Toward Modeling and Reasoning with Words Based on Hedge Algebra

Toward Modeling and Reasoning with Words Based

on Hedge Algebra

Nguyen Van Han

_{, Phan Cong Vinh}

1_{FacultyofInformationTechnology,CollegeofScience,HueUniversity}

77NguyenHuestreet,PhuNhuanward,Huecity,Vietnam [email protected]

2_{HoChiMinhCityIndustryAndTradeCollege.}

20TangNhonPhustreet,PhuocLongBWard,District9,HoChiMinhcity.Vietnam 3_{FacultyofInformationTechnology,NguyenTatThanhUniversity}

300ANguyenTatThanhstreet,Ward13District4,HoChiMinhcity.Vietnam [email protected]

Abstract

In this paper, we introduce a method for reasoning with words based on hedge algebra using linguistic cognitive map. Our computing method consists of static and dynamic reasoning. In static reasoning, inferring on causal path of graph drives fuzzy linguistic value between any vertices and edges. With dynamic reasoning, system behaves as a dynamical system and convolution as automata property.

Received on 03 December 2018; accepted on 09 December 2018; published on 10 December 2018

Keywords: fuzzy logics, linguistic variable, hedge algebra, linguistic cognitive map, modeling with words, reasoning with words

Copyright © 2018 Nguyen Van Han and Phan Cong Vinh, licensed to EAI. This is an open access article distributed under the terms of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/), which permits unlimited use, distribution and reproduction in any medium so long as the original work is properly cited.

doi:10.4108/eai.6-2-2019.156535

A preliminary version of this work under the title “Modeling with Words Based on Hedge Algebra” has appeared in the proceedings of the 7th EAI International Conference, ICCASA2018and 4thEAI International Conference, ICTCC 2018, November 22–23,2018,VietTriCity,Vietnam[9].

1

Introduction

. In everyday life, people use natural language (NL) for analysing, reasoning, and finally,make their decisions. Computing with words (CWW) [5] is a mathematical solution of computational problems stated in an NL. CWW based on fuzzy setandfuzzylogic,introducedbyL.A.Zadehisan approximatemethodoninterval[0,1].Inlinguistic domain,linguistichedgesplayanimportantrolefor generatingsetoflinguisticvariables.Awellknown application of fuzzy logic (FL) is fuzzy cognitive map(FCM),introducedbyB.Kosko [1],combined

fuzzy logic with neural network. FCM has a lots

of applications in both modeling and reasoning fuzzy knowledge [3, 4] on interval [0,1] but not in linguistic values, However, many applications cannot model in numerical domain [5], for example, linguistic summarization problems. To solve this problem, in [9] , we used an abtract algebra, called hedge algebra (HA) as a tool for

modeling with words without computing on the graph. In the paper we study two method for reasoning with words on our model.

The remainder of paper is organized as follows. Section 2 reviews some main concepts of computing with words based on HA in subsection 2.1 and

describes several primary concepts for FCM in

next subsection 2.2. In section 3, we review our approach technique to modeling with words using

HA. Section 4 presents reasoning with words using

hedge algebra. Section 5 outlines conclusion and future work.

2

Preliminaries

This section presents basic concepts of HA and

FCM usedinthepaper.

2.1

Hedge

algebra

. In this section, we review some HA knowledges related to our research paper and give basic definitions.Firstdefinitionofa HA isspecifiedby 3-Tuple HA =(X,H,≤_{)in[6].In [7]toeasilysimulate} fuzzy knowledge, twoterms G and C areinserted to 3-Tuple so HA =(X,G,C,H,≤_{) where H ,} ∅_{, G=}{_c+_{, c}−}_{, C=}{_0,W,1}_{. Domain of X is} L ₌

Dom(X)={_δc|_c∈_{G, δ}∈_H∗_{(hedgestringoverH)}} _, {_L,≤} _{is a POSET (partial order set) and x=}

hnhn−₁...h₁c is said to be a canonical string of linguisticvariablex.

Example 1. Fuzzy subset X is Age, G={_c+ ₌

young; c−=old}_{, H =}{_{less;more;very}} _{so term-set} oflinguisticvariableAgeXis L(X)or L forshort:

L ={verylessyoung;lessyoung;young;moreyoung ;veryyoung;veryveryyoung...}

Fuzzinesspropertiesofelementsin HA,specified byfm(fuzzinessmeasure) [7]asfollows:

Definition2 .1.Amappingfm: L → _{[0,1]issaidto} bethefuzzinessmeasureof L if:

1. P

c∈{_c+_,c−_}fm(c) = 1, fm(0) =fm(w) =fm(1) = 0. 2. P

hi∈Hfm(hix) =fm(x), x=hnhn−1. . . h1c, the

canonical form.

3. fm(hnhn−₁. . . h₁c) =Qn_i=1fm(h_i)×µ(x).

2.2

Fuzzy

cognitive

map

Fuzzycognitivemap(FCM)isfeedbackdynamical

system for modeling fuzzy causal knowledge, introducedbyB.Kosko [1]. FCM isasetofnodes, whichpresentconceptsand aset ofdirectededges tolinknodes.Theedgesrepresentthecausallinks betweentheseconcepts.Mathematically,a FCM bis definedby.

Definition2 .2.A FCM isa4-Tuple:

FCM={C, E,C, f} (1)

In which:

1. C={_{C1, C2, . . . , Cn}} _{is the set of N concepts} forming the nodes of a graph.

2. E: (Ci, Cj)−→eij ∈ {−1, 0, 1} is a function

associatingeij with a pair of concepts (Ci, Cj),

so that eij = “weight of edge directed from

Ci to Cj. The connection matrix E(N×N) =

{_e

ij}N×_N

3. The map:C_:C

i −→Ci(t)∈[0,1], t∈N

4. With C(0) = [C1(0, C2(0), . . . , Cn(0)]∈[0,1]N

is the initial vector, recurring transformation functionf defined as:

Cj(t+ 1) =f( N

X

i=1

eijCi(t)) (2)

Example 2. Fig.1 shows a medical problem from expert domain of strokes and blood clotting involv-ing. Concepts C={blood stasis (stas), endothelial injury ( inju), hypercoagulation factors (HCP and HCF)} [2]. The conection matrix is:

E= (eij)4×4 =              

0 1 1 1 0 0 1 0 0 1 0 1 0 0 0 0

             

stas

HCP

inju

HCF

1

1 1

1 1

1

Fig. 1. AsimpleFCM

FCMshaveplayedavitalroleintheapplications

ofscientificareas,includingexpertsystem,robotics, medicine, education, information technology, pre-diction,etc [3,4].

3

Modeling

with

words

Our model, based on linguistic variables, is constructed from linguistic hedge of HA. The

followingaredefinitionsinourreseachpaper.

Definition3.1(Linguisticl attice).With L asinthe section2,set{∧_, ∨}_{arelogicaloperators,definedin} [6, 7],alinguisticlatticeL_isatuple:

Property 3.1.The following are some properties for L_:

1. L_{is a linguistic-bounded lattice.}

2. (L, ∨) and (L, ∧) are semigroups.

Proof. Without loss of generality, let

L={ρ c+|ρ∈(H+)∗∧c+ ∈G}. W is the neutral

element inHA, we have:

1. 0< w < c+ <1 and for

∀_ρ∈_(H+₎∗_{: ρ0< ρw < ρc}+_{< ρ1. Because}

ρ0 = 0;ρw=w;ρ1 = 1. This is equivalent to: 0< ρc+ <1 orL_{is bounded}

2. Let◦₌∧_or◦₌∨_{be operators inHAand} {_{p, q, r}} ∈_{X. Applying definitions of operators} ∧_and∨_{from [8]:}

p◦_(q◦_r)∧₍◦₌∨_{) =max}{_{p, max}{_{q, r}}}₌

max{_{p, q, r}}_{= (p}◦_q)◦_r∧₍◦₌∨₎

Definition 3.2. A linguistic cognitive map (LCM)

is a 4- Tuple:

LCM={C, E,C, f} (4)

In which: 1. C={_C

1, C2, . . . , Cn}is the set of N concepts

forming the nodes of a graph.

2. E: (Ci, Cj)−→eij ∈L;eij = “weight of edge

directed fromCi toCj. The connection matrix

E(N×_{N) =}{_e

ij}N×_N ∈LN×N

3. The map:C_:C

i−→Ci(t)∈L, t∈N

4. WithC(0) = [C1(0), C2(0), . . . , Cn(0)]∈LN is

the initial vector, recurring transformation functionf defined as:

Cj(t+ 1) =f( N

X

i=1

eijCi(t))∈L (5)

Example 3. Fig.2shows a simpleLCM. Let

HA=<X =truth;c+ =true;H={L,M,V}> (6)

be a_HAwith order asL <M <V (L for less,M for more andV for very are hedges ).

C={_c

1, c2, c3, c4}is the set of 4 concepts with corresponding values

C₌{_true,M_true,L_true,V_true}

cLtrue

3

CVtrue

4

cM₂ true

ctrue

1 L

true

V

true

Mtrue

M

true

Ltrue

Fig. 2. A simpleLCM

Square matrix:

M= (mij ∈L)4×4=

0 Ltrue 0 0

0 0 0 Mtrue

0 Mtrue 0 Vtrue

Ltrue 0 0 0

.

0

1 2

istheadjacencymatrixof LCM.Causalrelation betweenciandcjismij,forexampleifi=1,j=2

thencausalrelationbetweenc1andc2is:“ifc1 is

truethenc2is M trueis L true“orletP=”ifc1 is

truethenc2 is M true”beapropositionthen

truth(P₎₌ L _true

Definition3 .3.A LCM iscalledcompleteif betweenanytwonodesalwayhavingaconnected edge(withoutloopingedges).

4

Reasoning

with

words

Givestatespace:

C={C}n₌{C0_, C1_,...,Cn} WhereCi₌{_Ci_{, C}i_{,..., C}i

N},i=0,n.Inferenceon LCM consistsofstaticreasoning SR anddynamic reasoningDR.

Definition4 .1.AlinguisticreasoningRin LCM is asequenceoftransitionswherethesourceofeach isthedestinationofthepreviousone,whichcanbe writtenas:

The stateO₀is thesourceof the reasoningR, and

O_nitsdestination. The length of reasoning isn. Static properties allow deduction between concepts in a specific case ofCi∈_{C. In equation (7) by} definition (4.1), substitute objectsO_iwith conceptsCi (Ci Oi) for∀i∈1, nto driveSR Definition 4.2. Static reasoning on stateCi∈_{Cis a} process:

SR,C₁i −−−−→`1

LCM C i 2

`2

−−−−→

LCM . . . `n

−−−−→

LCM C i

n; `i∈L∀i= 1, n

(8)

We writeLCM`SRϕN(C1i, C2i, . . . , CNi) to mean

that causal relationϕ2(C_ji, C_ki)∈_{Lfor 1}≤_j ≤_k ≤_N on causal path:C₁i →_Ci

2→. . .→CNi by applying SRinterpretation rules.

Theorem 4.1.For everyC_ji∈ Ci_{; 1}≤_j ≤_N:

LCM`SRϕ2(C1i, Cji)∈L (9)

Proof. We use inductive mathematical method to prove theorem 4.1.QEDis short for the Latin

phrasequod erat demonstrandum, which means “which was to be proved”. Justification is placed in

.

Statement

Justification

1.LCM`SRϕ2(C1i, C2i)∈L By definition 4.2 2.LCM`SRϕk(C1i, Ci2, . . . , Cki)

Premise of inductive hypothesis 3.LCM`SRϕ2(C1i, Cki)∈L

Infer from 2

4.LCM`SRϕ2(C_ki, Ci_k+1)∈L By definition 4.2 5.QED

3,4,Fuzzy hypothetical Syllogism

Example 4. Considering the set of fuzzy concepts: 1. If a student studying possible hard or his

university is high-ranking, then he will be a good employee is more very true.

2. The university where Mary studies is very high-ranking is possibly very true.

3. Mary is studying very hard is more true.

Concepts are written in fuzzy propotions:

C1 Student studying Possible hard stud(x,PHard)

C2 University is high-ranking isUniv(x, Hi-rank)

C3 Good employee emp(x, good)

Causal relation between (Ci, Cj),1≤i,j ≤3 are

ϕ2(C1, C3) =M V true, ϕ2(C2, C3) =M V true Dynamic properties appear between states

Ci_{, for}∀_{i= 1, nin state spaceC(}Ci∈_{C). In} equation (7) by definition (4.1), substitute objects

O_i with conceptsCi(Ci←Oi) for∀i∈0, nto drive DR

Definition 4.3. Dynamic reasoning on state space

Cis a sequences:

DR,C0 −−`→1

C C 1 `2

−−→

C C

2_{. . .−−}`n_→ C C

n_{; `}

i ∈L∀i = 1, n

(10)

Example 5. Let us consider theF P _{set in example}

4, using inverse mapping for normalizing input propositions, sayI, to conceptsF P_:

For proposition: “Mary is studying very hard is more true” is formalized as:

Mary is studying very hard is more true ,(stud(Mary,VHard),M true) = (stud(Mary,V−VHard),V M true) = (stud(Mary, Hard),V M true) = (stud(Mary,PHard),P−V M true) = (stud(Mary,PHard),PV V true)

Letα=PV V truebe an input thenα∈_I. Proposition "The university where Mary

true,(isUniv(Mary,VHi-rank ),PVtrue)” (isUniv(Mary,VHi-rank ),PVtrue)

= (isUniv(Mary,V−VHi-rank ),V PVtrue) = (isUniv(Mary, Hi-rank ),V PVtrue) andβ=V PVtrue∈_I.

We assume that, concepts are initated to neutral value:C0₌{_C0

1 =W, C20 =W, C30=W}, next state C1_{is in example 6. State spaceCbehaves as an} automataA ,A(I,Q,F) in the following property.

Property 4.1.

C`A_(I,Q_,F_{) (11)} In which:

1. Iis a finite set of input symbols.

2. Qis the internal states of the system.

3. F is a state transition function.:

F :Q×_I→Q

(12)

Proof. SetQ⊆_{C,Iis nomalized concepts as in} example 5andF ={_f(PN

i=1eijCit)}as in equation

(5) thenA(I,Q,F) is an automata.

Example 6. Let input setI={_{α, β}}_,C0_{be in} example 5.f(.) =W

is the max function and matrix

M= (mij)3×₃=

0 0 M Vtrue

0 0 M Vtrue

0 0 0

.

• With input{_α}_,C1 1

W_{

α,C0W

mi1, i = 1,3} =W_{PV V

true,0}₌PV V true, drives C1₌{_C1

1, C12, C31}={PV V true, W, W} • With input{_β}_,C2

2 W_{

β,C1W

mi2, i = 1,3} =W_{V PV

true,PV V true}₌PV V _true,

drivesC2 ₌ {_C2

1, C22, C32}{PV V true, PV V true, W} If we writeF ={_σA}

σ∈_I:

αA = W W W

PV V true W W

!

,

βA = PV V true W W

PV V true PV V true W

!

Then automataA in example 6 becomes:

A ={C0_, C1_, C2}_, {_{α, β}}_,n_αA_{, β}Ao

0

5

Conclusion

and

future

work

Wehaveintroducedanewgraphicalmodelfor representingfuzzyknowledgeusinglinguistic variablesfrom HA.Ourmodel,called LCM, extendedfrom FCM ,isadynamicalsystemwith twoproperties:staticanddynamic.Static

propertiesallowforwardorwhat-ifinferencing betweenconceptsonlinguisticdomain.Especially, weindicateinverseproportionrelationship

betweenlengthofhedgesstringandanumberof partitionsinrepresentingfuzzyknowledge. Dynamicbehaviorsaretransformationstatesin statespaceCn₌{C}n₌{C_(0), C_(1),...,C_(n)}_,where C_(i)={_C

1(i), C2(i),..., CN(i)},i=0,n.Wealso

provethetheoremaboutthenumberofstatesin statespaceis|Cn|₌|_~|N×|_~|

,thisistheimportant theoremtodecidewhetherornotinstallable computerprograms.Ournextstudyisasfollow: Let

A={_~n_{: ~}n_=h

nhn−₁...h₁h₀withh_i ∈H, i=0,n} beastringofhedges.AssumeI =C_{(0),T =}C_(n) andT ⊂C×_A×C_{inorderareinitial,finaland} transitionstates.Wewillprovethat LCM actions arefuzzyautomataA_=<A, C_{, I,} T_{, T >.}

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