On Typical Hesitant Fuzzy Languages and Automata
Valdigleis S. Costa, Benjamín C. Bedregal, Regivan H. N. Santiago
aa r X i v : . [ c s . F L ] F e b Appeared in Fundamenta Informaticae 178(1) : 1–17 (2021). 1Available at IOS Press through https://doi.org/10.3233/FI-2021-0001
On Typical Hesitant Fuzzy Languages and Automata
Valdigleis S. Costa
Universidade Federal do Vale do S˜ao FranciscoColegiado de Ciˆencia da Computac¸ ˜aoSalgueiro-PE, [email protected]
Benjam´ın C. Bedregal * Regivan H. N. Santiago † Universidade Federal do Rio Grande do NorteDepartamento de Inform´atica e Matem´atica AplicadaNatal-RN, [email protected]@dimap.ufrn.br
Abstract.
The idea of nondeterministic typical hesitant fuzzy automata is a generalization ofthe fuzzy automata presented by Costa and Bedregal. This paper, presents the sufficient andnecessary conditions for a typical hesitant fuzzy language to be computed by nondeterministictypical hesitant fuzzy automata. Besides, the paper introduces a new class of Typical HesitantFuzzy Automata with crisp transitions, and we will show that this new class is equivalent to theoriginal class introduced by Costa and Bedregal.
Keywords:
Typical Hesitant Fuzzy Sets, Fuzzy Languages, Automata, Nondeterminism
Address for correspondence: [email protected] * Thanks to CNPQ for the research funding granted through the project 311429/2020-3. † Thanks to CNPQ for the research funding granted through the project 312053/2018-5.
Valdigleis, Benjam´ın & Regivan / On THFL and Automata
1. Introduction
The fuzzy computation theory emerged as a model based on fuzzy sets [1] capable of extrapolatingthe Church’s thesis [2]. The machine models more studied by fuzzy computation theory are fuzzyTuring machines [2, 3] and the fuzzy finite automata [4]. Finite automata are a computational modelthat has finite memory limitation. This model is widely used for modeling applications in hardwareand software [5], and it is also essential for building compilers [4, 6]. In recent years, several general-izations of the concept of finite automata have been presented, such as fuzzy automata [7, 8, 9, 10, 11],probabilistic automata [12, 13, 14], and quantum automata [15, 16, 17, 18, 19].With the development of the various extensions for fuzzy sets, some generalizations for fuzzyautomata have been presented, for example, interval-valued fuzzy automata [20, 21] and intuitionis-tic fuzzy automata [22]. Recently, Costa and Bedregal in [23] using the concepts of typical hesitantfuzzy set [24, 25], introduced the concept of Nondeterministic Typical Hesitant Fuzzy Automata andpresented a subclass, called Deterministic Typical Hesitant Fuzzy Automata, which generalizes thenotion of deterministic finite automata. They showed how it is possible to obtain a Deterministic Typ-ical Hesitant Fuzzy Automata from a Nondeterministic Typical Hesitant Fuzzy Automata. However,the removal of the nondeterminism presented by Costa and Bedregal does not preserve the language.Costa and Bedregal defined nondeterministic typical hesitant fuzzy automata as machines capableto compute typical hesitant fuzzy language [23]. However, it does not characterize a class for theselanguages. Thus an open question exists, every typical hesitant fuzzy language can be computed by atypical hesitant fuzzy automaton ?In this article, the theory of typical hesitant fuzzy automata will be strengthened, characterizingthe languages computed by nondeterministic typical hesitant fuzzy automata. Moreover, we will showthe nondeterminism does not increase the power of typical hesitant fuzzy automata. This work hasthe following division, first this introduction, then in Section 2, presents the mathematical basis forthis work. Section 3, presents a characterization for the languages computed by typical hesitant fuzzyautomata. Section 4, display a new class of typical hesitant fuzzy automata and some results.
2. Preliminaries
In this section, we present all the basic definitions and notations sed throughout the text.
As said in [4], an alphabet is any finite non-empty set Σ . The elements from Σ are called letters,and a word on Σ is any finite sequence of letters. The symbol λ denotes the empty word, i.e., theword without letters from Σ . The set Σ ∗ is a free monoid generated by Σ concerning the operation ofconcatenation [23]. The set Σ + = Σ ∗ − { λ } , and any L ⊆ Σ ∗ is called language. Definition 1. [4] A deterministic finite automaton (DFA) is a quintuple A = h Q, Σ , δ, q , F i where Q is finite non-empty set of states, δ : Q × Σ → Q is a the transition function , q ∈ Q is the initial This paper always assumes complete (N)DFA, meaning that the transition function is total. aldigleis, Benjam´ın & Regivan / On THFL and Automata state and F ⊆ Q is the set of final states.The transition function δ can extend into a function b δ : Q × Σ ∗ → Q by the following recursion: b δ ( q, λ ) = q (1) b δ ( q, wa ) = δ ( b δ ( q, w ) , a ) (2)where a ∈ Σ and w ∈ Σ ∗ . Definition 2.
The language computed by a DFA A is the set L ( A ) = { w ∈ Σ ∗ | b δ ( q, w ) ∈ F } .As said in [26], if w = a · · · a n ∈ Σ + and w ∈ L ( A ) for some DFA A , then there exists a finitesequence of states q , · · · , q n such that δ ( q , a ) = q , · · · , δ ( q n − , a n ) = q n with q n ∈ F . On theother hand, λ ∈ L ( A ) if and only if q ∈ F . Definition 3. [19] A language L is regular, whenever L is finite or L is obtained from regular lan-guages L and L by either finite union, concatenation, or Kleene closure . Theorem 1. [19] A language L is regular if and only if a DFA compute it. Remark 1.
Notice that by the Chomsky’s hierarchy [11], as mentioned in proof of Theorem 2 in [12],there is only an enumerable set of regular languages.Another well-known type of finite automata is the nondeterministic finite automaton, defined be-low.
Definition 4. [4] A nondeterministic finite automaton (NFA) is a quintuple N = h Q, Σ , δ N , q , F i where Q, Σ , q and F are equal to Definition 1, and δ N : Q × Σ → Q is the nondeterministictransition function.As discussed by Hopcroft et al. in [4], it is clear that every AFD is an AFN where the inequality δ ( q, a ) ≤ is satisfied for every pair ( q, a ) ∈ Q × Σ , where denote the cardinality of sets. Thefunction δ N can be extended into a function c δ N : Q × Σ ∗ → Q by the following recursion: c δ N ( q, λ ) = q (3) c δ N ( q, wa ) = [ q ∈ c δ N ( q,w ) { δ N ( q, a ) } (4)for all a ∈ Σ and w ∈ Σ ∗ . Definition 5.
The language computed by NFA A is the set L ( A ) = { w ∈ Σ ∗ | b δ ( q, w ) ∩ F = ∅} .According to [27], for any n ∈ N , there are n -state NFAs recognizing languages which cannot berecognized by any DFA with less than n states. Theorem 2. [4] A language L is regular if and only if a NFA compute it. For more details about union, concatenation and Kleene closure see [4]
Valdigleis, Benjam´ın & Regivan / On THFL and Automata
According to [24, 28], a HFS is defined in terms of a function which return sets of membership degreesfor each element of their domain
U 6 = ∅ . In 2014, Bedregal et al. [25] introduced a particular case ofHFS, called Typical Hesitant Fuzzy Set, or simply THFS, which considers some restrictions. Definition 6. [25, Definition 8] Let H ⊆ [0 , be the set of all finite non-empty subsets of the interval[0,1], and let U be a non-empty set. A THFS on U is a function ψ : U → H . Remark 2.
Here will be considered that h [0 , , ∨ , ∧ , , i is a distributive lattice concerning the usualorder ≤ on real numbers.Each X ∈ H is called a Typical Hesitant Fuzzy Element (THFE). The set H = { X ∈ H | X = 1 } is called of degenerate elements set. Several operators on H were proposed in [25, 28,29, 30, 31, 32, 33]. In particular, Costa and Bedregal [23], have presented the inf-combination andsup-combination. Definition 7. [23, Definition 3.1.] For
X, Y ∈ H the function ⊗ : H × H → H is computed by, X ⊗ Y = { x ∧ y | x ∈ X, y ∈ Y } (5)is called inf-combination of X and Y . Definition 8. [23, Definition 3.2.] For
X, Y ∈ H the function ⊔ : H × H → H is computed by, X ⊔ Y = { x ∨ y | x ∈ X, y ∈ Y } (6)is called sup-combination of X and Y .The operations ∧ and ∨ are, respectively, the operations of infimum and supremum on the dis-tributive lattice [0 , , for complete notions related to partial order and lattice theory, the reader canrefer to [34]. According to [23], H has the following properties:( H
1) The structures h H , ⊗ , { }i and h H , ⊔ , { }i are commutative and idempotent monoids.( H
2) The element { } is an annihilator of ⊔ , and { } is an annihilator of ⊗ .( H ⊗ distribute over ⊔ and ⊔ distribute over ⊗ .According to [23] since h H , ⊔ , { }i is a monoid, the operation ⊔ is extended for the n -dimensionalcase. Definition 9. [23] Given X , X , · · · , X n ∈ H , n G i =1 X i = (cid:16) n − G i =1 X i (cid:17) ⊔ X n (7) aldigleis, Benjam´ın & Regivan / On THFL and Automata Remark 3.
Since ⊗ and ⊔ are both commutative, associative, and idempotent, given any finite non-empty set κ ⊂ H , the set of elements generated by the (sup) inf-combination on the set κ is alsofinite.The ordering problem THFS and THFE have yet been studied in [31, 32, 35, 36]. Now we proposea new relation on H based in the sup-combination, and we will show that this order generalizes theusual order of real numbers. Definition 10.
Given
X, Y ∈ H , X ⊑ Y ⇐⇒ X ⊔ Y = Y Proposition 1. ⊑ is a partial order on H . Proof:
Consider that
X, Y, Z ∈ H so:(i) we have that, X ⊔ X ( H = X . So ⊑ is reflexive;(ii) suppose that X ⊑ Y and Y ⊑ X , then X Hyp. = Y ⊔ X ( H = X ⊔ Y Hyp. = Y so ⊔ is anti-symmetricand(iii) suppose that X ⊑ Y and Y ⊑ Z , thus we have that, X ⊔ Z Hyp. = X ⊔ ( Y ⊔ Z ) ( H = ( X ⊔ Y ) ⊔ Z Hyp. = Y ⊔ Z Hyp. = Z therefore, X ⊑ Z , so ⊑ is transitive.Since that ⊑ is reflexive, anti-symmetric and transitive, the relation ⊑ is a partial order on H . ⊓⊔ Theorem 3.
The order ⊑ generalize the usual order ≤ on [0 , . Proof:
Let H the set of degenerate elements of H , for all { x } , { y } ∈ H we have that, { x } ⊑ { y } ⇐⇒ { x } ⊔ { y } = { y } ⇐⇒ x ∨ y = y ⇐⇒ x ≤ y and x, y ∈ [0 , , thus completing the proof. ⊓⊔ Moreover, the following properties are easily verified.( R
1) If X ⊑ Y , then ( X ⊔ Z ) ⊑ ( Y ⊔ Z ) .( R
2) If X ⊑ Y , then ( X ⊗ Z ) ⊑ ( Y ⊗ Z ) .( R { } ⊑ X ⊑ { } for all X ∈ H . Proposition 2.
For ◦ ∈ {⊗ , ⊔} , if X ⊑ Y , then X ⊑ ( X ◦ Y ) . Proof:
Suppose that X ⊑ Y so by ( R and ( R we have that ( X ◦ X ) ⊑ ( X ◦ Y ) but by ( H , X ◦ X = X ,therefore, X ⊑ ( X ◦ Y ) . ⊓⊔ Valdigleis, Benjam´ın & Regivan / On THFL and Automata
Remark 4.
Notice that for all
X, Y ∈ H , X ⊔ ( X ⊔ Y ) = X ⊔ Y . Therefore, X ⊑ ( X ⊔ Y ) . Theorem 4.
Let { X i } i ∈ I be a finite family of elements of H , then for any X j , with j ∈ I , X j ⊑ F i ∈ I X i . Proof:
By remark 4, commutativity and associativity in ( H . ⊓⊔ Definition 11.
Let U be a nonempty set, f : U → H and k ∈ H . Then we have the following sets:(i) R f = { X ∈ H | f ( x ) = X, x ∈ U } .(ii) S kf = { x ∈ U | k ⊑ f ( x ) } Definition 12.
A Typical Hesitant Fuzzy Language, or simply THFL, is any THFS f : Σ ∗ → H . Theset of all Typical Hesitant Fuzzy Languages is denoted by T .Recently, Costa and Bedregal in [23] introduced a new generalization of fuzzy automata calledNondeterministic Typical Hesitant Fuzzy Automata. Definition 13. [23] A Nondeterministic Typical Hesitant Fuzzy Automaton, or simply NTHFA, isa quintuple M = h Q, Σ , ψ, q , F i where Q is a finite non-empty set of states, Σ is an alphabet, ψ : Q × Σ × Q → H is a THFS, q is initial state and F : Q → [0 , is the THFS on Q of final states. Definition 14. [23] For any NTHFA M the functions ψ is extended into a function b ψ : Q × Σ ∗ × Q → H using the recursion: b ψ ( q, λ, q ′ ) = ( { } , if q = q ′ { } , else (8) b ψ ( q, wa, q ′ ) = G q ′′ ∈ Q (cid:16) b ψ ( q, w, q ′′ ) ⊗ ψ ( q ′′ , a, q ′ ) (cid:17) (9) Definition 15. [23] Let M be a NTHF, then M computes the THFL f M : Σ ∗ → H is defined by, f M ( w ) = G q ∈ Q (cid:16) b ψ ( q , w, q ) ⊗ F ( q ) (cid:17) (10) Remark 5.
By definition 15 it’s posible to deduce that, for all M , f M ( λ ) = F ( q ) . aldigleis, Benjam´ın & Regivan / On THFL and Automata
3. A Characterization of the Languages Computed by NTHFA
This paper will denote the set of all the THFL computed by NTHFA by T R . Now consider thefollowing definition. Definition 16.
Let Σ be an alphabet and let f : Σ ∗ → H and L : Σ ∗ → H two be THFL such that f , f ∈ T R , the H -union of f and f , denoted by f ⊎ f , is given by, f ⊎ f ( w ) = f ( w ) ⊔ f ( w ) (11)for all w ∈ Σ ∗ . Theorem 5. If f , f ∈ T R , then f ⊎ f ∈ T R . Proof:
Assume that f : Σ ∗ → H and f : Σ ∗ → H belongs to T R , so there exists M = h S, Σ , ψ , s , F i and M = h P, Σ , ψ , p , F i such that f = f M and f = f M . Moreover, without loss of generality,assume that S ∩ P = ∅ , now define a new NTHFA M = h Q, Σ , ψ, q , F i where,(a) Q = S ∪ P ∪ { q } with q / ∈ ( S ∪ P ) .(b) For all q ∈ Q we have, F ( q ) = F ( q ) , if q ∈ S F ( q ) , if q ∈ P F ( s ) ⊔ F ( p ) , if q = q (12)(c) For all q, q ′ ∈ Q and a ∈ Σ we have, ψ ( q, a, q ′ ) = ψ ( q, a, q ′ ) , if q, q ′ ∈ Sψ ( s , a, q ′ ) , if q = q , q ′ ∈ Sψ ( q, a, q ′ ) , if q, q ′ ∈ Pψ ( p , a, q ′ ) , if q = p , q ′ ∈ P { } , otherwise (13)it is evident that M is NTHFA. Now notice that, f M ( λ ) Rem. F ( q ) Eq. ( ) = F ( s ) ⊔ F ( p ) (14) Rem. F M ( λ ) ⊔ F M ( λ )= f M ⊎ f M Valdigleis, Benjam´ın & Regivan / On THFL and Automata and for all w ∈ Σ + with w = a a · · · a n , by ( H we have that H is a monoid thus, f M ( a a · · · a n ) = F q f ∈ Q (cid:16) b ψ ( q , a a · · · a n , q f ) ⊗ F ( q f ) (cid:17) = F (cid:16) F q f ∈ Q −{ q } (cid:16) b ψ ( q , a a · · · a n , q f ) ⊗ F ( q f ) (cid:17) , ( b ψ ( q , a a · · · a n , q ) ⊗ F ( q )) (cid:17) . (15)But by equations (9) and (13), and also by ( H it is clear that, b ψ ( q , a a · · · a n , q ) = { } . Therefore, L M ( a a · · · a n ) = G (cid:16) G q f ∈ Q −{ q } (cid:16) b ψ ( q , a a · · · a n , q f ) ⊗ F ( q f ) (cid:17) , (cid:0) { } ⊗ F ( q ) (cid:1)(cid:17) ( H = G q f ∈ Q −{ q } (cid:16) b ψ ( q , a a · · · a n , q f ) ⊗ F ( q f ) (cid:17) (16) = G (cid:16) G q f ∈ S (cid:16) b ψ ( q , a a · · · a n , q f ) ⊗ F ( q f ) (cid:17) , G q f ∈ P (cid:16) b ψ ( q , a a · · · a n , q f ) ⊗ F ( q f ) (cid:17)(cid:17) = f M ( w ) ⊔ f M ( w )= f ⊎ f ( w ) . Hence, by equations (14) and (16), f M = f M ⊎ f M , completing the proof. ⊓⊔ The result of the above theorem shows that the H -union is a closure for the set T R , and this resultis generalized as follows. Corollary 1.
Let { f i } i ∈ I be a finite family of THFL such that f i ∈ T R for all i ∈ I , then there existsa NTHFA M such that f M = U i ∈ I f i . Proof:
Using induction on I and the theorem 5. ⊓⊔ The next result presents a characterization of the languages computed by NTHFA, i.e., the nextresult establishes the sufficient and necessary conditions for a THFL f belongs to T R . Theorem 6.
Let f : Σ ∗ → H be a THFL. Then the following statements are equivalent. ( i ) f ∈ T R . ( ii ) R f is finite and for each k ∈ R f the set S kf is a regular language. aldigleis, Benjam´ın & Regivan / On THFL and Automata Proof: ( i ) ⇒ ( ii ) Suppose that f : Σ ∗ → H belongs to T R , thus there exists a NTHFA M = h Q, Σ , ψ, q , F i such that f = f M , i.e. for all w ∈ Σ ∗ we have, f ( w ) = G q n ∈ Q (cid:16) b ψ ( q , w, q n ) ⊗ F ( q n ) (cid:17) . Nevertheless, by definition of M we have that, Q × Σ × Q is finite. Therefore, R ψ and R F are finitesets. But by Remark 3 we have that R b ψ is finite. Hence, the set R f is finite. Now for each k ∈ R f wedefine an NFA A k = h Q, Σ , q , δ k , F k i where: δ k ( q, a ) = p ⇐⇒ k ⊑ ψ ( q, a, p ) (17)and q ∈ F k ⇐⇒ k ⊑ F ( q ) . (18)with q, p ∈ Q and a ∈ Σ . Now we have that for any w ∈ Σ ∗ , w ∈ L ( A k ) ⇐⇒ b δ k ( q , w ) ∈ F k ⇐⇒ ∃ q , · · · , q n − , q n ∈ Q , such that q n ∈ F k and δ k ( q , a ) = q , · · · , δ k ( q n − , a n ) = q nEq. ( ) , ( ) ⇐⇒ ∃ q , · · · , q n − , q n ∈ Q , such that k ⊑ F ( q n ) and k ⊑ ψ ( q , a , q ) , · · · , k ⊑ ψ ( q n − , a n , q n ) ⇐⇒ ∃ q n ∈ Q , such that k ⊑ F ( q n ) and k ⊑ b ψ ( q , w, q n ) ( R ⇐⇒ ∃ q n ∈ Q , such that k ⊑ (cid:16) F ( q n ) ⊗ b ψ ( q , w, q n ) (cid:17) T heo. ⇐⇒ k ⊑ G q n ∈ Q (cid:16) b ψ ( q , w, q n ) ⊗ F ( q n ) (cid:17) ⇐⇒ k ⊑ f ( w ) ⇐⇒ w ∈ S kf Hence, L ( A k ) = S k L . Since A k is a NFA, by Theorem 2 we have that S k L is a regular language. ( ii ) ⇒ ( i ) Assume that R f = { k , · · · , k n } for some n ∈ Z ∗ + and that for each k ∈ R f theset S k L is a regular language. Hence, by Theorem 1 there exists a finite family of DFA { A k } k ∈ R f ,where A k = h Q k , Σ , δ k , q k , F k i such that L ( A k ) = S kf . Now for each k ∈ R f define an NTHFA M k = h Q k , Σ , ψ k , q k , F k i such that for each q, q ′ ∈ Q k and a ∈ Σ : ψ k ( q, a, q ′ ) = ( { } , if δ k ( q, a ) = q ′ { } , else (19)and F k ( q ) = ( k, if q ∈ F k { } , else (20) Valdigleis, Benjam´ın & Regivan / On THFL and Automata
By the construction above we have that for all w ∈ Σ ∗ , w ∈ L ( A k ) ⇒ f M k ( w ) = k and w / ∈ L ( A k ) ⇒ f M k ( w ) = { } so by definition f M k ∈ T R . Since R f is finite there exists a finite family { f M k } k ∈ R f and, by corollary1, there exists an NHTFA M such that, f M = ] k ∈ R f f M k . Moreover, it is clear that f M = f , therefore, f ∈ T R . ⊓⊔ But by Theorem above, it is possible to conclude the following result.
Corollary 2.
There exists THFL f : Σ ∗ → H such that f / ∈ T R . Proof:
The THFL f : Σ ∗ → H , defined by: f ( w ) = | w | [ i =0 n i + 1) o is such that R f is infinite. Hence, by Theorem 6 we have that f / ∈ T R . ⊓⊔ The characterization present by Theorem 6 and the Corollary 2 induce the inclusion described infigure 1. TT R Figure 1. Inclusion between T and T R . aldigleis, Benjam´ın & Regivan / On THFL and Automata Theorem 7.
The set T R is nondenumerable. Proof:
For each x ∈ [0 , define a THFA M x = h{ q x , q x } , Σ , ψ, q x , F x i such that for all q, q ′ ∈ { q x , q x } and a ∈ Σ we have: ψ ( q, a, q ′ ) = { x } and F x ( q ) = { x } now for each w ∈ Σ ∗ it is clear that f M x ( w ) = { x } . Now define the set Θ = { f M x | x ∈ [0 , } ,clearly Θ ⊂ T R , moreover, there exists a bijection from Θ into [0 , , so Θ is nondenumerable.Therefore, T R is nondenumerable. ⊓⊔
4. Crisp Typical Hesitant Fuzzy Automata
In this section, the paper is showing that the existence of a THFS of transitions in the definition ofNTHFA is not essential to compute a THFL. For this, we will introduce below a new class of typicalhesitant fuzzy automata.
Definition 17.
A Crisp Nondeterministic Typical Hesitant Fuzzy Automaton, or simply CNTHFA, isa quintuple N = h Q, Σ , δ N , q , F i , where Q, Σ , q o , F is equal to definition 13 and δ N : Q × Σ → Q is equal to definition 4. Definition 18.
A CNTHFA M computes the THFL f M : Σ ∗ → H define by, f M ( w ) = G q ∈ c δ N ( q ,w ) F ( q ) (21)First, the research show that any THFL that an CNTHFA computes, also is computed by a NTHFA. Theorem 8.
Let N be an CNTHFA, then there exists an NTHFA M such that f N = f M . Proof:
Given a CNTHFA N = h Q, Σ , δ N , q , F i , define a new NTHFA M = h Q, Σ , ψ, q , F i where: ψ ( q, a, q ′ ) = ( { } , if q ′ ∈ δ N ( q, a ) { } , else (22)for all q, q ′ ∈ Q and a ∈ Σ . Now For any w ∈ Σ ∗ it is clear that f N ( w ) = f M ( w ) . Hence, f N = f M . ⊓⊔ On the other hand, the computational power of NTHFA is equivalent to the power of CNTHFA. Valdigleis, Benjam´ın & Regivan / On THFL and Automata
Theorem 9.
Let M be an NTHFA, then there exist a CNTHFA N such that N has one more statethan M and f M = f N . Proof:
Without loss of generality, by the proof of item ( ii ) in Theorem 6 assume that M = h Q, Σ , ψ, q , F i is a NTHFA with the restriction ψ ( q, a, q ′ ) = { } or ψ ( q, a, q ′ ) = { } for all q, q ′ ∈ Q and a ∈ Σ .Now define the following CNTHFA N = h Q ∪ { q ℵ } , Σ , δ N , q , F N i where q ℵ / ∈ Q , therefore, N hasone more state than M , moreover, δ N is defined by the rules: ( r If ψ ( q, a, q ′ ) = { } , then q ′ ∈ δ N ( q, a ) for any q, q ′ ∈ Q and a ∈ Σ . ( r If ψ ( q, a, q ′ ) = { } , then q ℵ ∈ δ N ( q, a ) for any q, q ′ ∈ Q and a ∈ Σ . ( r δ N ( q ℵ , a ) = { q ℵ } for all a ∈ Σ .Finally define F N as being: F N ( q ) = ( F ( q ) , if q ∈ Q { } , else (23)By this construction, it is easy to see that f M ( w ) = f N ( w ) for all w ∈ Σ ∗ , completing the proof. ⊓⊔ The above theorem shows that CNTHFA needs an extra state to compute the same language as anNTHFA. These results together present a new way to characterize the set T R . Corollary 3.
Let f : Σ ∗ → H be a THFL. We have that f ∈ T R if and only if there exists CNTHFA N such that f = f N . Proof:
Straightforward by theorems 8 and 9. ⊓⊔ Definition 19.
A Crisp Deterministic Typical Hesitant Fuzzy Automaton, or simply CDTHFA, is aquintuple D = h Q, Σ , δ, q , F i , where Q, Σ , q o , F is equal to definition 13 and δ : Q × Σ → Q is equalto definition 1. Let D be a CDTHFA, the THFL computed by D is exactly the THFL f D : Σ ∗ → H define as: f D ( w ) = F ( b δ ( q , w )) (24) Theorem 10.
For all CDTHFA D there exists a CNTHFA N such that f D = f N . Proof:
Is obvious, since CDTHFA can be seen as a special instance of CNTHFA with δ ( q, a ) ≤ for all ( q, a ) ∈ Q × Σ . ⊓⊔ Theorem 11. if f ∈ T R , then there exists a CDTHFA D such that f = f D . aldigleis, Benjam´ın & Regivan / On THFL and Automata Proof:
Suppose that f ∈ T R , so there exists a CNTHFA N = h Q, Σ , δ N , q , F i such that f = f N , now it issufficient to construct the CDTHFA D = h Q , Σ , δ, { q } , F D i where for all ( X, a ) ∈ Q × Σ we havethat: δ ( X, a ) = [ q ∈ X δ N ( q, a ) (25)and F D ( X ) = G q ∈ X F ( q ) , if X = ∅{ } , else (26)moreover, it is not difficult to verify that b δ ( { q } , w ) = c δ N ( q , w ) for all w ∈ Σ ∗ . Hence, we have that, f D ( w ) = F D ( b δ ( { q } , w ))= G q ∈ b δ ( { q } ,w ) F ( q )= G q ∈ c δ N ( q ,w ) F ( q )= f N ( w )= f ( w ) completing the proof. ⊓⊔ The Theorem 11 shows that nondeterminism is not essential to compute THFL. Moreover, thislemma said that questions about T R elements could be seen as questions about CDTHFA. Definition 20.
Let f : Σ ∗ → H and f : Σ ∗ → H two be THFL the H -intersection of f and f ,denoted by f ⋓ f , is the THFL define by: f ⋓ f ( w ) = f ( w ) ⊗ f ( w ) (27) Theorem 12. If f , f ∈ T R on the same alphabet Σ , then f ⋓ f ∈ T R . Proof:
Suppose that f , f ∈ T R so by Theorem 11 there exists two CDTHFAs D = h Q , Σ , δ , q , F i and D = h Q , Σ , δ , p , F i such that f = f D and f = f D . Now without loss of generality assumethat Q ∩ Q = ∅ , then define D = h Q × Q , Σ , δ, ( q , p ) , F i where: δ (( q, p ) , a ) = ( δ ( q, a ) , δ ( p, a )) (28)with ( q, p ) ∈ Q × Q , a ∈ Σ and F (( q, p )) = F ( q ) ⊗ F ( p ) (29) Valdigleis, Benjam´ın & Regivan / On THFL and Automata it is easy to see that D is a CDTHFA and that for all ( q, p ) ∈ Q × Q and w ∈ Σ ∗ : b δ (( q, p ) , w ) = ( b δ ( q, w ) , b δ ( p, w )) (30)hence for all w ∈ Σ ∗ , f D ( w ) = F ( b δ (( q , p ) , w ) Eq. ( ) = F (( b δ ( q , w ) , b δ ( p , w ))) Eq. ( ) = F ( b δ ( q , w )) ⊗ F ( b δ ( p , w ))= f ( w ) ⊗ f ( w )= f ⋓ f ( w ) since D is a CDTHFA, by Theorems 10 and 11 f ⋓ f ∈ T R . ⊓⊔ The above theorem result shows that the H -intersection is a closure for the set T R and this resultcan be generalized as follows. Corollary 4.
Let { f i } i ∈ I be a finite family of THFL such that f i ∈ T R for all i ∈ I , then there existsa CDTHFA D such that f D = \ \ i ∈ I f i . Proof:
Using induction on I and the theorem 12. ⊓⊔
5. Conclusions
This paper presents a characterization for the languages computed by nondeterministic typical hesitantfuzzy automata, i.e., here we prove the sufficient and necessary conditions for that a typical nondeter-ministic hesitant fuzzy automaton compute a typical hesitant fuzzy language. Besides, we show thattypical hesitant fuzzy transitions and also that nondeterminism are not attributes necessary to com-pute typical hesitant fuzzy languages. This result corrects the previous result presented in [23], whichpresented the nondeterministic typical fuzzy automata as not equivalent to the deterministic counter-part. This paper also presents the initial results about closure operators for the class of typical hesitantfuzzy languages computed by typical nondeterministic hesitant fuzzy automata. Here we prove thatthe H -union and the H -intersection are both closed for this class of languages. It is intended in futurework to study the process of approximating typical hesitant fuzzy automata based on the idea of totaladmissible orders [36] and also based on partial order ⊑ introduced in this paper. References [1] Zadeh LA. Fuzzy sets.
Information and Control , 1965. (3):338–353. aldigleis, Benjam´ın & Regivan / On THFL and Automata [2] Farias ADS, Lopes LRA, Bedregal B, Santiago RHN. Closure properties for fuzzy recursively enumerablelanguages and fuzzy recursive languages. Journal of Intelligent & Fuzzy Systems , 2016. (3):1795–1806.[3] Bedregal BC, Figueira S. On the computing power of fuzzy Turing machines. Fuzzy Sets and Systems ,2008. (9):1072–1083.[4] Hopcroft JE, Motwani R, Ullman JD. Automata theory, languages, and computation.
International Edi-tion , 2006. :19.[5] Farias ADS, Costa VS, Santiago RH, Bedregal B. A residuated function in a class of Mealy type L-valuedfinite automaton. In: Annual Conference of the North American Fuzzy Information Processing Society -NAFIPS. 2017 pp. 1–6. doi:10.1109/NAFIPS.2016.7851592.[6] Aho AV. Compilers: principles, techniques and tools (for Anna University), 2/e. Pearson Education India,2003.[7] Stanimirovi´c S, ´Ciri´c M, Ignjatovi´c J. Determinization of fuzzy automata by factorizations of fuzzy statesand right invariant fuzzy quasi-orders. Information Sciences , 2018. :79–100.[8] Wei X, Li Y. Fuzzy alternating automata over distributive lattices.
Information Sciences , 2018. :34–47.[9] Mordeson JN, Malik DS. Fuzzy automata and languages: theory and applications. CRC Press, 2002.[10] Mizumoto M, Toyoda J, Tanaka K. Some considerations on fuzzy automata.
Journal of Computer andSystem Sciences , 1969. (4):409–422.[11] Costa V, Bedregal B. Fuzzy linear automata and some equivalences. Tendˆencias em Matem´atica Aplicadae Computacional , 2018. (1):127–145.[12] Rabin MO. Probabilistic automata. Information and Control , 1963. (3):230–245.[13] Paz A. Introduction to probabilistic automata. Academic Press, 2014.[14] Abney S, McAllester D, Pereira F. Relating probabilistic grammars and automata. In: Proceedings of the37th Annual Meeting of the Association for Computational Linguistics. 1999 pp. 542–549.[15] Ying M. Automata theory based on quantum logic II. International Journal of Theoretical Physics , 2000. (11):2545–2557.[16] Moore C, Crutchfield JP. Quantum automata and quantum grammars. Theoretical Computer Science ,2000. (1-2):275–306.[17] Qiu D. Automata theory based on quantum logic: some characterizations.
Information and Computation ,2004. (2):179–195.[18] Qiu D. Automata theory based on complete residuated lattice-valued logic.
Science in China Series:Information Sciences , 2001. (6):419–429.[19] Hirvensalo M. Quantum automata theory - A review. Lecture Notes in Computer Science (includingsubseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) , 2011. :146–167. doi:10.1007/978-3-642-24897-9 7.[20] Ravi K, Alka C. Interval-valued fuzzy regular language.
Journal of Applied Mathematics & Informatics ,2010. (3 4):639–649.[21] Ravi K, Choubey A. Myhill-Nerode Theorem for Interval-valued Fufzzy Regular Lanugage. In: AIPConference Proceedings, volume 1324 (1). American Institute of Physics, Chandigarh, 2010 pp. 30–33. Valdigleis, Benjam´ın & Regivan / On THFL and Automata [22] Choubey A, Ravi K. Intuitionistic fuzzy automata and intuitionistic fuzzy regular expressions.
J. Appl.Math. & Informatics , 2009. (1-2):409–417.[23] Costa VS, Bedregal B. On typical hesitant fuzzy automata. Soft Computing , 2020. (12):8725–8736.[24] Torra V. Hesitant fuzzy sets. International Journal of Intelligent Systems , 2010. :529–539.[25] Bedregal B, Reiser R, Bustince H, Lopez-Molina C, Torra V. Aggregation functions for typical hesitantfuzzy elements and the action of automorphisms. Information Sciences , 2014. :82–99.[26] Levelt WJ. An introduction to the theory of formal languages and automata. John Benjamins Publishing,2008.[27] Eilenberg S. Automata, languages, and machines. Academic press, 1974.[28] Matzenauer ML, Reiser R, Santos H, Bedregal B. Typical hesitant fuzzy sets: Evaluating strategies inGDM applying consensus measures. In: 2019 Conference of the International Fuzzy Systems Associationand the European Society for Fuzzy Logic and Technology (EUSFLAT 2019). Atlantis Press. ISBN 978-94-6252-770-6, 2019/08 pp. 438–445.[29] Xia M, Xu Z. Hesitant fuzzy information aggregation in decision making.
International Journal ofApproximate Reasoning , 2011. (3):395–407.[30] Rodr´ıguez RM, Bedregal B, Bustince H, Dong Y, Farhadinia B, Kahraman C, Mart´ınez L, Torra V, Xu Y,Xu Z, et al. A position and perspective analysis of hesitant fuzzy sets on information fusion in decisionmaking. Towards high quality progress. Information Fusion , 2016. :89–97.[31] Garmendia L, Campo RG, Recasens J. Partial orderings for hesitant fuzzy sets. International Journal ofApproximate Reasoning , 2017. :159–167.[32] Santos H, Bedregal B, Santiago R, Bustince H, Barrenechea E. Construction of typical hesitant triangularnorms regarding Xu-Xia-partial order. In: 2015 Conference of the International Fuzzy Systems Associ-ation and the European Society for Fuzzy Logic and Technology (IFSA-EUSFLAT-15). Atlantis Press,2015 pp. 953–959.[33] Bedregal B, Santiago RH, Bustince H, Paternain D, Reiser R. Typical hesitant fuzzy negations. Interna-tional Journal of Intelligent Systems , 2014. (6):525–543.[34] Gr¨atzer G. Lattice theory: foundation. Springer Science & Business Media, 2011.[35] Xu Z, Xia M. Distance and similarity measures for hesitant fuzzy sets. Information Sciences , 2011. (11):2128–2138.[36] Matzenauer M, Reiser R, Santos H, Bedregal B, Bustince H. Strategies on admissible total orders overtypical hesitant fuzzy implications applied to decision making problems.
Int. J. Intell. Sys. (In Press) ,2021. doi:10.1002/int.22374.[37] Huang L. Advanced dynamic programming in semiring and hypergraph frameworks. In: Coling 2008:Advanced Dynamic Programming in Computational Linguistics: Theory, Algorithms and Applications-Tutorial notes. 2008 pp. 1–18.[38] Zeng W, Li D, Yin Q. Distance and similarity measures between hesitant fuzzy sets and their applicationin pattern recognition.
Pattern Recognition Letters , 2016. :267–271. doi:10.1016/j.patrec.2016.11.001.URL http://dx.doi.org/10.1016/j.patrec.2016.11.001 .[39] Hamers L, et al. Similarity measures in scientometric research: The Jaccard index versus Salton’s cosineformula. Information Processing and Management , 1989. (3):315–18. aldigleis, Benjam´ın & Regivan / On THFL and Automata [40] Rabin MO, Scott D. Finite automata and their decision problems. IBM Journal of Research and Develop-ment , 1959. (2):114–125.[41] Moore EF. Gedanken-experiments on sequential machines. Automata studies , 1956. :129–153.[42] Mealy GH. A method for synthesizing sequential circuits. The Bell System Technical Journal , 1955. (5):1045–1079.[43] McCulloch WS, Pitts W. A logical calculus of the ideas immanent in nervous activity. The Bulletin ofMathematical Biophysics , 1943. (4):115–133.[44] Kleene SC. Representation of events in nerve nets and finite automata. Technical report, RAND PROJECTAIR FORCE SANTA MONICA CA, 1951.[45] Zhou NL, Hu BQ. Axiomatic approaches to rough approximation operators on complete completelydistributive lattices. Information Sciences , 2016.348