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MalayaJournalofMatematik,Vol. 08,No. 04,2263-2266,2020 https://doi.org/10.26637/MJM0804/0160 Cartesian composition of Γ− reset single valued neutrosophic automata N. Mohanarao1 and V. Karthikeyan2* Abstract Cartesian composition of Γ− reset single valued neutrosophic automata(SVNA) are introduce, prove that cartesiancompositionofΓ−resetSVNAisΓ−resetSVNA. Keywords SingleValuedneutrosophicset,SingleValuedneutrosophicautomaton,Γ−reset,CartesianComposition. AMSSubjectClassification 03D05,20M35,18B20,68Q45,68Q70,94A45. 1DepartmentofMathematics,GovernmentCollegeofEngineering,Bodinayakkanur,Tamilnadu,India. 2DepartmentofMathematics,GovernmentCollegeofEngineering,Dharmapuri,Tamilnadu,India. *Correspondingauthor:[email protected]; [email protected] ArticleHistory:Received13October2020;Accepted21December2020 (cid:13)c2020MJM. Contents cartesiancompositionofΓ−resetSVNAisΓ−resetSVNA. 1 Introduction......................................2263 2. Preliminaries 2 Preliminaries.....................................2263 Definition2.1. [1]AfuzzyautomataistripleF =(S,A,α) 3 SingleValuedNeutrosophicAutomata ..........2264 whereS,Aarefinitenonemptysetscalledsetofstatesand 4 Γ−ResetSingleValuedNeutrosophicAutomata2264 set of input alphabets and α is fuzzy transition function in 5 CartesianCompositionofΓ−ResetSingleValuedNeu- S×A×S→[0,1]. trosophicAutomata .............................2264 Definition2.2. [1]AfuzzyautomataistripleF =(S,A,α) 5.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . 2264 whereS,Aarefinitenonemptysetscalledsetofstatesand 5.2 Definition . . . . . . . . . . . . . . . . . . . . . . . . . 2264 set of input alphabets and α is fuzzy transition function in 6 Conclusion.......................................2266 S×A×S→[0,1]. References.......................................2266 Definition 2.3. Neutrosophic Set [2] LetU be the universe of discourse. A neutrosophic set (NS) N in U is 1. Introduction characterized by a truth membership function ηN, an inde- terminacymembershipfunctionζ andafalsitymembership N Thetheoryofneutrosophyandneutrosophicsetwas functionρ ,whereη ,ζ ,andρ arerealstandardornon- N N N N introducedbyFlorentinSmarandachein1999[2]. Thefuzzy standardsubsetsof]0−,1+[.Thatis setswasintroducedbyZadehin1965[5]. Aneutrosophicset N={(cid:104)x,(η (x),ζ (x),ρ (x))(cid:105),x∈U, N N N N is a set where each element of the universe has a degree η ,ζ ,ρ ∈]0−,1+[}andwiththecondition N N N of truth, indeterminacy and falsity respectively and which 0−≤supη (x)+supζ (x)+supρ (x)≤3+. liesinthenonstandardunitinterval]0−,1+[.Wangetal.[3] N N N weneedtotaketheinterval[0,1]fortechnicalapplications introducedthenotionofsinglevaluedneutrosophicsets. insteadof]0−,1+[. The fuzzy automaton was introduced Wee [4]. Theconceptofsinglevaluedneutrosophicfinitestatemachine Definition2.4. SingleValuedNeutrosophicSet[2] wasintroducedbyTahirMahmood[1].Cartesiancomposition LetU betheuniverseofdiscourse. Asinglevaluedneutro- ofΓ−resetsinglevaluedneutrosophicautomata(SVNA)are sophicset(NS)NinU ischaracterizedbyatruthmembership introduce and studied their properties. Finally, prove that functionη ,anindeterminacymembershipfunctionζ anda N N CartesiancompositionofΓ−resetsinglevaluedneutrosophicautomata—2264/2266 falsitymembershipfunctionρ . Definition4.4. LetΘ=p ,p ,...,p beapartitionof N 1 2 z N={(cid:104)x,(ηN(x),ζN(x),ρN(x))(cid:105),x∈U, ηN,ζN,ρN∈ [0,1].} thestatessetSsuchthatifηN∗(qi,x,qj)>0.ζN∗(qi,x,qj)< 1,ρN∗(qi,x,qj)<1.forsomex∈Athenqi∈psandqj∈ps+1. Then Θ will be called periodic partition of order z ≥ 2. 3. Single Valued Neutrosophic Automata An SVNA F is periodic of period z≥2 if and only if z= Maxcard(Θ)wherethismaximumistakenoverallperiodic Definition3.1. [1] partitionsΘofF. IfF hasnoperiodicpartition, thenF is F=(S,A,N)iscalledsinglevaluedneutrosophic calledaperiodic. Throughoutthispaperweconsideraperi- automaton(SVNAforshort),whereSandAarenon-emptyfi- odicSVNA. nitesetscalledthesetofstatesandinputsymbolsrespectively, and N ={(cid:104)ηN(x),ζN(x),ρN(x)(cid:105)} is an SVNS in S×A×S. Definition4.5. LetF=(S,A,N)beanSVNA.Wesay ThesetofallwordsoffinitelengthofAisdenotedbyA∗.The thatFissaidtobeΓ−resetifthereexistsawordw∈A∗, q ∈ j emptywordisdenotedbyε,andthelengthofeachx∈A∗is SandarealnumberΓwithΓ∈(0,1]suchthatηN∗(qi,w,qj)≥ denotedby|x|. Γ>0,ζN∗(qi,w,qj)≤Γ<1,ρN∗(qi,w,qj)≤Γ<1 ∀qi∈S. Definition3.2. [1] F=(S,A,N)beanSVNA.DefineanSVNSN∗= 5. Cartesian Composition of Γ− Reset {(cid:104)ηN∗(x),ζN∗(x),ρN∗(x)(cid:105)}inS×A∗×Sby Single Valued Neutrosophic Automata (cid:40) 5.1 Definition 1 ifq =q ηN∗(qi, ε, qj)= i j Let Fi =(Si,Ai,Ni) be SVNA, i=1,2 and let A1∩ 0 ifq (cid:54)=q i j A =φ.LetF ◦F =(S ×S ,A ∪A ,N ◦N ),where 2 1 2 1 2 1 2 1 2 (cid:40)  ζN∗(qi, ε, qj)= 01 iiffqqii=(cid:54)=qqjj ηηifNx1((∈qqi,,Axx,1,q,qkq))j=ql (η ◦η )((q,q ),x,(q ,q))= N2 j l ρN∗(qi, ε, qj)=(cid:40)01 iiffqqi(cid:54)==qqj N1 N2 i j k l 0oifthxer∈wiAse2,qi=qk i j ηN∗(qi,xy,qj)=∨qr∈Q[ηN∗(qi,x,qr)∧ηN∗(qr,y,qj)],  ζρ∀NNq∗i∗,((qqqjii,,∈xxyyS,,q,qjj))==∧∧qqrr∈∈QQ[[ζρNN∗∗((qqii,,xx,,qqrr))∨∨ζρNN∗∗((qqrr,,yy,,qqjj))]],, ζζifNx1((qq∈i,,xAx,,1qq,kq))j=ql x∈A∗andy∈A. (ζN1◦ζN2)((qi,qj),x,(qk,ql))=0oifNthx2er∈wjiAse2,lqi=qk 4. Γ− Reset Single Valued Neutrosophic Automata  ADsaeitfidhnetroiteiboeenxdi4se.t1tse.rumniiqnuiseticstLaSetVteNFqA=jtsh(ueScn,hAfo,thNra)etabηcehNa∗qn(iqS∈iV,xNQ,Aqa.jn)Idf>Fx0∈is. ρρifNx1((∈qqi,,Axx,1,qq,kq))j=ql ζsDaNei∗dfi(nqtoiit,bixoe,nqcoj4)n.2<n.ec1t,eρdNS∗V(NqLiA,exit,fFqfoj=)r<a(lSl1q,.A,,qN)∈bSetahnerSeVeNxiAst.sIfxF∈iAs (ρN1◦ρN2)((qi,qj),x,(qk,ql))=0oiftNhx2er∈wjiAse2,lqi=qk i j suchthatηN(qi,x,qj)>0.ζN∗(qi,u,qj)<1,ρN∗(qi,u,qj)< 1.(or)ηN(qj,x,qi)>0.ζN∗(qj,x,qi)<1,ρN∗(qj,x,qi)<1. ∀(qi,qj),(qk,ql)∈S1×S2, x∈A1∪A2.ThenF1◦F2is calledtheCartesiancompositionofF ◦F . 1 2 Definition4.3. LetF=(S,A,N)beanSVNA.IfF is saidtobestronglyconnectedSVNAifforallq,q ∈Sthere 5.2 Definition i j existsu∈A∗ suchthatηN∗(qi,u,qj)>0.ζN∗(qi,u,qj)<1, Let Fi =(Si,Ai,Ni) be SVNA, i=1,2 and let A1∩ ρN∗(qi,u,qj)<1.Equivalently,F isstronglyconnectedifit A2=φ.LetF1◦F2=(S1×S2,A1∪A2,N1◦N2),bethecarte- hasnopropersubautomaton. siancompositionofF andF .Then∀w∈A∗∪A∗,w(cid:54)=ε. 1 2 1 2 2264 CartesiancompositionofΓ−resetsinglevaluedneutrosophicautomata—2265/2266 ηN∗(qi,w1,qk) ∧ ηN∗(qj,w2,ql)=0, 1 2  ζN∗(qi,w1,qk) ∨ ζN∗(qj,w2,ql)=1and (ηN1∗◦ηN2∗)((qi,qj),w,(qk,ql))=ηi0oηifftNNhww21∗∗e((rqq∈∈wij,i,swAAwe,∗2∗1,q,,qklqq))ij==qqkl INρfNox11∗w(=,qi(ε,ηwaN1n1∗,dq◦kyη)(cid:54)=N∨2∗ε)(ρo(Nqr22∗ix,(qq(cid:54)=jj),ε,wwa21,nwqdl2)y,=(=qk1ε,.qthl)e)n=theresultisholds. ∨(qr,qs)∈S1×S2{(ηN1∗◦ηN2∗)((qi,qj),w1,(qr,qs))∧ (ηN∗◦ηN∗)((qr,qs),w2,(qk,ql))}  1 2 (ζN1∗◦ζN2∗)((qi,qj),w,(qk,ql))=iζ0oiζffNNthww21∗∗e((rqq∈∈wij,,iwswAAe,,∗2∗1qq,,klqq))ij==qqkl (=ηN∨1∗q◦rη∈NS2∗1){(((ηqNr,1=∗q◦lη)η,NNw1∗2∗(2)q,((i(,qqwki,,1qq,qlj))k),)}w∧1wη2,N(2∗q(rq,jq,lw))2∧,ql). (ζN∗◦ζN∗)((qi,qj),w1w2,(qk,ql))= 1 2  (ρN1∗◦ρN2∗)((qi,qj),w,(qk,ql))=ρi0oρifftNNhww21∗∗e((rqq∈∈wij,,iwsAAwe,,∗2∗1qq,,klqq))ij==qqkl ((∧=ζζ(NNq∧11∗∗r,q◦◦qrsζζ)∈∈NNSS22∗∗11))×{((S(((2qqζ{rrN,,(1∗qqζ=◦slN))ζ1∗ζ,,NwNw◦2∗1∗2ζ2)(,,N(q(((2∗iqqq,)kkwi(,,,(qq1qq,llji)q)),)),qk}}w)j)1∨,ww2ζ1,,N((2∗qq(rrq,,qjq,ls)w)))2∨∨,ql). ∀(q,q ),(q ,q)∈S ×S , w∈A∗∪A∗.ThenF ◦F is i j k l 1 2 1 2 1 2 calledtheCartesiancompositionofF ◦F . 1 2 Theorem5.1. LetF =(S,A,N)beSVNA,i=1,2 i i i i (ρN∗◦ρN∗)((qi,qj),w1w2,(qk,ql))= andletA1∩A2=φ. 1 2 LetF1◦F2=(S1×S2,A1∪A2,N1◦N2),bethecartesiancom- ∧(qr,qs)∈S1×S2{(ρN1∗◦ρN2∗)((qi,qj),w1,(qr,qs))∨ positionofF1andF2. (ρN∗◦ρN∗)((qr,qs),w2,(qk,ql))} Then∀w ∈A∗,w ∈A∗ 1 2 (ηN∗◦ηN1∗)((q1i,q2j),w12w2,(qk,ql))=ηN∗(qi,w1,qk)∧ =∧qr ∈S1{(ρN1∗◦ρN2∗)((qi,qj),w1w2,(qr,ql))∨ 1 2 1 ηN∗(qj,w2,ql) (ρN∗◦ρN∗)((qr,ql),w2,(qk,ql))} 2 1 2 (ζN1∗◦ζN2∗)((qi,qj),w1w2,(qk,ql))=ζN1∗(qi,w1,qk)∨ =ρN∗(qi,w1,qk) ∨ρN∗(qj,w2,ql). ζN∗(qj,w2,ql) 1 2 2 (ρN∗◦ρN∗)((qi,qj),w1w2,(qk,ql))=ρN∗(qi,w1,qk)∨ 1 2 1 ρN∗(qj,w2,ql) 2 Proof. Letw ∈A∗, w ∈ A∗, 1 1 2 2 (qi,qj),(qk,ql) ∈ S1 × S2. Theorem5.2. LetFi=(Si,Ai,Ni)beSVNA,i=1,2 Ifw1=ε =w2,thenw1w2=ε.Suppose(qi,qj)=(qk,ql). and let A1∩A2 =φ. If F1 and F2 are Γ− reset SVNA then Thenqi=qk andqj=ql.Hence, cartesiancompositionofF1◦F2isΓ−resetSVNA. (ηN∗◦ηN∗)((qi,qj),w1w2,(qk,ql))=1=ηN∗(qi,w1,qk)∧ 1 2 1 ηN2∗(qj,w2,ql) Proof. LetFi=(Si,Ai,Ni),i=1,2beΓ−resetSVNA. (ζN1∗◦ζN2∗)((qi,qj),w1w2,(qk,ql))=0=ζN1∗(qi,w1,qk) ∨ Thenthereexistsawordw1∈A∗, qj∈S1andw2∈A∗, ql ∈ ζN∗(qj,w2,ql) S2arealnumberΓwithΓ∈(0,1]suchthat 2 (ζN1∗◦ζN2∗)((qi,qj),w1w2,(qk,ql))=0=ζN1∗(qi,w1,qk)∨ ηN1∗(qi,w1,qk)≥Γ>0,ζN1∗(qi,w1,qk)≤Γ<1, ζN2∗(qj,w2,ql) ρN1∗(qi,w,qk)≤Γ<1 ∀qi∈S1. Suppose (qi,qj)(cid:54)=(qk,ql). Then either qi (cid:54)=qk or qj (cid:54)=ql. ηN∗(qj,w2,ql)≥Γ>0,ζN∗(qj,w2,ql)≤Γ<1, 2 2 Then ρN∗(qj,w2,ql)≤Γ<1 ∀qj∈S2. 2 2265 CartesiancompositionofΓ−resetsinglevaluedneutrosophicautomata—2266/2266 Now, (ηN∗◦ηN∗)((qi,qj),w,(qk,ql))= 1 2 (ηN∗◦ηN∗)((qi,qj),w1w2,(qk,ql)),w=w1w2 1 2 =∨(qr,qs)∈S1×S2{(ηN1∗◦ηN2∗)((qi,qj),w1,(qr,qs))∧ (ηN∗◦ηN∗)((qr,qs),w2,(qk,ql))} 1 2 =∨qr ∈S1{(ηN1∗◦ηN2∗)((qi,qj),w1,(qr,qj))∧ (ηN∗◦ηN∗)((qr,qj),w2,(qk,ql))} 1 2 =ηN∗(qi,w1,qk) ∧ ηN∗(qj,w2,ql). 1 2 Hence,thecartesiancompositionofF ◦F isΓ−resetSVNA. 1 2 6. Conclusion CartesiancompositionofΓ−resetsinglevaluedneutrosophicautomata(SVNA)areintroduce,provethatcartesian compositionofΓ−resetSVNAisΓ−resetSVNA. 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