS) A o n X. Let the measure of p A ( x ) be denoted by h A ( s ) , then

is called the connection degree of A o n x, and the parameter 0 is called the threshold value of A. Definition 2.2 Let 3 be a collection of all LVS on X. Then for all A, B E 8 and for all x E X , we define

P A U B ( ~ ) = PA(^) UPB(Z), (2.2)

P A ~ B ( ~ ) = P A @ ) ~ P B ( z ) , (2.3)

P A ( 5 ) = PA(z ) , (2.4)

where U, n, and - are called parallel connection, cas- cade connection, and reflexive phase respectively.

Theorem 2.1 In Definitions 2.1 and 2.2, let

x = {(.,I) : x E x}, #= { ( z , $ ) : x EX}.

If ($, U, n,-, X , 4) is a Boolean algebra, then (3, U, n,-, X,#) also forms a Boolean algebra (called fuzzy Boolean algebra (FBA) hereafter).

Definition 2.3 A FBA is said to be normal , if

I = [ ; I , e = o ,

or I = [-'1], B = O .

Example 2.1 If we set

B = { $ , I } , I = , 6 = 0 , LI then for all A E and for all x E X, we have WA(x) E (0, I), and

is a binary-valued switch dgebra.

Example 2.2 If we set

then for all A E 8 and for all z E X, we have W A ( L ) E { - l , O , l}, and

wAnA(") = -1, (2.9)

wA(Z) = -WA(Z). (2.10)

In this case, 3 is a trinary-valued switch algebra in which the connection degree can be negative.

3. Integration Operator of LVS For FBA, we have the following proposition:

Proposition 3.1 Let

A and

Then for all A,B E C8 and all x E X , we have

wB(Z) wB(z)=wAnB(x)-

The reasonableness of integrated definition o f FLS and NNS is based on the following proposition: Proposition 3.2 In normal FBA, for all A, B E and x E X, we have

w A ( 2 ) U wB(Z)

-wA(z), if "A,-,*((") = -1; (3.6) W A ( X ) = 1 - W A ( Z ) , if "An*(") = 0. (3.7)

Thus, we can define the following: for FLS ( F , A, V):

35 a

Iuput Lays Hidden Layer Output Laver If Fig.3.1 is changed to Fig.3.3, then the inte-

gration operator ”0” has the following matrix repre- sentation:

(i) for ( N , +, .):

= ( Y l , Y 2 ) , (3.8)

(ii) for ( F , V, A):

where the matrix (3.5’)

(3.7’) (3.10)

is called the connection degree matrix. Similarly, we can define a n x m connection degree matrix.

(3.2’)

Iuput Layer Hidden Layer

- - - --. (3.4’)

+ - ---L----’

(3.6‘)

In many NNS, its switches can be divided into three layers, namely, the input layer {sj}, hidden layer {wij}, and output layer {yi}. This kind of NNS is known as layered FNN (feedforward neural network). When the network graph of FNN shows the switches in the hidden layer (in the ”black box”) explicitly, then it is called an explicit network graph (see Fig.3.1) in which the switches {wij} are repre- sented by nodes. Otherwise it is called an implicit network graph (see Fig.3.2) in which the switches {wij} are represented by the connection degrees as- sociated with edges.

Iuput Layer Output Laver

Fig.3.2 Implicit network graph.

D Yl Y l

output Layer

Fig.3.3 Explicit matrix network graph.

In ( F , v, A), because the integration operator ” 0” is closed, its operation is simple, and its ”hidden layer” is transparent. Hence its internal structure is easy to understand and update. But in ( N , +, .), be- cause the integration operator ”0” is not closed, it is necessary to add a close func t ion C such as hard limiter, threshold, and sigmoidal functions, e.g.

(3.11) 1, if C ( a ) 2 1; 0, if C ( a ) < 1. C ( a ) =

4. Transformation from NNS to FLS Since FLS and NNS are both examples of LVSS,

we expect that they can be transformed from one to the other, although they have different integration operators.

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Definition 4.1 A FLS ( F , V, A) is said to be normal if the longitudinal interval and 6 are

. 1 1 1 0.5 0.5 0.5- 0.5 1 1 0.5 0.5 0.5 0.5 0.5 1 1 0.5 0 0.5 0.5 0.5 1 0.5 1 0.5 0 0 0.5 1 1

-0.5 0.5 0.5 1 0.5 1 - and a NNS ( N , +, .) is said to be normal if the lon- gitudinal interval and 0 are

.

I N = [:1], e = 1,

where 6 is the threshold value of FLS or NNS.

Proposition 4.1 If W and W are the connection degree matrices of normal FLS and NNS respectively, then we have

wN = 2w$ - 1, wi, F = (wi j N + 1)/2. (4.1) v

The relation between w$ and w$ are shown in the table below.

0.5 -1

Example 4.1 Consider a normal NNS ( N , +, e) with

1 1 0 0 0 1; 1 1 0 0 0 1

J 0 - 1 - 1 0 1 1 0 0 0 1 0 1

and with close function C given by (3.11). We wish to find the associative memory of the input X ( 0 ) = (1 0 0 0 0 0) such that X(t + 1) = X ( t ) for some positive integer t.

From (3.8) and (3.11), we obtain

X ( 0 ) 0 W N = (1 1 1 0 0 0) = x(1),

X(1) 0 W N = (1 1 1 1 0 0) = x(2), X(2) 0 W N = (I 1 1 1 0 0) = X ( 3 ) = X(2).

Thus, the solution is (1 1 1 1 Example 4.2 The NNS problem in Example 4.1 can be changed to one in FLS using (4.1). In this case,

X ( 0 ) = (I 0.5 0.5 0.5 0.5 0.5),

Using (3.9), we obtain

X ( 0 ) o W F = (1 1 1 0.5 0.5 0.5) = X(1),

X ( l ) o W F = (1 1 1 1 0.5 0.5) = X(2),

X(2) o W F = (1 1 1 1 0.5 0.5) = X(3) = X(2)

Thus, the solution is (1 1 1 1 0.5 0.5). Comparing Examples 4.1 and 4.2, it is evident

that although the computational methods associated with NNS and FLS are quite different, sometimes their results are consistent.

5. From FLS to FBSS Now, we introduce a new switching system called

the furzy Boolean switching system (FBSS) ( B , U, n) (see Fig.l.1) which is a special LVSS. To clarify ideas, we consider a simple example. Example 5.1 Let f be an input-output X-Y func- tion whose graph is shown by the dashed curve in Fig. 5.1.

X

Fig.5.1 Graph of f.

In FLS, we have the following usual fuzzy rea- soning rules:

' if z is -B, then y is LOW;

if z is -S, then y is if z is +S, then y is HIGH; if z is + M , then t/ is if 3: io +B, then y is E

if z is - M , then y is IUM;

. Next, we give a method which tra lem in FES into one in FBSS. This method is base

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Fig.5.2 Boolean partition of the step-switch.

Fig.5.3 Boolean step-switch.

on partitioning the step-switches A I , . . . , Am by d e cision boundaries b l , . . . , b m - l . Figure 5.2 shows the partitions for m = 6 .

Let A l , . . . , A, be step-switches which are par- titioned into disjoint Boolean step-switches A I , . . . , A,, and X is partitioned into disjoint subsets XI, . . . , X, (see Fig.5.3). Definition 5.1 Let { X i , i = 1, . . . , m } be a partition of XI and each Xi be represented by the unit column vector

ei = [O,. . . , l(i-th component), . . . , O I T ,

where [-IT denotes transposition. The column vec- tor [Xi] = ei is called the Boolean vector or B-vector of X i , and { X i , i = 1,. . . , m} is called the Boolean partition or B-parti t ion of X.

The main idea of FBSS is to use 1-valued blocks to cover the X-Y function (each 1-valued block is a fuzzy reasoning rule in FLS). In a Boolean neural network learner of FBSS (see Fig.l.l), we can obtain simplified 1-valued blocks (see Fig.5.4) from the B- vectors of X i and Yi.

Definition 5.2 Let f = f ( z ) denote the input- output function from X -+ Y ; { X i , i = 1,. . . , m} be the B-partition of X ; Y, = f ( X i ) , and [Yi] the B-vector of Yi. Then the matrix W defined by

a

m m

is called the perspective drawing of the graph of f, and Bi = ( X i , Yi) is called the 1-valued block corre- sponding to X i and Y,.

Fig. 5.4 1-valued block.

Example 5.2 Applying (5.1) to the input- output function shown in Fig.5.4, we obtain

0 0 # l - - l* ,o 0 w = 0.1 0 0 1 .0 . (5.2)

[ V d 0 0 0 k l l

Evidently, W depicts the graph of f .

6. FBSS Design Method Here, we propose a method for designing FBSS.

To illustrate the basic ideas, we consider a single- input-single-output system shown in Fig.5.1. The basic steps of the proposed method are as follows:

Step 1. Data Base: Let the domain of defini- tion of f = f (z ) be X which is partitioned equally into disjoint subintervals XI,, . , , Xn. Let z: and z:+l be the endpoints of Xi. From the physically measured data, we obtain

Y: = f(z:), Y:+l = fb:+d?

Step 2. 1-valued block simplification: Let K = [--oo,+-oo] be partitioned into disjoint subin- tervals, for example,

6

K = ux,, i= 1

where

K1 = [-cQ, -1), K2 = [-1, -0.5), K3 = [-0.5,0), w Wi 4 c((Yi] o [ X i l T ) (5.1)

i = l i=l

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If ki,. . . , E K,,, TI E (1,. . . ,6}, then we ob- tain the simplified 1-valued block B1 = (XI, Y I ) (see Fig.6.1), where

9 S

X I = xi, Y1= U Y i . t=l i=l

Similarly, if then we obtain B2 = ( X 2 , Y2), where

. . . , k$+, E K,,, 7-2 E (1,. . . ,6},

s+t s+t

i=s+l i=s+l

Continuing this process, we obtain m simplified 1- valued blocks B1,. . . , Bm. Since each Bi corresponds to a fuzzy reasoning rule, (B1, . . . , B m } corresponds to m fuzzy reasoning rules.

1 X

Unsimplified 1-valued Block

Simplified I-valued Block

Fig.6.1 1-valued block simplification.

Step 3. Fuzzification and Defuzzification: From the simplified Bi, we obtain

where zi and zi+l are the endpoints of X i ; and for all z E Xi, we have approximately

Thus, fuzzification and defuzzification in FBSS are extremely simple. They are given by

hzzification: a E X i , Defuzzification: y = !(xi) -+ k i ( z - si). The above scheme greatly simplifies the fuzzifi-

cation and defuzzification processes in applications.

7. Concluding Remarks The main objective of this paper was to cre-

ate a more generalized framework for fuzzy logic and neural networks. First, we showed that both fuzzy- logic and neural network based systems have a uni- fied mathematical representation in the form of LVS. Next, we introduced a method for designing fuzzy Boolean switching systems. Also, we introduced two methods for obtaining nonlinear input-output map- ping from (i) the data base (as shown in Fig.l.1 and discussed in Sec.6), and (ii) from a knowledge base (as discussed in Sec.5). These methods are particu- larly useful in neuro-fuzzy control system modeling and design. The proposed approach also permits a connection between fuzzy logic based controls and controls based on precise data. A detailed discussion of specific applications will be given elsewhere.

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