AdvancesinMathematics:ScientificJournal 9 (2020),no.4,1653–1659 ISSN:1857-8365(printed);1857-8438(electronic) https://doi.org/10.37418/amsj.9.4.21Spec.IssueonNCFCTA-2020
SUBSYSTEMSOFINTERVALNEUTROSOPHICAUTOMATA
V.KARTHIKEYAN 1 ANDR.KARUPPAIYA
ABSTRACT.Inthispaper,weintroducesubsystemofintervalneutrosophicautomatonwithexample.Weintroducedifferenttypesofsubsystemsofinterval neutrosophicautomatonanddiscussthepropertiesofsubsystemsofinterval neutrosophicautomatonandestablishtheconnectionbetweendifferenttypes ofsubsystems.
1.INTRODUCTION
ThetheoryofneutrosophyandneutrosophicsetwasintroducedbyFlorentin Smarandachein1999[2].Aneutrosophicset N isclassifiedbyaTruthmembershipfunction TN , Indeterminacymembershipfunction IN , andFalsitymembershipfunction FN , where TN ,IN , andFN arerealstandardandnon-standard subsetsof ]0 , 1+[. Wang etal. [3]introducedthenotionofinterval-valuedneutrosophicsets.Theconceptofintervalneutrosophicfinitestatemachinewas introducedbyTahirMahmood[1].Inthispaper,weintroducedtheconceptof subsystemofintervalneutrosophicautomata.
Also,weintroducedsomeothersubsystemsofintervalneutrosophicautomata anddiscussedtheirproperties.Weestablishanecessaryandsufficientcondition forintervalneutrosophicsubset NQ of Q tobeasubsystemofintervalneutrosophicautomaton.
1correspondingauthor 2010 MathematicsSubjectClassification. 03D05,20M35,18B20,68Q45,68Q70,94A45. Keywordsandphrases. Neutrosophicset,intervalneutrosophicset,intervalneutrosophic automaton,subsystem. 1653
2.PRELIMINARIES
Definition2.1. [2]Let U betheuniverseofdiscourse.Aneutrosophicset(NS) N in U is N = { x, (TN (x),IN (x),FN (x)) ,x ∈ U,TA,IA,FA ∈ ]0 , 1+[ } andwith thecondition 0 ≤ supTN (x)+ supIN (x)+ supFN (x) ≤ 3+ Weneedtotakethe interval [0, 1] fortechnicalapplicationsinsteadof ]0 , 1+[
Definition2.2. [3]Let U beauniversalset.Anintervalneutrosophicset(INS forshort)isoftheform N ={ αN (x),βN (x),γN (x) |x ∈ U } = { x, [inf αN (x), sup αN (x)], [inf βN (x), sup βN (x)], [inf γN (x), sup γN (x)] |x ∈ U }, where αN (x),βN (x), and γN (x) ⊆ [0, 1] andtheconditionthat 0 ≤ sup αN (x)+sup βN (x)+sup γN (x) ≤ 3.
3.INTERVAL NEUTROSOPHIC AUTOMATA
Definition3.1. [1] M =(Q, Σ,N ) iscalledintervalneutrosophicautomaton (INAforshort), where Q and Σ arenon-emptyfinitesetscalledthesetofstates andinputsymbolsrespectively,and N = { αN (x),βN (x),γN (x) } isan INS in Q × Σ × Q. Thesetofallwordsoffinitelengthof Σ isdenotedby Σ∗ . Theempty wordisdenotedby , andthelengthofeach x ∈ Σ∗ isdenotedby |x|.
Definition3.2. [1] M =(Q, Σ,N ) bean INA.Definean INS N ∗ = { αN ∗ (x),βN ∗ (x),γN ∗ (x) } in Q × Σ∗ × Q by αN ∗ (qi, ,qj )= [1, 1] if qi = qj [0, 0] if qi = qj βN ∗ (qi, ,qj )= [0, 0] if qi = qj [1, 1] if qi = qj γN ∗ (qi, ,qj )= [0, 0] if qi = qj [1, 1] if qi = qj
αN ∗ (qi,w,qj )= αN ∗ (qi,xy,qj )= ∨qr ∈Q[αN ∗ (qi,x,qr ) ∪ αN ∗ (qr ,y,qj )], βN ∗ (qi,w,qj )= βN ∗ (qi,xy,qj )= ∨qr ∈Q[βN ∗ (qi,x,qr ) ∪ βN ∗ (qr ,y,qj )], γN ∗ (qi,w,qj )= γN ∗ (qi,xy,qj )= ∨qr ∈Q[γN ∗ (qi,x,qr ) ∪ γN ∗ (qr ,y,qj )], ∀qi,qj ∈ Q,w = xy,x ∈ Σ∗ and y ∈ Σ.
Definition3.3. Let M =(Q, Σ,N ) beanintervalneutrosophicautomaton.Let NQ beaintervalneutrosophicsubsetof Q, andforeach qi ∈ Q, NQ ={ αNQ (qi),βNQ (qi),γNQ (qi) } = qi, [inf αNQ (qi), sup αNQ (qi)], [inf βNQ (qi), sup βNQ (qi)], [inf γNQ (qi), sup γNQ (qi)] . Then (Q,NQ, Σ,N ) iscalledasubsystemof M anditisdenotedby NQ if ∀qi,qj ∈ Q and x ∈ Σ suchthat αNQ (qj ) ≥∨qi∈Q{αNQ (qi) ∧ αN (qi,x,qj )},βNQ (qj ) ≤ ∧qi∈Q{βNQ (qi) ∨ βN (qi,x,qj )} and γNQ (qj ) ≤∧qi∈Q{γNQ (qi) ∨ γN (qi,x,qj )}.
Definition3.4. Let M =(Q, Σ,N ) beanintervalneutrosophicautomatonand NQ = { αNQ ,βNQ ,γNQ } beanintervalneutrosophicsubsetof Q. Let qj ∈ Q and forall x ∈ Σ∗ , defineanintervalneutrosophicsubset NQx of Q by, (αNQ x)(qj )= ∨qi ∈ Q{αNQ (qi) ∧ αN (qi,x,qj )} (βNQ x)(qj )= ∧qi ∈ Q{βNQ (qi) ∨ βN (qi,x,qj )} and (γNQ x)(qj )= ∧qi ∈ Q{γNQ (qi) ∨ γN (qi,x,qj )}
4.PROPERTIESOF SUBSYSTEMSOF INTERVAL NEUTROSOPHIC AUTOMATA
Theorem4.1. Let M =(Q, Σ,N ) beanintervalneutrosophicautomaton.Let NQ = αNQ ,βNQ ,γNQ beanintervalneutrosophicsubsetof Q. Then NQ isa subsystemof M ifandonlyif ∀qi,qj ∈ Q, ∀x ∈ Σ∗ , αNQ (qj ) ≥∨qi∈Q{αNQ (qi) ∧ αN (qi,x,qj )}, βNQ (qj ) ≤∧qi∈Q{βNQ (qi) ∨ βN (qi,x,qj )} and γNQ (qj ) ≤∧qi∈Q{γNQ (qi) ∨ γN (qi,x,qj )} Proof. Suppose NQ isasubsystemof M. Let qi,qj ∈ Q and x ∈ Σ∗ Weprove theresultbyinductionon |x| = n. If n =0, then x
Thus,
αNQ (qj ) ≥∨qi∈Q{αNQ (qi) ∧ αN ∗ (qi,x,qj )}
∧qi∈Q{βNQ (qi) ∨ βN ∗ (qi,x,qj )} = ∧qi∈Q{βNQ (qi) ∨ βN ∗ (qi,ya,qj )} = ∧qi∈Q{βNQ (qi) ∨{∧qk ∈Q {βN ∗ (qi,y,qk ) ∨ βN (qk ,a,qj )}}} = ∧qk ∈Q{∧qi∈Q βNQ (qi) ∨ βN ∗ (qi,y,qk ) ∨ βN (qk ,a,qj ) } ≥∧qk ∈Q βNQ (qk ) ∨ βN (qk ,a,qj ) ≥ βNQ (qj )
∧qi∈Q{βNQ (qi) ∨ βN (qi,x,qj )}≥ βNQ (qj )
Thus,
βNQ (qj ) ≤∧qi∈Q{βNQ (qi) ∨ βN ∗ (qi,x,qj )} ∧qi∈Q{γNQ (qi) ∨ γN ∗ (qi,x,qj )} = ∧qi∈Q{γNQ (qi) ∨ γN ∗ (qi,ya,qj )} = ∧qi∈Q{γNQ (qi) ∨{∧qk ∈Q {γN ∗ (qi,y,qk ) ∨ γN (qk ,a,qj )}}} = ∧qk ∈Q{∧qi∈Q γNQ (qi) ∨ γN ∗ (qi,y,qk ) ∨ γN (qk ,a,qj ) } ≥∧qk ∈Q γNQ (qk ) ∨ γN (qk ,a,qj ) ≥ γNQ (qj ) ∧qi∈Q{γNQ (qi) ∨ γN ∗ (qi,x,qj )}≥ γNQ (qj ) Thus, γNQ (qj ) ≤∧qi∈Q{γNQ (qi) ∨ γN ∗ (qi,x,qj )
(βNQ x)(qj )= ∧ βNQ (qi) ∨ βN ∗ (qi,x,qj ) | qi ∈ Q ≥ βNQ (qj ), (γNQ x)(qj )= ∧ γNQ (qi) ∨ γN ∗ (qi,x,qj ) | qi ∈ Q ≥ γNQ (qj )
Hence, αNQ x ⊆ αNQ ,βNQ x ⊇ βNQ , and γNQ x ⊇ γNQ ∀x ∈ Σ∗
Conversely,suppose αNQ x ⊆ αNQ ,βNQ x ⊇ βNQ , and γNQ x ⊇ γNQ ∀x ∈ Σ∗ . Let qj ∈ Q and x ∈ Σ∗ Now, αNQ (qj ) ≥ (αNQ x)(qj )= ∨qi∈Q αNQ (qi) ∧ αN ∗ (qi,x,qj )
Thus, αNQ (qj ) ≥∨qi∈Q{αNQ (qi) ∧ αN ∗ (qi,x,qj )}
NQ (qj ) ≤ (βNQ x)(qj )= ∧qi∈Q βNQ (qi) ∨ βN ∗ (qi,x,qj ) Thus, βNQ (qj ) ≤∧qi∈Q{βNQ (qi) ∨ βN ∗ (qi,x,qj )}, γNQ (qj ) ≤ (γNQ x)(qj )= ∧qi∈Q γNQ (qi) ∨ γN ∗ (qi,x,qj ) , γNQ (qj ) ≤∧qi∈Q{γNQ (qi) ∨ γN ∗ (qi,x,qj )} Hence NQ isasubsystemof M.
Theorem4.3. Let M =(Q, Σ,N ) beanintervalneutrosophicautomaton.Let NQ1 , and NQ2 besubsystemsof M. Thenthefollowingconditionshold: (i) NQ1 ∧ NQ2 isasubsystemof M ; (ii) NQ1 ∨ NQ2 isasubsystemof M. Proof. Here, NQ1 and NQ2 aresubsystemsofanintervalneutrosophicautomaton M. (i)Nowwehavetoprove NQ1 ∧ NQ2 isasubsystemof M. Thatis (αNQ1 ∧ αNQ2 )(qj ) ≥∨qi∈Q{(αNQ1 ∧ αNQ
Thus, (αNQ1 ∧ αNQ2 )(qj ) ≥{∨qi∈Q{(αNQ1 ∧ αNQ2 )(qi) ∧ αN (qi,x,qj )}} (4.1) (βNQ1 ∧ βNQ2 )(qj )=(βNQ1 (qj ) ∧ βNQ2 (qj ))
≤{∧qi∈Q{βNQ1 (qi) ∨ βN (qi,x,qj )}}∧{∧qi∈Q{βNQ2 (qi) ∨ βN (qi,x,qj )}} = {∧qi∈Q{βNQ1 (qi) ∧ βNQ2 (qi) ∨ βN (qi,x,qj )}} = {∧qi∈Q{(βNQ1 ∧ βNQ2 )(qi) ∨ βN (qi,x,qj )}}
Thus, (βNQ1 ∧ βNQ2 )(qj ) ≤{∧qi∈Q{(βNQ1 ∧ βNQ2 )(qi) ∨ βN (qi,x,qj )}} (4.2) γNQ1 ∧ γNQ2 )(qj )=(γNQ1 (qj ) ∧ γNQ2 (qj )) ≤{∧qi∈Q{γNQ1 (qi) ∨ γN (qi,x,qj )}}∧{∧qi∈Q{γNQ2 (qi) ∨ γN (qi,x,qj )}} = {∧qi∈Q{γNQ1 (qi) ∧ γNQ2 (qi) ∨ γN (qi,x,qj )}} = {∧qi∈Q{(γNQ1 ∧ γNQ2 )(qi) ∨ γN (qi,x,qj )}}
Thus, (4.3) (γNQ1 ∧ γNQ2 )(qj ) ≤{∧qi∈Q{(γNQ1 ∧ γNQ2 )(qi) ∨ γN (qi,x,qj )}} From(4.1),(4.2)and(4.3) NQ1 ∧ NQ2 isasubsystemofanintervalneutrosophic automaton M. (ii)Nowtoprove NQ1 ∨NQ2 isasubsystemofintervalneutrosophicautomaton M. Now, (αNQ1 ∨ αNQ2 )(qj )=(αNQ1 (qj ) ∨ αNQ2 (qj )) ≥{∨qi∈Q{αNQ1 (qi) ∧ αN (qi,x,qj )}}∨{∨qi∈Q{αNQ2 (qi) ∧ αN (qi,x,qj )}} = {∨qi∈Q{αNQ1 (qi) ∨ αNQ2 (qi) ∧ αN (qi,x,qj )}} = {∨qi∈Q{(αNQ1
Thus, (βNQ1 ∨ βNQ2 )(qj ) ≤{∧qi∈Q{(βNQ1 ∨ βNQ2 )(qi) ∨ βN (qi,x,qj )}} (4.5) (γNQ1 ∨ γNQ2 )(qj )=(γNQ1 (qj ) ∨ γNQ2 (qj )) ≤{∧qi∈Q{γNQ1 (qi) ∨ γN (qi,x,qj )}}∨{∧qi∈Q{γNQ2 (qi) ∨ γN (qi,x,qj )}} = {∧qi∈Q{γNQ1 (qi) ∨ γNQ2 (qi) ∨ γN (qi,x,qj )}} = {∧qi∈Q{(γNQ1 ∨ γNQ2 )(qi) ∨ γN (qi,x,qj )}} Thus, (4.6) (γNQ1 ∨ γNQ2 )(qj ) ≤{∧qi∈Q{(γNQ1 ∨ γNQ2 )(qi) ∨ γN (qi,x,qj )}} From(4.4),(4.5),and(4.6), NQ1 ∨ NQ2 isasubsystemofintervalneutrosophic automata M.
REFERENCES
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