Weigh ed Fuzzy Spiking Neu al P Sys ems
Jun Wang, Peng Shi, Senio Membe , IEEE, Hong Peng, Ma io J. P´e ez-Jim´enez, and Tao Wang
Abs ac
Spiking neu al P sys ems (SN P sys ems) a e a new class o
compu ing models inspi ed by he neu ophysiological be-ha io o
biological spiking neu ons. In o de o make SN P sys- ems
capable o ep esen ing and p ocessing uzzy and unce ain
knowledge, we p opose a new class o spiking neu al P sys ems in
his pape called weigh ed uzzy spiking neu al P sys ems (WFSN
P sys ems). New elemen s, including uzzy u h alue, ce ain
ac o , weigh ed uzzy logic, ou pu weigh , h eshold, new i ing
ule, and wo ypes o neu ons, a e added o he o iginal de ini ion
o SN P sys ems. This allows WFSN P sys ems o adequa ely
cha ac e ize he ea u es o weigh ed uzzy p oduc ion ules in a
uzzy ule-based sys em. Fu he mo e, a weigh ed uzzy
backwa d easoning algo i hm, based on WFSN P sys ems, is
de eloped, which can ac-complish dynamic uzzy easoning o a
ule-based sys em mo e lexibly and in elligen ly. In addi ion, we
compa e he p oposed WFSN P sys ems wi h o he knowledge
ep esen a ion me hods, such as uzzy p oduc ion ule, concep ual
g aph, and Pe i ne s, o demons a e he ea u es and ad an ages
o he p oposed ech-niques.
Index Te ms
Spiking neu al P sys ems (SN P sys ems), weigh ed
uzzy p oduc ion ules, weigh ed uzzy easoning, weigh ed
uzzy spiking neu al P sys ems (WFSN P sys ems).
I. INTRODUCTION
NATURAL Compu ing is a no el ield o compu e sci-
ence esea ch ha uses compu a ional pa adigms
inspi ed
om a ious well-known na u al phenomena in physics, chem-
is y, and biology. The e a e se e al ields in Na u al Com-
pu ing ha a e now well es ablished, such as gene ic algo-
i hms [1], [2], a i icial neu al ne wo ks [3]–[6], pa icle
swa m
op imiza ion [7]–[11], DNA-based molecula compu ing [12].
Memb ane compu ing, which is pa o molecula compu ing,
was in oduced in [13] unde he assump ion ha he p ocesses
aking place in he compa men al s uc u e o a li ing cell can
be in e p e ed as compu a ions. Since hen, a la ge numbe o
a ian s ha e been conside ed, and he de ices o he models
a e gene ally called P sys ems [14].
A new class o dis ibu ed and pa allel compu ing de ices,
spiking neu al P sys ems (SN P sys ems), p esen ed in [15], was
inspi ed by he neu ophysiological beha io o neu ons sending
elec ical impulses (spikes) along axons o o he neu ons. An
SN P sys em can be iewed as a se o neu ons placed in he
nodes o a di ec ed g aph whose a cs ep esen he synap ic con-
nec ions among he neu ons, and each neu on con ains a numbe
o copies o single objec ype as well as a lo o i ing/spiking
and o ge ing ules. The ules in each neu on a e used in a
sequen ial manne , bu neu ons unc ion wi h each o he in pa -
allel. Mo e ecen ly, a la ge numbe o a ian s o SN P sys ems
ha e been de eloped (see, [16]–[23] and he e e ences he ein).
In addi ion o dis ibu ed and pa allel compu ing abili ies, SN
P sys ems inhe en ly ea u e: 1) high unde s andabili y (due o
hei di ec ed g aph s uc u e); 2) dynamic beha io (i seems o
be sui able o model dynamic beha io s o a sys em on he basis
o neu on’s i ing/spiking mechanisms); 3) synch oniza ion (i
seems o be sui able o desc ibe concu en e en s o ac i i ies);
4) nonlinea i y (i is capable o dealing wi h nonlinea p oblem);
and 5) nonde e minis ic. No doub , hese ea u es will be a ac-
i e o a lo o eal-wo ld p oblems, such as p ocess con ol,
expe sys ems, aul diagnosing, in es men ad ising sys ems,
and e en he new in elligen wi eless senso ne wo ks.
Howe e , many success ul applica ions ha e de e mined ha
he e is a g ea deal o uzzy and unce ain in o ma ion in he
a o emen ioned eal-wo ld a eas and ou compu ing models a e
o en equi ed o be capable o dealing wi h uzzy and unce ain
knowledge. I is well known ha uzzy knowledge e ie ed by
human expe s o ex ac ed by uzzy neu al ne wo ks (FNNs) is
usually ep esen ed by uzzy p oduc ion ules [24]–[27]. Ne -
e heless, uzzy p oduc ion ules a e no s aigh o wa d and
hei uzzy easoning is usually a complica ed p ocess. Fo his
eason, some knowledge ep esen a ion me hods we e de el-
oped, such as concep ual g aph [28], seman ic ne wo ks [29],
and uzzy Pe i ne s [30]–[33]. Meanwhile, weigh ed uzzy p o-
duc ion ules and weigh ed uzzy logics we e de eloped in
o de o p ocess uzziness and unce ain y in he knowledge
base [34]–[39].
As s a ed p e iously, some signi ican ea u es possessed by
SN P sys ems a e a ac i e o eal-wo ld applica ions. Un o u-
na ely, exis ing SN P sys ems and hei a ian s lack he abili y
o p ocess uzzy and unce ain knowledge so a . The main mo-
i a ion behind ou s udy is o build a b idge be ween SN P
sys ems and a ious eal-wo ld p oblems such ha he SN P
sys ems can se e as a new model in eal-wo ld p oblems. Fo
his eason, we will ex end SN P sys ems so ha hey a e capable
o dealing wi h uzzy and unce ain in o ma ion and ep esen -
ing weigh ed uzzy p oduc ion ules. In his pape , we p opose
weigh ed uzzy spiking neu al P sys ems (WFSN P sys ems)
by inco po a ing some new elemen s in o he o iginal de ini-
ion o SN P sys ems, including a new ype o neu on, uzzy
u h alue, ce ain ac o , weigh ed uzzy logic, ou pu weigh ,
h eshold, and new i ing/spiking ules. WFSN P sys ems a e
especially sui able om exp essing weigh ed uzzy p oduc ion
ules in g aphical o m. In addi ion, a uzzy backwa d eason-
ing algo i hm, based on he WFSN P sys ems, is de eloped. The
main ad an ages o e ed by he p oposed WFSN P sys ems can
be summa ized as ollows.
1) Because o he g aphical na u e o WFSN P sys ems, he
s uc u e o weigh ed uzzy p oduc ion ules in a uzzy
knowledge base can be easily modeled and isualized,
and he model is ela i ely simple and legible.
2) Thedynamic i ingmechanismo neu onsinWFSNPsys-
ems a e capable o ca ying ou dynamic uzzy easoning
p ocess mo e in elligen ly.
3) Based on he pa allel compu ing mechanism o WFSN P
sys ems, he p oposed easoning algo i hm is an e icien
easoning algo i hm wi h pa allel easoning abili y.
The es o his pape is o ganized as ollows. In Sec ion II,
we p o ide he o iginal de ini ion o SN P sys ems, and p opose
WFSN P sys ems and simpli ied e sions. In Sec ion III, we
pe o m weigh ed uzzy knowledge ep esen a ion based on he
WFSN P sys ems. A uzzy backwa d easoning algo i hm based
on WFSN P sys ems o a ule-based sys em is p esen ed in
Sec ion IV. In Sec ion V, we compa e WFSN P sys ems wi h
o he knowledge ep esen a ion me hods o show he ad an ages
o ou esul s. Finally, Sec ion VI gi es he conclusions.
II. WEIGHTED FUZZY SPIKING NEURAL PSYSTEMS
A. Spiking Neu al P Sys ems
In his sec ion, we b ie ly e iew SN P sys ems in s anda d
o m and in a compu ing e sion (i.e., able o ake an inpu
and p o ide an ou pu ). (A mo e de ailed desc ip ion o SN P
sys ems can be ound in [16]–[22]).
De ini ion 1: A compu ing SN P sys em o deg ee m≥1isa
cons uc o he o m
Π=(O,σ1,...,σ
m,syn,in,ou )
whe e
1) O={a}single on alphabe ( he objec ais called spike);
2) σ1,...,σ
mneu ons, o he o m σi=(ni, i)wi h 1 ≤
i≤m, whe e:
a) ni≥0 ini ial numbe o spikes con ained in neu on
σi;
b) i ini e se o ules o he ollowing wo o ms:
1) E/ac→a;d, whe e Eis a egula exp ession
o e a, and c,d≥0 a e na u al numbe s;
2) as→λ, whe e s≥1 is a na u al numbe , wi h
es ic ion ha o each ule E/ac→a;do
ype (i) om i,weha eas∈ L(E);
3) syn ⊆{σ1,σ
2,...,σ
m}×{σ1,σ
2,...,σ
m}wi h i=j
o all (σi,σ
j)∈syn, 1 ≤i, j ≤m(synapses be ween
neu ons);
4) in,ou ∈{σ1,σ
2,...,σ
m}inpu and ou pu neu ons,
espec i ely.
In he a o emen ioned de ini ion, he ule o ype (1) is called
he i ing/spiking ule, and ha o ype (2) is called he
o ge ing ule. The i ing mechanism o neu ons in SN P
sys ems can be desc ibed as ollows. I a neu on σicon ains k
spikes, ak∈L(E)and k≥c, he i ing/spiking ule E/ac→
a;d∈ iin neu on σiis enabled and can be applied. This means
ha cspikes a e consumed, k−cspikes emain in he neu on,
he neu on i es, and hen i p oduces a spike a e d ime uni s.
I d=0, he spike is emi ed immedia ely. In he case d≥1, i
he uleisuseda s ep , he neu on is “closed” and “blocked”
a s eps , +1,..., +d−1, and i canno ecei e new spikes
om o he neu ons. A s ep ( +d), he neu on emi s a spike
and becomes again open; hence, o he neu ons can ecei e he
spike. The spike emi ed by neu on σiis eplica ed and i goes o
all neu ons σjsuch ha (σi,σ
j)∈syn (each such neu on σjo
hose ecei es a spike). A o ge ing ule ac→λis applicable o
a neu on whe he he neu on con ains exac ly cspikes, and hen,
all cspikes a e emo ed. No e ha i all ules o a sys em ha e
d=0, i.e., no delay is in ol ed, he pa ame e dis omi ed.
SN P sys ems a e synch onized because a global clock is
assumed, ma king he ime o he whole sys em. Besides, SN P
sys emsa enonde e minis ic because wo ulesE1/ac1→a;d1
and E2/ac2→a;d2can ha e L(E1)∩L(E2)=∅. The e o e,
i is possible ha wo o mo e ules o he sys em can be enabled
in a neu on. In his case, one o hem is nonde e minis ically
chosen o be used. Mo eo e , in each ime uni , i a neu on can
use a ule, he ule mus be used. Each neu on deals wi h i s
spikes in a sequen ial manne , only using one ule in each ime
uni , bu he ules a e used in pa allel o all neu ons o he
sys em.
An ins an aneous desc ip ion o a con igu a ion a any ins an
o an SN P sys em is desc ibed by bo h he numbe o spikes in
each neu on and he s a e o he neu on, o mo e p ecisely, by
he numbe o s eps o coun down un il i becomes open ( his
numbe is ze o i he neu on is al eady open). The ini ial con-
igu a ion is desc ibed by he numbe o spikes ini ially placed
in each neu on, n1,n
2,...,n
m, wi h all neu ons being open. A
con igu a ion is a hal ing con igu a ion i all neu ons a e open
and no ule o he sys em is applicable o i . Using he ules
desc ibed p e iously, one can de ine ansi ions among con ig-
u a ions. We say ha con igu a ion C1yields con igu a ion C2
in one ansi ion s ep, which is deno ed by C1⇒ΠC2,i we
can pass om C1 o C2by applying he ules om he sys em
ollowing he p e ious ema ks.
Acompu a ion o Πis a ( ini e o in ini e) sequence o con-
igu a ions such ha :
1) he i s e m o he sequence is he ini ial con igu a ion o
he sys em;
2) each nonini ial con igu a ion o he sequence is ob ained
om he p e ious con igu a ion by a ansi ion s ep;
3) i he sequence is ini e (called hal ing compu a ion), hen
he las e m o he sequence is a hal ing con igu a ion.
Wi h any compu a ion (hal ing o no ), we can associa e a
spike ain, which is a sequence o symbols 0, and 1, desc ibing
he beha io o he ou pu neu on. I he ou pu neu on spikes,
hen we w i e 1; o he wise, we w i e 0. In addi ion, we can
also associa e o he o ms o compu a ion esul s acco ding o
di e en compu ing pu poses, such as he dis ance be ween wo
consecu i e s eps when he e a e spikes ha exi he sys em.
B. Weigh ed Fuzzy Spiking Neu al P Sys ems
The in oduc ion o uzzy elemen s in P sys ems is an in-
e es ing and open issue. This pape will pay a en ion o
his issue bu be limi ed o he discussion o SN P sys ems
o p ocessing uzzy and unce ain knowledge. Thus, we will
ex end he de ini ion o SN P sys ems and p opose a class
o ex ended SN P sys em models, called WFSN P sys ems.
The mo i a ion o his is o model weigh ed uzzy p oduc-
ion ules in a uzzy knowledge base and pe o m weigh ed
uzzy easoning by using WFSN P sys ems in a mo e in elligen
manne .
De ini ion 2: A compu ing WFSN P sys em o deg ee m≥1
is a cons uc o he o m
Π=(O,Np,N
,syn,IN,OUT)
whe e
1) O={a}single on alphabe ( he objec ais called spike);
2) Np={σp1,σ
p2,...,σ
pm}p oposi ion neu on se , whe e
σpi is i s i h p oposi ion neu on associa ed wi h a uzzy
p oposi ion in a uzzy knowledge base, 1 ≤i≤m. Each
p oposi ion neu on σpi has he o m σpi =(αi,ωi,λi, i),
whe e
a) αi∈[0,1]po en ial alue o pulse con ained in
p oposi ion neu on σpi.αiis used o exp ess uzzy
u h alue o a p oposi ion associa ed wi h p opo-
si ion neu on σpi.
b) ωi=(ωi1,ω
i2,...,ω
isi)ou pu weigh ec o o
he neu on σpi, whe e componen ωij ∈[0,1] is
he weigh on j h ou pu synapse (a c) o he neu-
on, 1 ≤j≤si, and siis he numbe o all ou pu
synapses (a c) o he neu on.
c) i ini e se o i ing/spiking ules o he o m
E/aα→aα;d, whe e α∈[0,1], and d≥0isa
na u al numbe . E={α≥λi}is called he i ing
condi ion, i.e., i α≥λi, hen he i ing ule will
be enabled, whe e λi∈[0,1)is called he i ing
h eshold.
3) N ={σ 1,σ
2,...,σ
n} uleneu on se , whe e σ i is i s
i h ule neu on associa ed wi h a weigh ed uzzy p oduc-
ion ule in a uzzy knowledge base, 1 ≤i≤n. Each ule
neu on σ i has he o m σ i =(αi,γ
i,νi,τ
i, i), whe e
a) αi∈[0,1]po en ial alue o pulse con ained in ule
neu on σ i.
b) γi∈[0,1]ce ain ac o . I ep esen s he s eng h o
belie o aweigh ed uzzyp oduc ion uleassocia ed
wi h ule neu on σ i.
c) νi=(νi1,ν
i2,...,ν
i i)ou pu weigh ec o o he
neu on σ i, whe e componen νij ∈[0,1]is he
weigh on j h ou pu synapse (a c) o he neu-
on, 1 ≤j≤ i, and iis he numbe o all ou pu
synapses (a c) o he neu on.
d) i ini e se o i ing/spiking ules o he o m
E/aα→aβ;d, whe e α∈[0,1],β∈[0,1], and
d≥0 is a na u al numbe . E={α≥τi}is called
he i ing condi ion, i.e., i α≥τi, hen he i ing
ule will be enabled, whe e τi∈[0,1)is called he
i ing h eshold.
4) syn ⊆(Np×N )(N ×Np)synapses be ween bo h
p oposi ion neu ons and ule neu ons. No e ha he e a e
no synapse connec ions be ween any wo p oposi ion neu-
ons o be ween any wo ule neu ons;
5) IN,OUT ⊆Npinpu neu on se and ou pu neu on se ,
espec i ely.
In he ollowing, we illus a e how WFSN P sys ems a e
ex ended om he o iginal de ini ion o SN P sys ems. Fi s ,
WFSN P sys ems consis o wo ypes o neu ons: p oposi ion
neu ons and ule neu ons. The in ui i e pu pose o in oducing
he wo ypes o neu ons is o exp ess uzzy p oposi ions and
weigh ed uzzy p oduc ion ules in a uzzy knowledge base.
Second, con en o he neu on is now deno ed by a uzzy u h
alue ins ead o he numbe o spikes as in SN P sys ems. I can
be in e p e ed as he (po en ial) alue o spike om he iew-
poin o biological neu on. Fo a p oposi ion neu on, i s con en
is used o exp ess he uzzy u h alue o a uzzy p oposi ion
associa ed wi h i . When a neu on i es and emi s a spike, he
(po en ial) alue o he spike is ansmi ed in o all successi e
neu onsconnec ed wi h he neu on. Thi d, eachp oposi ion neu-
on is assigned an ou pu weigh ec o ω =(ω1,ω
2,...,ω
s).
This indica es ha he j h ou pu synapse o he p oposi ion
neu on has he ou pu weigh ωj. The e o e, when a p oposi ion
neu on i es and emi s a spike wi h alue α, i s j h successi e
neu on will ecei e a spike wi h alue α⊗ωj om i s ou pu
synapse, whe e “⊗” is he mul iplica ion ope a o o uzzy u h
alues. Simila ly, each ule neu on is also assigned an ou pu
weigh ec o ν =(ν1,ν
2,...,ν
). When a ule neu on i es
and emi s a spike wi h alue α, i s j h successi e neu on will
ecei e a spike wi h alue (ανj)⊗γ, whe e “” is he di i-
sion ope a o o uzzy u h alues. Fou h, because p oposi ion
neu ons and ule neu ons in WFSN P sys ems a e used o cha -
ac e ize uzzy p oposi ions and weigh ed uzzy p oduc ion ules
in a uzzy knowledge base, espec i ely, bo h inpu neu on se
and ou pu neu on se only consis o p oposi ion neu ons, while
ule neu ons a e in e connec o o p oposi ion neu ons. Mo e-
o e , he e a e no di ec connec ions be ween wo p oposi ion
neu ons o be ween wo ule neu ons. Fi h, WFSN P sys ems
use he new i ing condi ion E={α≥λi}o E={α≥τi}
a he han he o iginal egula exp ession in SN P sys ems, and
his con ols whe he he co esponding neu on i es o no . I
a p oposi ion neu on con ains a leas a spike and i s alue o
spike is wi h α≥λi, hen i i es. Likewise, i a ule neu on
con ains a leas a spike and i s alue o spike is wi h α≥τi,
hen i i es. Finally, when a neu on ecei es spikes om i s
se e al p edecesso neu ons, (po en ial) alues o he ecei ed
Fig. 1. P oposi ion neu on in S-WFSN P sys ems.
Fig. 2. Rule neu on in S-WFSN P sys ems.
spikes will be calcula ed by using some logical ope a o s unlike
he neu on in SN P sys ems ha simply accumula e he num-
be o spikes ecei ed by i . The p oposi ion neu on calcula es
(po en ial) alues o spikes ecei ed by i om i s p edecesso
neu ons h ough logical “OR” ope a o “∨,” whe eas ule neu-
on calcula es po en ial alues o spikes ecei ed by i h ough
addi ion ope a o “⊕” (see Figs. 1 and 2).
In addi ion o se e al aspec s desc ibed p e iously, o he o ig-
inal mechanisms in SN P sys ems a e e ained in WFSN P
sys ems, o ins ance, ime delay d, synch oniza ion, nonde e -
minacy, and so o h.
As s a ed p e iously, he pu pose o p oposing he WFSN
P sys ems is o model weigh ed uzzy p oduc ion ules in a
knowledge base and de elop a mo e in elligen weigh ed uzzy
easoning algo i hm. Some elemen s in WFSN P sys ems how-
e e , a e edundan , such as ime delay d, and i ing h esholds
λiand τi. Hence, we simpli y he WFSN P sys ems by emo -
ing he edundan elemen s and deno e he simpli ied e sion o
WFSN P sys ems as S-WFSN P sys ems.
Compa ed wi h WFSN P sys ems, S-WFSN P sys ems ha e
he ollowing di e ences. Fi s , ime delay dis omi ed; hence,
all neu ons a e always open in S-WFSN P sys ems. Second,
i ing h esholds λiand τiin WFSN P sys ems a e emo ed.
The e o e, i a neu on con ains a leas a spike and i s alue
o spike αi>0, hen i i es. Thi d, any neu on (p oposi ion
neu on o ule neu on), con ains only a i ing ule. Finally, each
ule neu on has only an ou pu weigh ac o νi, i.e., all i s ou pu
synapses a e assigned he same weigh .
We desc ibe he ope a ing p inciple o S-WFSN P sys ems
as ollows. Ini ially, he sys em p o ides a spike o each in-
pu neu on in IN (o each inpu neu on in IN ecei es a spike
om he en i onmen as i s inpu ), whe e he alue o he spike
equals he uzzy u h alue o he co esponding p oposi ion.
When he sys em hal s, he con en s con ained in ou pu neu-
ons (o esul s expo ed by ou pu neu ons) a e ega ded as i s
compu ing esul s. In S-WFSN P sys ems, each neu on con ains
only a i ing/spiking ule and i s i ing p inciple is explained as
ollows. Fi s , i a p oposi ion neu on has kp edecesso ule
neu ons and i ecei es kspikes om hem, he (po en ial) alue
o he ecei ed kspikes is calcula ed as i s con en α h ough
Fig. 3. Example o S-WFSN P sys ems: Π0.
logical “OR” ope a o “∨.” When α>0, he neu on i es and
i s i ing/spiking ule E/aα→aαcan be applied. Applying he
i ing/spiking ule E/aα→aαmeans ha he spike con ained
in he neu on is consumed, and hen, i p oduces a spike wi h
alue α, which will be weigh ed by he co esponding weigh
ac o . In his pape , we deno e a p oposi ion neu on by a ci -
cle, as shown in Fig. 1. He e, α=x1∨x2∨...∨xk, and i s
ou pu s a e α⊗ωi(i=1,2,...,s), espec i ely. Second, i a
ule neu on has kp edecesso p oposi ion neu ons, hen i i es
and i s i ing/spiking ule E/aα→aβcan be applied when i
ecei es kspikes om i s all p edecesso p oposi ion neu ons.
The alue o he ecei ed kspikes is calcula ed as i s con en
α h ough addi ion ope a o “⊕.” Applying he i ing/spiking
ule E/aα→aβmeans ha he spike con ained in he neu on
is consumed, and hen, i p oduces a spike wi h alue βwhe e
β=(αν)⊗γ. In his pape , we deno e a ule neu on by a
ec angle, as shown in Fig. 2. He e, α=x1⊕x2⊕...⊕xk,
and all i s ou pu s a e (αν)⊗γ.
Example 1: Fig. 3 shows an example o S-WFSN P sys ems,
which can be o mally desc ibed as ollows. Π0=({a},
{σp1,σ
p2,σ
p3,σ
p4,σ
p5},{σ 1,σ
2,σ
3,σ
4},syn,IN,OUT),
whe e
1) σpj =(αj,ω
j,
j)(j=1,...,5)p oposi ion neu ons.
The weigh s o p oposi ion neu ons σp1,σ
p3, and σp4
a e ω1=1.0,ω
3=0.8, and ω4=0.7, espec i ely, while
p oposi ion neu on σp2has wo weigh s ω21 =1.0 and
ω22 =1.0;
2) σ i =(αi,γ
i,ν
i,
i)(j=1,...,4) ule neu ons. The ce -
ain ac o s o ule neu ons σ 1,σ
2,σ
3, and σ 4a e
γ1=0.85,γ
2=0.90,γ
3=0.95, and γ4=0.90, espec-
i ely. The weigh s o ule neu ons σ 1,σ
2,σ
3, and σ 4
a e ν1=ω1=1.0,ν
2=ω22 =1.0,ν
3=ω21 =1.0, and
ν4=ω3⊕ω4=1.5, espec i ely;
3) syn={(σp1,σ
1),(σp2,σ
2),(σp2,σ
3),(σp3,σ
4),(σp4,
σ 4),(σ 1,σ
p2),(σ 2,σ
p3),(σ 3,σ
p4),(σ 4,σ
p54)};
4) IN ={σp1}, OUT ={σp5}.
In addi ion o modeling weigh ed uzzy p oduc ion ules by
using WFSN P sys ems, we will de elop a uzzy easoning
algo i hm based on WFSN P sys ems in his pape . In o de o
con enien ly desc ibe ou weigh ed uzzy easoning algo i hm,
we i s , de ine se e al concep s ( e minologies) he e. Le σ s
TABLE I
IMMEDIATE RULE-INCIDENCE TABLE OF ALL PROPOSITION
NEURONS IN EXAMPLE 1
be a ule neu on and σpi,σpj, and σpk be h ee p oposi ion
neu ons.
De ini ion 3: Immedia ely backwa d ule incidence.I
(σpi,σ
s)∈syn and (σ s,σ
pk)∈syn, i.e., σ s is he in e con-
nec o be ween σpi and σpk, hen σpi is called an immedia ely
backwa d ule incidence o σpk.
In Example 1, σp1is an immedia ely backwa d ule incidence
o σp2, and σp2is an immedia ely backwa d ule incidence o
σp3and σp4, while σp3and σp4a e immedia ely backwa d ule
incidences o σp4.
De ini ion 4: Backwa d ule incidence.I σpi is an immedi-
a ely backwa d ule incidence o σpj, and σpj is an immedia ely
backwa d ule incidence o σpk, hen σpi is called a backwa d
ule incidence o σpk.
F om Example 1, we can obse e ha σp1is a backwa d
ule incidence o σp3and σp4, espec i ely. Mo eo e , σp2is a
backwa d ule incidence o σp5.
De ini ion 5: Immedia ely backwa d ule incidence se .Fo
a p oposi ion neu on σpk, i s immedia ely backwa d ule inci-
dence se is de ined as ollows:
IBRIS(σpk)={σpi ∈Np|σpi is an immedia ely backwa d
ule incidence o σpk}.
De ini ion 6: Immedia ely backwa d ule incidence able.Fo
WFSNPsys ems,i simmedia elybackwa d ule incidence able
is de ined as ollows:
IRIT={(σpk,IBRIS(σpk),σ
i)| o ∀σpk ∈Npand ∀σpj ∈
IBRIS(σpk),∃σ i ∈N such ha (σpj,σ
i)∈syn and (σ i,
σpk)∈syn}.
Fo Example 1, Table I gi es he immedia ely backwa d
ule-incidence able o Π0. F om Table I, we can see ha
IBRIS(σp2)={σp1}, IBRIS(σp3)={σp2}, IBRIS(σp4)=
{σp2}, while IBRIS(σp5)={σp3,σ
p4}, whe e σp3and σp4a e
adjacen p oposi ion neu ons wi h espec o ule neu on σ 4.
III. WEIGHTED FUZZY KNOWLEDGE REPRESENTATION
A. Weigh ed Fuzzy P oduc ion Rules
The weigh ed uzzy p oduc ion ules ha a e discussed he e
a e simila o con en ional uzzy p oduc ion ules. Howe e , a
weigh ac o (o ec o ) is assigned o each p oposi ion in he
an eceden pa in a uzzy p oduc ion ule, and a ce ain y ac o
is also assigned o he ule. Weigh ac o o a p oposi ion indi-
ca es he deg ee o i s impo ance con ibu ing o he consequen
when compa ing wi h o he p oposi ion in he an eceden pa .
Ob iously, when he e is only one p oposi ion in he an eceden
o a uzzy p oduc ion ule, weigh is meaningless o he ule.
Gene ally, weigh ed uzzy p oduc ion ules can be ca ego-
ized in o ou ypes as ollows:
Type 1: Ri:IFpjTHEN pk(CF =γi), ω.
This ype ule is a simple uzzy p oduc ion ule. In he ule Ri,
pjand pka e p oposi ions, γiis ce ain y ac o o he ule,
and ωis he weigh o p oposi ion pj. Since pjis only one
p oposi ion in he an eceden pa o he ule Ri, i s weigh ωis
meaningless o he ule. The e o e, we can se ω=1.
Type 2: Ri:IFp1AND p2AND ... AND pk−1THEN pk
(CF =γi), ω1,ω
2,...,ω
k−1
whe e ω1,ω
2,...,ω
k−1a e he weigh s o p oposi ions
p1,p
2,...,p
k−1in he an eceden pa o he ule Ri, espec-
i ely. This is a composi e conjunc i e uzzy p oduc ion ule.
Type 3: Ri:IFp1THEN p2AND p3AND ... AND pk
(CF =γi), ω
whe e ωis he weigh o p oposi ion p1in he an eceden pa
o he ule Ri. Simila ly, we can se ω=1 since weigh ωin he
ule is meaningless.
Type 4: Ri:IFp1OR p2OR ... OR pk−1THEN pk
(CF =γi), ω1,ω
2,...,ω
k−1.
This is a composi e disjunc i e uzzy p oduc ion ule. In
he ule Ri,ω1,ω
2,...,ω
k−1a e he weigh s o p oposi-
ions p1,p
2,...,p
k−1in he an eceden pa o he ule Ri,
espec i ely.
B. Mapping Weigh ed Fuzzy P oduc ion Rules In o Weigh ed
Fuzzy Spiking Neu al P Sys ems
In o de o model weigh ed uzzy p oduc ion ules in a uzzy
knowledge base by using S-WFSN P sys ems, we ha e o
map he a o emen ioned weigh ed uzzy p oduc ion ules in o
S-WFSN P sys ems. The basic p inciple is o map each uzzy
p oposi ion in uzzy knowledge base in o one p oposi ion neu-
ono S-WFSNPsys emsand o map each uzzy p oduc ion ule
in o one ule neu on o se e al ule neu ons. Thus, he weigh ed
uzzy p oduc ion ules o ou ypes desc ibed p e iously and
hei uzzy easoning p ocesses can be modeled as ollows.
Fo a ule o Type 1, assume ha he uzzy u h alue o
p oposi ions pjis αjand ce ain y ac o o he ule is γi. Hence,
he ule o Type 1 can be modeled by he ollowing S-WFSN P
sys em Π1, as shown in Fig. 4(a):
Π1=({a},{σpj,σ
pk},{σ i},syn,IN,OUT), whe e
1) σpj andσpk wop oposi ionneu onsassocia edwi h uzzy
p oposi ions pjand pk, espec i ely. σpj =(αj,ω
j,
j)
and σpk =(αk,ω
k,
k). Since he an eceden pa o he
ule has only one p oposi ion, se ωj=1;
2) σ i ule neu on associa ed wi h he uzzy p oduc ion ule
Ri.σ i =(αi,γ
i,ν
i,
i), whe e νi=1;
3) syn ={(σpj,σ
i ),(σ i,σ
pk )},IN ={σpj},OUT =
{σpk}.
Fu he mo e, Fig. 4(a) shows he dynamic uzzy eason-
ing p ocess modeled by Π1. The uzzy easoning p ocess is

(b)
(c)
(d)
(a)
……
…
Fig. 4. Fou weigh ed uzzy ule p esen a ions wi h S-WFSN P sys ems and
hei ule easoning. (a) Type 1. (b) Type 2. (c) Type 3. (d) Type 4.
au oma ically ca ied ou ia h ee ime uni s. Ini ially, a spike
wi h alue αjis assigned in o p oposi ion neu on σpj. Thus,
σpj i es and emi s a spike wi h alue αj. Nex , ule neu on σ i
ecei es he spike and i es, and hen, i sends a spike wi h alue
αj⊗γi o p oposi ion neu on σpk. Finally, p oposi ion neu on
σpk ecei es he spike and i s alue o spike αj⊗γiis ega ded
as he esul compu ed by he S-WFSN P sys em Π1,asshown
in Fig. 4(a).
Fo a ule o Type 2, assume ha he uzzy u h alues o
p oposi ions p1,p2,...,pk−1a e α1,α2,...,αk−1, espec i ely,
and ce ain y ac o o he ule is γi. Hence, he ule o Type
2 can be modeled by he ollowing S-WFSN P sys em Π2,as
shown in Fig. 4(b):
Π2=({a},{σp1,...,σ
pk−1,σ
pk},{σ i},syn,IN,OUT),
whe e
1) σp1,σp2,...,σpk−1, and σpk p oposi ion neu ons as-
socia ed wi h uzzy p oposi ions p1,p2,...,pk−1, and
pk, espec i ely. σpj =(αj,ω
j,
j),j=1,2,...,k−
1,k.ω1,ω
2,...,ω
k−1a e weigh s o he p oposi ions
p1,p
2,...,p
k−1in he an eceden pa o he ule, espec-
i ely;
2) σ i ule neu on associa ed wi h he uzzy p oduc ion ule
Ri.σ i =(α i,γ
i,ν
i,
i), whe e νi=ω1⊕ω2⊕...⊕
ωk−1;
3) syn ={(σp1,σ
i),...,(σpk−1,σ
i),(σ i,σ
pk)},
IN ={σp1,σ
p2,...,σ
pk−1}, OUT ={σpk}.
Fig. 4(b) shows he dynamic uzzy easoning p ocess mod-
eled by Π2. The uzzy easoning p ocess is au oma ically pe -
o med as ollows. Ini ially, a spike is p o ided o each p opo-
si ion neu on σpj in IN and hei alues a e α1,α
2,...,α
k−1,
espec i ely. These p oposi ion neu ons concu en ly i e, and
hen, each o hem emi s a spike wi h alue αj⊗ωj.Nex ,
ule neu on σ i ecei es spikes om hese p oposi ion neu ons
and alues o he spikes a e compu ed by addi ion ope a o
“⊕” as i s con en , i.e., α i =(α1⊗ω1)⊕(α2⊗ω2)⊕...⊕
(αk−1⊗ωk−1). Thus, he ule neu on i es, and hen, i
sends a spike wi h alue [α i νi]⊗γi o p oposi ion neu-
on σpk. Finally, p oposi ion neu on σpk will ecei e he spike
as i s con en , as shown in Fig. 4(b). The e o e, he esul
compu ed by Π2is αk={[(α1⊗ω1)⊕(α2⊗ω2)⊕...⊕
(αk−1⊗ωk−1)] (ω1⊕ω2⊕...⊕ωk−1)}⊗γi.
Fo a ule o Type 3, assume ha he uzzy u h alue o
p oposi ions p1is α1and ce ain y ac o o he ule is γi. Hence,
he ule o Type 3 can be modeled by he ollowing S-WFSN P
sys em Π3, as shown in Fig. 4(c):
Π3=({a},{σp1,...,σ
pk−1,σ
pk},{σ i},syn,IN,OUT),
whe e
1) σp1,σp2,...,σpk−1, and σpk p oposi ion neu ons associ-
a ed wi h uzzy p oposi ions p1,p2,...,pk−1, and pk, e-
spec i ely. σp1=(α1,ω
1,
1). Since p1is only one p opo-
si ion in he an eceden pa o he ule, se he weigh
ω1=1;
2) σ i ule neu on associa ed wi h he uzzy p oduc ion ule
Ri.σ i =(α i,γ
i,ν
i,
i), whe e νi=1;
3) syn ={(σp1,σ
i),(σ i,σ
p2),(σ i,σ
p3),...,(σ i,σ
pk)},
IN ={σp1}, OUT ={σp2,σ
p3,...,σ
pk}.
Fig. 4(c) shows dynamic uzzy easoning p ocess modeled
by Π3. The uzzy easoning p ocess is au oma ically pe o med
as ollows. Ini ially, a spike is p o ided o p oposi ion neu on
σp1and i s alues is α1. Thus, he p oposi ion neu on i es,
and hen, i emi s a spike wi h alue α1. Nex , ule neu on
σ i ecei es he spikes and i es, and hen, i sends a spike
wi h alue α1⊗γi o i s all successi e p oposi ion neu ons.
Finally, p oposi ion neu ons σp2,σ
p3,...,σ
pk will ecei e he
spike as hei con en s, as shown in Fig. 4(c). The e o e, he
esul s compu ed by Π3a e α2=α1⊗γi,α3=α1⊗γi,...,
αk=α1⊗γi.
Fo a ule o Type 4, assume ha uzzy u h alues o p opo-
si ions p1,p2,...,pk−1a e α1,α2,...,αk−1, espec i ely, and
he ce ain y ac o o he ule is γi. Hence, he ule o Type
4 can be modeled by he ollowing S-WFSN P sys em Π4,as
shown in Fig. 4(d):
Π4=({a},{σp1,...,σ
pk−1,σ
pk},{σ i},syn,IN,OUT),
whe e
1) σp1,σp2,...,σpk−1, and σpk p oposi ion neu ons asso-
cia ed wi h uzzy p oposi ions p1,p2,...,pk−1, and pk
espec i ely. σpj =(αj,ω
j,
j),j=1,2,...,k−1,k.
ω1,ω
2,...,ω
k−1a e he weigh s o he p oposi ions
p1,p
2,...,p
k−1in he an eceden pa o he ule, espec-
i ely;
2) σ 1,σ
2,...,σ
k−1 ule neu ons associa ed wi h he
uzzy p oduc ion ule Ri.σ j =(α j,γ
i,ν
i,
j),j =
1,2,...,k−1, whe e νi=ω1⊕ω2⊕...⊕ωk−1;
3) syn ={(σp1,σ
1),(σp2,σ
2),...,(σpk−1,σ
k−1),(σ 1,
σpk),...,(σ k−1,σ
pk)},IN={σp1,σ
p2,...,σ
pk−1},
OUT ={σpk}.
Fig.4(d)shows hedynamic uzzy easoningp ocess modeled
by Π4. The uzzy easoning p ocess is au oma ically pe o med
as ollows. Ini ially, a spike is p o ided o each p oposi ion
neu on σpj in IN and hei alues a e α1,α
2,...,α
k−1,
espec i ely. These p oposi ion neu ons σpj concu en ly i e,
and hen, each o hem emi s a spike wi h alue αj⊗ωjin o
he co esponding ule neu on σ j,j=1,2,...,k−1. Nex ,
each ule neu on σ j ecei es he co esponding spike and i es,
and hen, i sends a spike wi h alue [(αj⊗ωj)νi]⊗γi
o p oposi ion neu on σpk. Finally, p oposi ion neu on σpk
ecei es he spikes om he ule neu ons, and he alue o
he spikes is compu ed by logical “OR” ope a o “∨” as i s
con en , i.e., αk={[(α1⊗ω1)νi]⊗γi}∨{[(α2⊗ω2)
νi]⊗γi}∨...∨{[(αk−1⊗ωk−1)νi]⊗γi}. The e o e,
he esul compu ed by Π4is αk={[(α1⊗ω1)(ω1⊕ω2
⊕...⊕ωk−1)] ∨[(α2⊗ω2)(ω1⊕ω2⊕...⊕ωk−1)]
∨...∨[(αk−1⊗ωk−1)(ω1⊕ω2⊕...⊕ωk−1)]}⊗γi.
As is well-known, uzzy p oduc ion ules a e no s aigh o -
wa d and hei uzzy easoning is usually a complica ed p ocess.
Howe e , om he a o emen ioned discussions, we can see ha
he s uc u e o weigh ed uzzy p oduc ion ules modeled by
he p oposed WFSN P sys ems is isual and is easily com-
p ehended due o i s g aphical na u e. Mo eo e , owing o he
pa allel compu ing abili y o WFSN P sys ems and he neu on’s
i ing mechanism, he p oposed WFSN P sys ems a e able o
comple e uzzy easoning p ocess concu en ly and au oma i-
cally, and he compu ing p ocess only akes h ee ime uni s.
IV. WEIGHTED FUZZY REASONING ALGORITHM
In his sec ion, we will p esen a weigh ed uzzy easoning
algo i hm based on S-WFSN P sys ems. F om he p e ious dis-
cussion, we know ha o a uzzy knowledge base, p oposi ion
neu ons exp ess i s all uzzy p oposi ions, while ule neu ons
model i s weigh ed uzzy p oduc ion ules. Gene ally, we should
p o ide he uzzy u h alues o a pa o uzzy p oposi ions
be o e easoning, and he p oposi ion neu ons associa ed wi h
he pa o uzzy p oposi ions a e in ac inpu neu ons o he
S-WFSN P sys em model. The goal o uzzy easoning me hod
is o eason ou he uzzy u h alues o o he unknown uzzy
p oposi ions (p oposi ion neu ons) om known uzzy p opo-
si ions (inpu neu ons). These unknown uzzy p oposi ions a e
associa ed wi h ou pu neu ons o he S-WFSN P sys em model.
Suppose we modeled he weigh ed uzzy p oduc ion ules o a
uzzy knowledge base by an S-WFSN P sys em model Π.
Based on S-WFSN P sys ems, we de eloped a weigh ed uzzy
easoning algo i hm, which is called he weigh ed uzzy back-
wa d easoning algo i hm. The basis o he weigh ed uzzy
backwa d easoning algo i hm is he cons uc ion o a uzzy
“⊕-OR” (Addi ion-OR) ee, which is simila o he algo i hm
in [40]. The ee uses a special da a s uc u e, i.e., a iple
(σpk,IBRIS(σpk),α(σpk)) is used o exp ess a node in he ee,
whe e σpk is he k h p oposi ion neu on, IBRIS(σpk)is i s im-
media ely backwa d ule incidence se o σpk, and α(σpk)is
he uzzy u h alue o σpk. The weigh ed uzzy backwa d ea-
soning algo i hm (Algo i hm 1) consis s o h ee componen s:
1) building he immedia ely backwa d ule-incidence able IRIT
(Algo i hm 2); 2) gene a ing uzzy “⊕-OR” ee (Algo i hm 3);
and 3) compu ing uzzy u h alues (Algo i hm 4). Ini ially,
each inpu neu on is assigned an ini ial uzzy u h alue. When
he sys em hal s, he sys em’s ou pu s a e uzzy u h alues in
ou pu neu ons.
In he ollowing, we explain he basic ideas behind he h ee
componen s. Fi s Algo i hm 2 builds he immedia ely back-
wa d ule-incidence able IRIT acco ding o Π. Fo each p opo-
si ion neu on σpk in Π, he algo i hm will gene a e i s im-
media ely backwa d ule-incidence se IBRIS(σpk)and de e -
mine he co esponding ule neu on σ i acco ding o syn. Thus,
(σpk,IBRIS(σpk),σ
i)is composed o a uple o IRIT. Second,
Algo i hm 3 is used o gene a e a uzzy “⊕-OR” ee o Πbased
on he buil IRIT. This algo i hm s a s om ou pu neu ons
and hen, c ea es each node o uzzy “⊕-OR” ee acco ding o
backwa d connec ion ela ionship o p oposi ion neu ons. Each
c ea ed node has he s uc u e (σpk,IBRIS(σpk),−), whe e he
ma k “−” deno es ha he uzzy u h alue o he p oposi ion
neu on is unknown. In addi ion o he sides o i s le el in he
ee, o he sides a e labeled by he ce ain y ac o s associa ed
wi h uzzy p oduc ion ules. Finally, Algo i hm 4 compu es he
uzzy u h alue o each non e minal node o he uzzy “⊕-
OR” ee. He e, uzzy u h alues o e minal nodes, which a e
associa ed wi h inpu neu ons o Π, a e known. S a ing om
e minal nodes, uzzy u h alues o all non e minal nodes a e
backwa d compu ed on he basis o s ep 10 o s ep 12 o he
algo i hm.
The weigh ed uzzy backwa d easoning algo i hm and i s
h ee componen algo i hms a e lis ed in Tables II–V (Algo-
i hms 1–4), espec i ely. In he ollowing, we b ie ly discuss
he compu a ional complexi ies and con e gence o he a o e-
men ioned algo i hms. Fi s , Algo i hm 2 con ains iple loop
(m,n, and m imes, espec i ely); he e o e, i s ime complex-
i y is O(m2n), and space complexi y is O(m2). By analyzing
Algo i hm3,wecanconclude ha i s ime complexi yis O(m2),
and space complexi y is O(m2). Simila ly, ime complexi y and
space complexi y o Algo i hm 4 a e O(m2)and O(m2), e-
spec i ely. The e o e, he ime complexi y o Algo i hm 1 is
O(m2n), while i s space complexi y is O(m2). Finally, we ana-
lyze he con e gence o hese algo i hms. F om he desc ip ions
o hese algo i hms, we know ha he oles o Algo i hms 2 and
3 a e o cons uc a able IRIT and a ee T, espec i ely. The e-
o e, he con e gence o he p oposed weigh ed uzzy easoning
algo i hm mainly depends on he con e gence o Algo i hm 4.
Le lbe he la ges o he numbe s o p oposi ion neu ons in
all pa hs o Π om inpu neu ons o ou pu neu ons. We easily
conclude om Algo i hm 4 ha he algo i hm can deduce he
TABLE II
ALGORITHM 1: WEIGHTED FUZZY BACKWARD REASONING ALGORITHM
TABLE III
ALGORITHM 2: IRIT BUILDING ALGORITHM
alues o all unknown p oposi ion neu ons a s ep l, i.e., he
algo i hm will be con e ged a s ep l.
In o de o clea ly unde s and he algo i hms desc ibed p e-
iously, we use wo examples o illus a e he weigh ed uzzy
backwa d easoning p ocess. Fo Example 1 (Π0), Table I gi es
he immedia e ule-incidence able o i s all p oposi ion neu-
ons. By he uzzy “⊕-OR” ee gene a ing algo i hm, a uzzy
“⊕-OR” ee o Π0is gene a ed, as shown in Fig. 5. Assume ha
he u h alue o p oposi ion p1associa ed wi h inpu p opo-
si ion neu on σp1is 0.8. By pe o ming he uzzy u h alue
e alua ing algo i hm, he u h alues o ou pu p oposi ion neu-
ons in Π0a e ob ained, as shown in Fig. 5, om which i can
be clea ly seen ha he u h alue o ou pu p oposi ion neu on
σp5is 0.63.
Example 2: Le p1,p2,p3,p4,p5,p6,p7,p8, and p9be
nine p oposi ions. Assume he knowledge base o a ule-based
sys em con ains he ollowing weigh ed uzzy p oduc ion ules.
R1:IFp1THEN p5(CF =γ1), ω1.
R2:IFp2AND p3THEN p6(CF =γ2), ω2,ω31.
R3:IFp3AND p4THEN p7(CF =γ3), ω32,ω4.
R4:IFp5AND p6THEN p8(CF =γ4), ω5,ω6.
R5:IFp7THEN p9(CF =γ5), ω7.
He e, ue alues, ce ain y ac o s, and weigh s a e ex ended
o use iangula uzzy numbe s. Assume he ce ain y ac o s
γ1,γ
2,γ
3,γ
4, and γ5a e (0.80,0.90,1.0),(0.70,0.80,0.90),
TABLE IV
ALGORITHM 3: FUZZY “⊕-OR” TREE GENERATING ALGORITHM
(0.75,0.85,0.95),(0.85,0.95,1.0),and (0.80,0.90,1.0), e-
spec i ely.
Le ω1=(1.0,1.0,1.0),ω
2=(0.85,0.95,1.0),ω
31 =(0.70,
0.80,0.90),ω
32 =(0.85,0.95,1.0),ω
4=(0.75,0.85,0.95),
ω5=0.85,0.95,1.0),ω
6=(0.75,0.85,0.95),and ω7=(1.0,
1.0,1.0).
The weigh ed uzzy p oduc ion ules can be modeled by using
he ollowing WFSN P sys ems Π5, as shown in Fig. 6:
Π5=({a},{σp1,σ
p2,σ
p3,σ
p4,σ
p5,σ
p6,σ
p7,σ
p8,σ
p9},
{σ 1,σ
2,σ
3,σ
4,σ
5},syn,IN,OUT),
whe e
1) σp1,σ
p2,σ
p3,σ
p4,σ
p5,σ
p6,σ
p7,σ
p8, and σp9p oposi ion
neu ons;
2) σ 1,σ
2,σ
3,σ
4, and σ 5 ule neu ons. He e, ν1=
ω1=(1.0,1.0,1.0),ν2=ω2⊕ω31 =(1.55,1.75,1.90),
TABLE V
ALGORITHM 4: FUZZY TRUTH VALUE COMPUTING ALGORITHM
Fig. 5. Gene a ed uzzy “⊕-OR” ee o Π0and compu a ion o he uzzy
u h alues o he uzzy “⊕-OR” ee o Π0.
Fig. 6. Example 2 modeled by WFSN P sys ems Π5.
TABLE VI
IMMEDIATE RULE-INCIDENCE TABLE OF ALL PROPOSITION
NEURONS IN EXAMPLE 2
Fig. 7. Gene a ed uzzy “⊕-OR” ee o Π5and compu a ion o he uzzy
u h alues o he uzzy “⊕-OR” ee o Π5.
ν3=ω32 ⊕ω4=(1.6,1.8,1.95),ν4=ω5⊕ω6=(1.6,
1.8,1.95), and ν5=ω7=(1.0,1.0,1.0);
3) syn={(σp1,σ
1),(σp2,σ
2),(σp3,σ
2),(σp3,σ
3),(σp4,
σ 3),(σp5,σ
4),(σp6,σ
4),(σp7,σ
5),(σ 1,σ
p5),(σ 2,
σp6),(σ 3,σ
p7),(σ 4,σ
p8),(σ 5,σ
p9)};
4) IN ={σp1,σ
p2,σ
p3,σ
p4}, OUT ={σp8,σ
p9}.
Fo Example 2 (Π5), Table VI gi es he immedia e ule-
incidence able o all i s p oposi ion neu ons. By he uzzy
“⊕-OR” ee gene a ing algo i hm, a uzzy “⊕-OR” ee
o Π5is gene a ed, as shown in Fig. 7. Assume ha
he u h alues o p oposi ions p1,p
2,p
3, and p4associ-
a ed wi h inpu p oposi ion neu ons σp1,σ
p2,σ
p3, and σp4
a e (0.80,0.90,1.0),(0.70,0.80,0.90),(0.85,0.95,1.0), and
(0.75,0.85,0.95), espec i ely. By pe o ming he uzzy u h
alue compu ing algo i hm, he u h alues o ou pu p opo-
si ion neu ons in Π5a e ob ained, as shown in Fig. 7, om
which i can be easily seen ha he uzzy u h alues o
ou pu p oposi ion neu ons σp8and σp9a e uzzy numbe s
(0.381,0.714,1.246)and (0.395,0.691,1.130).