Characterizations of single valued neutrosophic automata using relations

2021·1.4 MB·English

by  V. Karthikeyan

MalayaJournalofMatematik,Vol. 9,No. 1,283-285,2021
https://doi.org/10.26637/MJM0901/0047
Characterizations of single valued neutrosophic
automata using relations
V. Karthikeyan1
Abstract
Reducibilityandstabilityrelationofsinglevaluedneutrosophicautomata(SVNA)areintroducedandprovedthat
stabilityrelationisequivalencerelationinsinglevaluedneutrosophicautomata.
Keywords
Neutrosophicset,SingleValuedneutrosophicset,SingleValuedneutrosophicautomaton,Stabilityrelation.
AMSSubjectClassification
03D05,20M35,18B20,68Q45,68Q70,94A45.
1DepartmentofMathematics,GovernmentCollegeofEngineering,Dharmapuri-636704,TamilNadu,India.
*Correspondingauthor:[email protected]
ArticleHistory:Received11November2020;Accepted22January2021 (cid:13)c2021MJM.
Contents
1 Introduction.......................................283
2 Preliminaries......................................283
3 CharacterizationofSVNAusingrelations........284
4 Conclusion........................................285
References........................................285
1. Introduction Wangetal.[7]introducedthenotionofsinglevaluedneutro-
sophicsets.
The theory of neutrosophy and neutrosophic set was intro-
ThenotionoftheautomatonwasfirstfuzzifiedbyWee
duced by Florentin Smarandache in 1999 [6]. The neutro-
[8]. The concept of single valued neutrosophic finite state
sophicsetisthegeneralizationofclassicalsets,fuzzyset[9],
machinewasintroducedbyTahirMahmood[5]. Inthispa-
intuitionsticfuzzyset[1],intervalvaluedintuitionisticfuzzy
per,theconceptofreducibiltyandstabilityrelationinsingle
sets[2]andsoon. Theconceptoffuzzysetandintuitionstic
valued neutrosophic automataare introduced. Alsoproved
fuzzysetunsuccessfulwhentherelationisindeterminate.
thatstabiltyrelationisanequivalencerelationinsinglevalued
ThetheoryoffuzzysetswasintroducedbyZadeh
neutrosophicautomata.
in1965[9]asageneralizationsofcrispsets. Sincethenthe
fuzzysetsandfuzzylogicareusedwidelyinmanyapplica-
tionsinvolvinguncertainty. Attanasovintroducedtheconcept 2. Preliminaries
ofintuitionisticfuzzysetsin1986[1]whichisanextension
Inthissection,werecallsomedefinitionsandbasicresults
offuzzyset. Inintuitionisticfuzzyset,eachelementofthe
whichwillbeusedthroughoutthepaper.
setrepresentingbyamembershipgradeandnon-membership
grade. Someothergeneralizationsoffuzzysetsarebipolar Definition2.1. [5]AfuzzyautomataistripleF =(S,A,α)
valuedfuzzyset[4],vagueset[3]andsoon. whereS,Aarefinitenonemptysetscalledsetofstatesand
AneutrosophicsetN isclassifiedbyaTruthmem- set of input alphabets and α is fuzzy transition function in
bershipfunctionT ,IndeterminacymembershipfunctionI , S×A×S→[0,1].
N N
and Falsitymembership functionF ,where T ,I , andF
N N N N
Definition 2.2. [6] LetU be the universe of discourse. A
arerealstandardandnon-standardsubsetsof]0−,1+[.
neutrosophicset(NS)N inU ischaracterizedbyatruthmem-
bershipfunctionT ,anindeterminacymembershipfunction
N
I andafalsitymembershipfunctionF ,whereT ,I ,and
N N N N
F arerealstandardornon-standardsubsetsof]0−,1+[.That
N
is
N={(cid:104)x,(T (x),I (x),F (x))(cid:105),x∈U,T ,I ,F ∈]0−,1+[}
N N N N N N
andwiththecondition
0−≤supT (x)+supI (x)+supF (x)≤3+.
N N N
weneedtotaketheinterval[0,1]fortechnicalapplications
insteadof]0−,1+[.
Characterizationsofsinglevaluedneutrosophicautomatausingrelations—284/285
Definition2.3. [6]LetU betheuniverseofdiscourse. Asin- Definition3.4. LetF=(S,A,N)beanSVNA.Anequivalence
glevaluedneutrosophicset(NS)N inU ischaracterizedbya relation R on a set S is said to be congruence relation if
truthmembershipfunctionT ,anindeterminacymembership ∀q,q ∈Qandx∈A, q Rq impliesthatthenthereexists
N i j i j
functionI andafalsitymembershipfunctionF . q,q ∈Ssuchthat
N N l k
N={(cid:104)x,(TN(x),IN(x),FN(x))(cid:105),x∈U, TN,IN,FN ∈ [0,1].} ηN∗(qi,x,ql)>0,ηN∗(qj,x,qk)>0
ζN∗(qi,x,ql)<1,ζN∗(qj,x,qk)<1and
Definition2.4. [5]F =(S,A,N)iscalledsinglevaluedneu- ρN∗(qi,x,ql)>0,ρN∗(qj,x,qk)<1.
trosophic automaton (SVNA forshort), where S and A are
non-emptyfinitesetscalledthesetofstatesandinputsymbols Definition 3.5. Let F = (S,A,N) be an SVNA. If qi and
respectively,andN={(cid:104)ηN(x),ζN(x),ρN(x)(cid:105)}isanSVNSin qj, qi,qj ∈S are said to be reducible relation and it is de-
S×A×S.ThesetofallwordsoffinitelengthofAisdenoted notedbyqiϒqj ifthereexistawordw∈A∗,qk∈Ssuchthat
byA∗.Theemptywordisdenotedbyε,andthelengthofeach ηN∗(qi,w,qk)>0⇔ηN∗(qj,w,qk)>0
x∈A∗isdenotedby|x|. (ζN∗(qi,w,qk)>0⇔ζN∗(qj,w,qk)<1
ρN∗(qi,w,qk)>0⇔ρN∗(qj,w,qk)<1
Definition 2.5. [5] F =(S, A, N) be an SVNA. Define an
SVNSN∗={(cid:104)ηN∗(x),ζN∗(x),ρN∗(x)(cid:105)}inS×A∗×Sby
Example3.6. LetF =(S,A,N)beansinglevaluedneutro-
(cid:40) sophicautomaton,where
1 ifq =q
i j
ηN∗(qi, ε, qj)= S={q1,q2,q3,q4}, A={x,y},andN aredefinedasbelow.
0 ifq (cid:54)=q
i j (ηN∗,ζN∗,ρN∗)(q1,x,q4)=[0.6,0.4,0.5]
(ηN∗,ζN∗,ρN∗)(q1,y,q2)=[0.3,0.5,0.4]
(cid:40) (ηN∗,ζN∗,ρN∗)(q2,x,q3)=[0.5,0.1,0.3]
ζN∗(qi, ε, qj)= 0 ifqi=qj (ηN∗,ζN∗,ρN∗)(q2,y,q4)=[0.7,0.4,0.3]
1 ifqi(cid:54)=qj (ηN∗,ζN∗,ρN∗)(q3,x,q2)=[0.1,0.7,0.5]
(ηN∗,ζN∗,ρN∗)(q3,y,q4)=[0.2,0.4,0.6]
(ηN∗,ζN∗,ρN∗)(q4,x,q1)=[0.6,0.2,0.4]
(cid:40)0 ifq =q (ηN∗,ζN∗,ρN∗)(q4,y,q3)=[0.3,0.4,0.5]
i j
ρN∗(qi, ε, qj)= Thestatesq2andq3arereduciblesinceηN∗(q2,xy,q4)>0⇔
1 ifq (cid:54)=q
i j ηN∗(q3,xy,q4)>0ζN∗(q2,xy,q4)<1⇔ζN∗(q3,xy,q4)<1,
andρN∗(q2,xy,q4)<1⇔ρN∗(q3,xy,q4)<1
ηN∗(qi,xy,qj)=∨qr∈Q[ηN∗(qi,x,qr)∧ηN∗(qr,y,qj)],
ζN∗(qi,xy,qj)=∧qr∈Q[ζN∗(qi,x,qr)∨ζN∗(qr,y,qj)], Definition3.7. LetF =(S,A,N)beanSVNA.Iftwostates
ρN∗(qi,xy,qj)=∧qr∈Q[ρN∗(qi,x,qr)∨ρN∗(qr,y,qj)], qi and qj are said to be stability related and it is denoted
∀qi,qj∈S, x∈A∗andy∈A. byqiΩqj ifforanywordw1∈A∗ thereexistsawordw2∈
A∗, q ∈Ssuchthat
k
ηN∗(qi,w1w2,qk)>0⇔ηN∗(qj,w1w2,qk)>0
3. Characterization of SVNA using ζN∗(qi,w1w2,qk)<1⇔ζN∗(qj,w1w2,qk)<1
relations ρN∗(qi,w1w2,qk)<1⇔ρN∗(qj,w1w2,qk)<1
Example3.8. LetF =(S,A,N)beansinglevaluedneutro-
Definition3.1. LetF =(S,A,N)beanSVNA.IfF issaidto
sophicautomaton,where
bedeterministicSVNAthenforeachq ∈Qandx∈Athereex-
i
S={q ,q ,q ,q }, A={x,y},andN aredefinedasbelow.
istsuniquestateqjsuchthatηN∗(qi,x,qj)>0.ζN∗(qi,x,qj)< 1 2 3 4
1,ρN∗(qi,x,qj)<1. (ηN∗,ζN∗,ρN∗)(q1,x,q4)=[0.3,0.4,0.5]
(ηN∗,ζN∗,ρN∗)(q1,y,q2)=[0.5,0.2,0.4]
Definition 3.2. Let Θ= p1,p2,...,pz be a partition of the (ηN∗,ζN∗,ρN∗)(q2,x,q3)=[0.7,0.1,0.4]
statessetSsuchthatifηN∗(qi,x,qj)>0.ζN∗(qi,x,qj)<1, (ηN∗,ζN∗,ρN∗)(q2,y,q4)=[0.1,0.6,0.3]
ρN∗(qi,x,qj)<1.forsomex∈Athenqi∈psandqj∈ps+1. (ηN∗,ζN∗,ρN∗)(q3,x,q2)=[0.2,0.5,0.4]
Then Θ will be called periodic partition of order z ≥ 2. (ηN∗,ζN∗,ρN∗)(q3,y,q4)=[0.5,0.2,0.3]
An SVNA F is periodic of period z≥2 if and only if z= (ηN∗,ζN∗,ρN∗)(q4,x,q1)=[0.6,0.3,0.3]
Maxcard(Θ)wherethismaximumistakenoverallperiodic (ηN∗,ζN∗,ρN∗)(q4,y,q3)=[0.7,0.3,0.2]
partitionsΘofF. IfF hasnoperiodicpartition, thenF is Foranywordw∈A∗,thereexistsawordxyy∈A∗ such
calledaperiodic. Throughoutthispaperweconsideraperi- that
odicSVNA. ηN∗(q1,wxyy,qk)>0⇔ηN∗(q4,wxyy,qk)>0
ζN∗(q1,wxyy,qk)<1⇔ζN∗(q4,wxyy,qk)<1
Definition3.3. LetF =(S,A,N)beanSVNA.ArelationR (ρN∗(q1,wxyy,qk)<1⇔ρN∗(q4,wxyy,qk)<1and
onasetSissaidtobeanequivalencerelationifitisreflexive, ηN∗(q2,wxyy,ql)>0⇔ηN∗(q3,wxyy,ql)>0.
symmetricandtransitive. ζN∗(q2,wxyy,ql)<1⇔ζN∗,(q3,wxyy,ql)<1.
284
Characterizationsofsinglevaluedneutrosophicautomatausingrelations—285/285
ρN∗(q2,wxyy,ql)>0⇔ρN∗(q3,wxyy,ql)<1. ρN∗(qi,u1u2u3,qn)<1⇔ρN∗(qk,u1u2u3,qn)<1.
Thestatesq ,q andq ,q arestabilityrelated. Now,chooseu u =u.
1 4 2 3 2 3
Foranywordu ∈A∗thereexistswordu∈A∗andq ∈Ssuch
Remark3.9. (i)Reducibilityrelationisnotanequivalence 1 n
relationinSVNA.Sincetransitiverelationdoesnotexists. thatηN∗(qi,u1u,qn)>0⇔ηN∗(qk,u1u,qn)>0
ζN∗(qi,u1u,qn)<1⇔ζN∗(qk,u1u,qn)<1
Example3.10. LetF =(S,A,N)beasinglevaluedneutro- ρN∗(qi,u1u,qn)<1⇔ρN∗(qk,u1u,qn)<1.
sophicautomaton,where Hence,q Ωq .
i k
S={q ,q ,q ,q }, A={x,y},andN aredefinedasbelow.
1 2 3 4
(ηN∗,ζN∗,ρN∗)(q1,x,q3)=[0.3,0.5,0.6] 4. Conclusion
(ηN∗,ζN∗,ρN∗)(q1,y,q1)=[0.5,0.2,0.4]
(ηN∗,ζN∗,ρN∗)(q2,x,q1)=[0.7,0.2,0.3] Inthispaperweintroducereducibilityandstabilityrelationin
(ηN∗,ζN∗,ρN∗)(q2,y,q1)=[0.1,0.5,0.4] singlevaluedneutrosophicautomata. Wehaveshownbyex-
(ηN∗,ζN∗,ρN∗)(q3,x,q4)=[0.3,0.4,0.5] amplethatreducibilityrelationisnotanequivalencerelation
(ηN∗,ζN∗,ρN∗)(q3,y,q4)=[0.5,0.3,0.4] andprovethatstabilityrelationisanequivalencerelation.
(ηN∗,ζN∗,ρN∗)(q4,x,q2)=[0.6,0.2,0.4]
(ηN∗,ζN∗,ρN∗)(q4,y,q4)=[0.7,0.3,0.2] References
In the above SVNA F ∃ a word yy ∈ A∗ such that
[1] K. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and
ηN∗(q1,yy,q1)>0⇔ηN∗(q2,yy,q1)>0
Systems,20(1986),87–96.
ζN∗(q1,yy,q1)<1⇔ζN∗(q2,yy,q1)<1
[2] K. Atanassov, Interval valued intuitionistic fuzzy sets,
(ρN∗(q1,yy,q1)<1⇔ρN∗(q2,yy,q1)<1
FuzzySetsandSystems,31(3)(1989),343–349.
Thereforethestatesq andq arereducible.
1 2
Alsothereexistsawordxy∈A∗ suchthatηN∗(q2,xy,q1)> [3] W. L. Gau and D. J. Buehrer, Vague sets, IEEE Trans.
Syst.ManCybern.23(2)(1993),610–614.
0⇔ηN∗(q4,xy,q1)>0
[4] K.M.Lee,Bipolar-valuedfuzzysetsandtheiroperations,
ζN∗(q2,xy,q1)<1⇔ζN∗(q4,xy,q1)<1
Proc.Int.Conf.onIntelligentTechnologies,(2000),307–
(ρN∗(q2,xy,q1)<1⇔ρN∗(q4,xy,q1)<1
312.
Thereforethestatesq andq arereduciblebutq andq are
2 4 1 4
notreducibleforanywordw∈A∗. [5] T.Mahmood,andQ.Khan,Intervalneutrosophicfinite
switchboard state machine, Afr. Mat. 20(2)(2016),191-
Hencereducibilityrelationisnottransitive.
210.
Theorem3.11. LetF =(S,A,N)beanSVNA.Stabilityrela- [6] F.Smarandache,AUnifyingFieldinLogics,Neutroso-
tiononSVNAF isanEquivalencerelation.
phy: NeutrosophicProbability,setandLogic,Rehoboth:
Proof. Let F =(S,A,N) be an SVNA. Clearly stability re- AmericanResearchPress,(1999).
lationonSVNAF isreflexiveandsymmetric. Forproving [7] H.Wang,F.Smarandache,Y.Q.Zhang,andR.Sunder-
stability relation is an equivalence relation it is enough to ramanSingleValuedNeutrosophicSets,Multispaceand
provethatitistransitive. Multistructure,4(2010),410–413.
LetqΩq andq Ωq . [8] W. G. Wee, On generalizations of adaptive algorithms
i j j k
ToproveqΩq ,weneedtoproveforanywordu ∈A∗,there andapplicationofthefuzzysetsconceptstopatternclas-
i k 1
existsawordu∈A∗,q ∈Ssuchthat sificationPh.D.Thesis,PurdueUniversity,(1967).
n
ηN∗(qi,w1w2,qn)>0⇔ηN∗(qk,u1u,qn)>0 [9] L. A. Zadeh, Fuzzy sets, Information and Control,
ζN∗(qi,w1w2,qn)<1⇔ζN∗(qk,u1u,qn)<1 8(3)(1965),338–353.
ρN∗(qi,w1w2,qn)<1⇔ρN∗(qk,u1u,qn)<1.
Sinceq Ωq foranywordu ∈A∗thereexistsawordu ∈A∗
i j 1 2 (cid:63)(cid:63)(cid:63)(cid:63)(cid:63)(cid:63)(cid:63)(cid:63)(cid:63)
andastateq ∈Ssuchthat
m
ISSN(P):2319−3786
ηN∗(qi,u1u2,qm)>0⇔ηN∗(qj,u1u2,qm)>0 MalayaJournalofMatematik
ζN∗(qi,u1u2,qm)<1⇔ζN∗(qj,u1u2,qm)<1
ISSN(O):2321−5666
ρN∗(qi,u1u2,qm)<1⇔ρN∗(qj,u1u2,qm)<1. (cid:63)(cid:63)(cid:63)(cid:63)(cid:63)(cid:63)(cid:63)(cid:63)(cid:63)
Sinceq Ωq wehaveforanywordu u ∈A∗ thereexistsa
j k 1 2
wordu ∈A∗,q ∈Ssuchthat
3 n
ηN∗(qj,u1u2u3,qn)>0⇔ηN∗(qk,u1u2u3,qn)<1
ζN∗(qj,u1u2u3,qn)<1⇔ζN∗(qk,u1u2u3,qn)>0
ρN∗(qj,u1u2u3,qn)<1⇔ρN∗(qk,u1u2u3,qn)<1.
ηN∗(qj,u1u2u3,qn)>0⇔ηN∗(qi,u1u2u3,qn)>0
ζN∗(qj,u1u2u3,qn)<1⇔ζN∗(qi,u1u2u3,qn)<1
ρN∗(qj,u1u2u3,qn)<1⇔ρN∗(qi,u1u2u3,qn)<1.[SinceqjΩqi].
ηN∗(qi,u1u2u3,qn)>0⇔ηN∗(qk,u1u2u3,qn)>0
ζN∗(qi,u1u2u3,qn)<1⇔ζN∗(qk,u1u2u3,qn)<1
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