algebraic aproach to meaning of linguistic terms, fuzzy logic and approximate reasoning

ALGEBRAIC APROACH TO MEANING OF LINGUISTIC TERMS, FUZZY LOGIC AND APPROXIMATE REASONING Nguyen Cat Ho Institute of Information Technology VAST E-mail: ncatho@hn.vnn.vn... Until recently

ALGEBRAIC APROACH TO MEANING OF LINGUISTIC TERMS, FUZZY LOGIC AND

APPROXIMATE REASONING

Nguyen Cat Ho

Institute of Information Technology

(VAST) E-mail: ncatho@hn.vnn.vn

However, up to now one has still not discovered the

mathematical structure of term-domains of linguistic variables,

as Rinks wrote in “A heuristic approach to aggregate production scheduling using linguistic variables, Proc of Inter Congr on

Appl Systems Research and Cybernetics, Vol VI (1981)” that

"verbal coding is a human way of repackaging material into a few chunks of rich information Natural language is rather unique

in this characteristic Until recently, a unified theory for

manipulating in a strict mathematical sense

non-numerical-valued variables, such as linguistic terms, did not exist."

A question is how can we find out a mathematical structure to

I Introduction

 1965 L.A Zadeh, in his paper

“ Fuzzy sets , Information and Control 8 (1965)”,

Fuzzy sets were introduced to model the meaning of

linguistic terms of natural languages.

In natural language there are vague terms such as

young, old, rather large, approximately 9, …

1

0.5

25 35 U

I Introduction

 1965 L.A Zadeh, in his paper

“Fuzzy sets, Information and Control 8 (1965)”,

Fuzzy sets were introduced to model the meaning of linguistic terms of natural languages

- Allow manipulate and process fuzzy, uncertain and inexact information;

- Establish a computation approach (analytic one) to simulate human reasoning methods, based on fuzzy sets and fuzzy logics;

- Open a new era of developing applications in uncertain

environment of industry, sciences, social-economy, …

- Now, we can find everywhere applications of fuzzy sets like in cars, washing machines, air conditioners, …

The achievements of fuzzy sets theory are very great and

incontrovertible However, it inherits still some problems

I Introduction

Some problems:

Representation of linguistic terms by fuzzy sets means that one should establish an embedding:  : Dom(X)  F(U,[0,1]) Since the image of Dom(X) under  has no mathematical

structure, operations on fuzzy sets are defined on the whole

space F(U,[0,1]) It seems to be unreasonable and, moreover, not correct, because

- Dom(X) is finite, but F(U,[0,1]) is infinite;

- We observe that Dom(X) has a semantics-based

ordering relation , but  does not preserve this relation

+ Maybe, by these reasons the efficiency of fuzzy sets-based methods is limited

+ The answer is affirmative

+ Since late 1980s: an algebraic approach to the structure of term-domains of linguistic variables has been introduced:

- Dom(X) = AX = (X,C,H,);

- A wide class of HAs: Lattices;

- Symmetrical HAs: algebraic foundation of non-classical

logics;

1 Introduction

We shall talk about

 An overview of fuzzy sets theory and its computation mechanism

 Hedge algebras – a semantics- based structure of terms-domains

 Quantification method of hedge algebras: Fuzziness Measure of linguistic terms, hedges and Semantically Quantifying Mappings

 Applicability of hedge algebras

 Some Conclusions

2 Fuzy Sets: An Overview

Definition of Fuzzy Sets and Their Operations

Example: Representation of the meaning of young

2 Fuzy Sets: An Overview

+ Fuzzy Sets Operations

Given fuzzy sets A and B

2 Fuzy Sets: An Overview

Cartesian Product: AB

AB (u) = A(u)  A(u)

+ Fuzzy Sets representing logical connectives

Given fuzzy sets A and B

- AND, OR, NEGATION by Union, Intersection and Complement

- If X is A then Y is B : A  B

AB (u,v) = A(u) * A(v) , where * is a logical implication

R B/A denotes fuzzy relation with R B/A(u,v) = AB (u,v)

+ Approximate Reasoning

2 Fuzy Sets: An Overview

+ Approximate Reasoning

E.g in fuzzy control one can model a dependency between

physical variables X j and Y linguistically:

If X 1 = A 11 and and X m = A 1m then Y = B 1

.

If X 1 = A n1 and and X m = A nm then Y = B n

It is called a fuzzy model representing expert knowledge

Approximate reasoning problem: FMCR problem

A method which solves this problem is called FMCR method

2 Fuzy Sets: An Overview

+ Main Steps of FMCR methods:

1) To construct appropriate fuzzy sets (membership problem)

2) To define a fuzzy relation R i = R Bi/Ai on U 1 U nV to represent the semantics of if-then sentence in the given fuzzy model, by

choosing suitable implication operator

3) To define a fuzzy relation R on U 1 U nV to represent the

semantics of the fuzzy model by choosing an aggregation operator

for aggregating fuzzy relations R i defined above:

R = Union {R 1, … , R n} or R = Intersection {R 1, … , R n}

4) Problem of determining an appropriate composition operator to

compute the output fuzzy set

5) To define a suitable defuzzification method to transform an output

2 Fuzy Sets: An Overview

+ Example: Aircraft Landing Problem (Ross T J., Fuzzy logic with engineering application, International Edition. Mc Graw-Hill,

Inc, 1997)



v h

Aim: Landing gently to avoid

Aim: Landing gently to avoid

h : High v : Vertical velocity

If a force f applied over a time t a

change in v will be v = ft/m

Choose t = 1.0 (sec) and m = 1.0

(lb-sec2/ft), we have v = f Using different notation, we have:

vi+1 = vi + fi and hi+1 = hi + vi (for cycle calculating).

2 Fuzy Sets: An Overview

Fuzzy control method : State

variables

vi+1 = vi + fi and hi+1 = hi + vi

1) Define a set of fuzzy rules

(FAM):

2) membership functions for

state variables h

v h

2 Fuzy Sets: An Overview

Fuzzy control method : State

variables

vi+1 = vi + fi and hi+1 = hi + vi

1) Define a set of fuzzy rules

(FAM):

2) membership functions for

state variables v = u

v h

Fuzzy control method : remember that : remember that v = f therefore:

vi+1 = vi + fi and hi+1 = hi + vi (*)

3) Define initial conditions and conduct a simulation for k cycles:

+ h0 = 1000 ft ; v0 = - 20 ft/s f0 to be computed ; High h fires L at 1.0 and M at 0.6; Velocity v fires only DL at 1.0.

Fuzzy control method :

vi+1 = vi + fi and hi+1 = hi + vi

3) Define initial conditions and conduct a simulation for k cycles:

+ Now, inputs for Cycle 1: h1 = 980, v1 = - 14.2 ft/s

High h fires L at 0.96 and M at 0.64; Velocity v fires only DS at 0.58 and DL at 0.42.

L (.96) AND DS (.58)  DS (.58); L (.96) AND DL (.42)  Z (.42)

M (.64) AND DS (.58)  Z (.58) ; M (.64) AND DL (.42)  US (.42)Compute f 0 = Centroid of the union of outputs = - 0.5 lbs

II Algebraic approach: Hedge algebras - Models of

terms- domains based on linguistic meaning

 Why we can and need introduce algebraic approach ?

 Consider a set of terms of the TRUTH variable:

True, V true, M True, A True, P True, VA True, MA True, VP True, MP True, VV true, VM True

False, V False, M False, A False, P False, VA False,

MA False, VP False, MP False, VV False, VM False

It can be seen that:

True ≤ V True, M True ≤ V True, …

A True and P True are incomparable !!

X - a linguistic; X = Dom( X ) - a set of terms

X := ( X , H , C , ≤) is at least a Poset (Partially Ordered Set)

II Hedge algebras - algebraic models of linguistic domains based on ling meaning

 An algebraic structure of X : AX = (Dom(X),C,LH,)

 X = Dom( X ) can be ordered based on meaning of terms: – X owns an ordering relation , induced by term meaning and called semantically ordering relation ;

– Several semantic properties of terms and hedges can be formulated in term of  :

 Positiveness and negativeness base term: c–  c+

 Positive hedges: c+  hc+ and c–  hc– The set H+

 Negative hedges: c+  hc+ and c–  hc– The set H– For ex old < Very old , young > Very young

while old > Possibly old , young < Little young

II Hedge algebras - algebraic models of linguistic domains based on ling meaning

 Positive hedges: c+  hc+ and c–  hc– The set H+

Negative hedges: c+  hc+ and c–  hc– The set H–

+ For ex old < Very old , young > Very young  V  H+

old < More old , young > More young  M  H+

while old > Possibly old , young < Possibly young  P  H–

old > Little old , young < Little young  L  H–

On each H we can define an ordering relation :

h ≤ k iff x ≤ hx implies that hx ≤ kx and

x  hx implies that hx  kx

II Hedge algebras

true  VP true  P true

true  L true  VL false

false  VA false  A false

V, M are positive w.r.t V, M, L negative w.r.t A, P , ML.

-II Hedge algebras

The algebraic sign of a term:

Based on these notions, we can define

a notion of algebraic sign of terms.

Definition (Sign function)

-II Hedge algebras

modifies only the meaning of term, it preserves essential

meaning of terms and we can formulate this property in term

of  :

For ex L true  A true 

true  L true  PL true  LA true  A true

Formulation of the property:

hx  kx  h’hx  k’kx so H(hx)  H(kx)

This means that h’ and k’ can not change the meaning of hx

and kx represented by hx  kx (!)

• Semantic inheritance of hedges:

H(V L A true)

H(ML L A true)

H(LA true)

Basic structure

II Hedge algebras

Axiomatization of Linear Hedge Algebras :

( A0 ) AX = ( Dom( X ) , C , H ,) : H and H+ and C are linearly ordered ( A1 ) The unit operation V in H+ is either positive or negative w.r.t any operations in H Particularly, V is positive w.r.t just V in H+

and the unit operation L in H.

( A2 ) If x  hx , then x  H ( hx ) If h  k and hx ≤ kx hold, then we have ohx ≤ o'kx , for any o and o' in UOS Moreover, if hx  kx

holds, then hx and kx are independent, i.e for  u  H ( hx ), u 

H ( kx ) and conversely, for  v  H ( kx ), v  H ( hx ).

( A3 ) If u  H ( v ) and u ≤ v ( u  v ), then u ≤ hv ( u  hv ), for each h

 UOS

Theorem Let x = hn h1u and y = km k1u be two arbitrary

canonical representations of x and y w.r.t u ,

respectively Then

(1) x = y iff m = n and hj = kj for all j  n

(2) If x  y then there exists an index j  min { m,n }+1 such

that hj' = kj', for all j'  j and

x  y iff hjxj  kjxj (!)

II Hedge algebras

II Hedge algebras

Theorem 2.1 Let AX = ( X,G,H, ) be a linear hedge algebra Then, X is a linearly ordered set.

In general case

Suppose that H- and H+ are Posets

In natural language, there are such terms:

Possibly Very false OR Approximately Very false ( ( P  A ) V false )

Possibly Very false AND Approximately Very false ( ( P  A ) V false )

Possibly Very true OR Approximately Very true ( ( P  A ) V true )

Possibly Very true AND Approximately Very true ( ( P  A ) V true ) Denote LH- and LH+ the lattices generated from H- and H+, respectively, and

LH = LH-  LH+

II Hedge algebras

Consider an abstract algebra

AX = ( X,C,LH, ) where X = LH ( C )

Axiomatization: A axioms systems which are semantic properties of linguistic terms can be established.

 X = Dom( X ) has reach enough algebraic structure

Theorem Let H and H+ of AX = ( Dom( X ) , C , LH ,) be modular lattices, where C = {0, c, W, c+, 1} Then, AX is a distributive lattice

II Hedge algebras

 Symmetrical Hedge algebras

II Hedge algebras

 Assume that x = hn h1a , where a  C \{ W }, is a representation of x

with respect to a An element y is said to be a contradictory

element of x if it can be represented as hn h1a' , with a'  C\{ W } and a'  a

Exam Little Very Possibly true Very Very Possibly false

 Definition : A hedge algebra AX = (X,C,LH,  ) , where C is defined

as above, is said to be symmetrical , provided every element x in X

has a unique contradictory element in X , denoted by x.

 Theorem: A hedge algebra AX = ( X,C,LH ,) is symmetrical iff AX satisfies the following condition:

(SYM) For every element x  X , x is a fixed point iff x is a fixed point.

II Hedge algebras

Theorem For every symmetrical hedge algebra AX =

( X,G,LH ,), the following statements hold:

( i ) ( hx )  = hx  , for every h  LH and x  X

( ii ) ( x ) = x , for every x  X

( iii ) hx > x iff hx  < x , for every h  LH and x  X

( iv ) hx > kx iff hx  < kx  , for any h , k  LH and x  X

( v ) x < y iff x  > y , for any x , y  X

( vi ) ( x  y ) = x  y  and ( x  y ) = x  y , for any x , y  X , where 

and  stand for join and meet, respectively, in AX.

II Hedge algebras

III Fuzziness measure and quantifying semantic mappings

Granularity information H ( x )

+ H ( x ) models fuzziness of x

+ The “size” of H ( x ) :

fuzziness measure of x

But first of all, we explain

more why we can use H ( x )

to model the fuzziness

III Fuzziness measure and quantifying semantic mappings

 Fuzziness measures of terms and hedges :

 Let AX be a linear hedge algebra with a base set X

Consider the class { H ( x ): x  X } Properties:

– H(x) = x , for x  { 0, W, 1 }

– H ( hx )  H ( x )

– H(x) = U{ H ( hx ) : h  H } and H ( hx )  H ( ky ) = 

→ So, the size of H ( x ) can model the fuzziness of term x

Let f : X → [0,1] Interpret the diameter of f [ H ( x )] the

fuzziness measure of term x, denoted by fm(x)

Now, we shall give a formalization:

III Fuzziness measure and quantifying semantic mappings

Fuzziness measures of terms and hedges:

 Formalization

Quantifying mapping : Let AX = ( X , C , H , ) be a free linear ComHA, where H = { h1, , hq}, with h1<h2< <hq, and H+ = { h1, , hp }, with h1< <hp Suppose that the mapping  : X  [0,1]

satisfying the following conditions :

( QM1 )  is one - to - one mapping and preserves the ordering relation on X , i.e x < y   ( x ) <  ( y );

( QM2 ) The image set  ( H ({ c, c+})) is dense in the unit interval

[0,1];

III Fuzziness measure and quantifying semantic mappings

 Formalization: Fuzziness measure of x = The SIZE of  ( H ( x )) :

III Fuzziness measure and quantifying semantic mappings

Fuzziness measures of terms and hedges:

Formalization: Fuzziness measure of x = The SIZE of (H(x)):

2) fm(h -2 c+) + fm(h -1 c+) + fm(h +1 c+) + fm(h +2 c+) = fm( c+)

) (

)

( )

(

)

x fm

x h

fm c

III Fuzziness measure and quantifying semantic mappings

 Definition : fm : X → [0,1] called a fuzziness measure if 1) fm is a full measure, i.e

(i) if c,c+ are all base terms then fm ( c) + fm ( c+) = 1; (ii) if H - set of all hedges, { fm ( hc ) ; h  H } = fm (c); 2) If x is a crisp term , i.e H ( x ) = { x }, then fm ( x ) = 0;

3)  x, y  X ,  h  H ,

called fuzziness measure of hedge h

) (

)

( )

(

)

(

y fm

hy

fm x

fm

hx fm

