Recognizability of languages via deterministic finite automata with values on a monoid: General Myhill-Nerode Theorem
aa r X i v : . [ c s . F L ] F e b Recognizability of languages via deterministic finiteautomata with values on a monoid:General Myhill-Nerode Theorem
Jos´e R. Gonz´alez de Mend´ıvil, Federico Fari˜na Figueredo
Departamento de Estad´ıstica, Inform´atica y Matem´aticasUniversidad P´ublica de Navarra31006 Pamplona (Spain)
Abstract
This paper deals with the problem of recognizability of functions ℓ : Σ ∗ → M that map words to values in the support set M of a monoid ( M, · , M -languages. M -languages are studied from the aspectof their recognition by deterministic finite automata whose components takevalues on M ( M -DFAs). The characterization of an M -language ℓ is basedon providing a right congruence on Σ ∗ that is defined through ℓ and a factor-ization on the set of all M -languages, L (Σ ∗ , M ) (in short L ). A factorization on L is a pair of functions ( g, f ) such that, for each ℓ ∈ L , g ( ℓ ) · f ( ℓ ) = ℓ ,where g ( ℓ ) ∈ M and f ( ℓ ) ∈ L . In essence, a factorization is a form of com-mon factor extraction. In this way, a general Myhill-Nerode theorem , whichis valid for any L (Σ ∗ , M ), is provided. Basically, ℓ ∈ L is recognized by an M -DFA if and only if there exists a factorization on L , ( g, f ) , such that theright congruence on Σ ∗ induced by the factorization ( g, f ) and f ( ℓ ) ∈ L , hasfinite index . This paper shows that the existence of M -DFAs guaranteesthe existence of natural non-trivial factorizations on L without taking ac-count any additional property on the monoid. In addition, the compositionof factorizations is also a new factorization, and the composition of naturalfactorizations preserves the recognition capability of each individual naturalfactorization. Keywords:
Recognizability, languages, Deterministic finite automata,Myhill-Nerode Theorem, monoid, factorization, right congruence.
Email addresses: [email protected] (Jos´e R. Gonz´alez de Mend´ıvil), [email protected] (Federico Fari˜na Figueredo)
Preprint submitted to Information Sciences February 12, 2021 . Introduction
In formal languages and automata [20], the
Myhill-Nerode Theorem [31][32] provides necessary and sufficient conditions for a language to be recog-nized by a deterministic finite automaton (DFA). In that case, the languageis said to be regular . Specifically, given a language ℓ ⊆ Σ ∗ (where Σ ∗ denotesthe set of all finite words on an alphabet Σ) a right congruence relation ≡ ℓ on Σ ∗ is defined in terms of the language ℓ , but with no regard to its rep-resentation. The recognizability of a language ℓ by a DFA is established byproving that: ℓ is a regular language if and only if ≡ ℓ has finite index . It isworth of mention that the states of the DFA based on ≡ ℓ , that recognizes ℓ , are the equivalence classes of that congruence. Furthermore, this DFA isminimal what makes Myhill-Nerode theorem to be considered in the topicof automata minimization.In order to cope with different domains of practical applications, re-searchers have presented in the literature effective generalizations of au-tomata using a wide diversity of algebraic structures. Fuzzy automata and weighted automata are ones of the best-known studied generalizations of au-tomata [30][5]. For weighted automata, values on the transitions of thoseautomata are usually taken from semirings [28], hemirings [6], or strong bi-monoids [4][7]. For fuzzy automata, values on transitions are taken fromcertain ordered structures like lattice-ordered monoids [25], lattice-orderedstructures [27], complete distributive lattices [1], general lattices [27], orcomplete residuated lattices [21][34][35].
Weighted deterministic finite state automata are called (sub)sequentialtransducers [8][37]. These transducers recognize languages called (sub) se-quential rational functions . Characterization of the (sub)sequential rationalfunctions in terms of a congruence relation in the flavour of Myhill-Nerodetheorem has been studied for different special cases of monoids like freemonoids [37], ( R +0 , + ,
0) [29], gcd monoids [39], or monoids based on sequen-tiable structures [10].It is remarkable that Gerdjikov has provided five algebraic axioms [11][12],based on the relation divisor of on the support set M of the monoid,for characterizing a wide class of monoids. Those axioms are satisfied bygroups, free monoids, sequentiable structures, tropical monoids (including Given a monoid ( M, · , a, b ∈ M , a is divisor of b if there is c ∈ M such that b = a · c . Q +0 , + , left cancellation and right cancellation considered in [12] avoid the existence of a zero element in M . In fuzzy languages and automata [42], Ignjatovi´c et al. [22] have pro-posed a Myhill-Nerode type theory for fuzzy languages with membershipvalues in an arbitrary set with two distinguished elements 0 and 1, ( M, , crisp ) languages in consideration. Thesefuzzy languages are studied in [22] from the aspect of their recognition by crisp determinist fuzzy automata . A crisp deterministic fuzzy automaton issimply an ordinary deterministic automaton equipped with a fuzzy subsetof final states [1]. It is worth of mention, that the Myhill-Nerode theoremfor fuzzy languages provided by Ignjatovi´c [22] includes the recognizabil-ity of fuzzy languages with membership values on some well-known struc-tures: G¨odel structure [38] [30](Chap.7) [33]; distributive lattices [36]; finitemonoids [2]; and general lattices [27]. However, all these previous works infuzzy languages have a severe limitation: they only consider recognizabilityof fuzzy languages of finite rank . In fact, a fuzzy language is recognized bya crisp deterministic fuzzy automaton if and only if it has a finite rank andall its kernel languages are recognizable (see Theorem 4.3 in [22]).In order to circumvent this restriction, a generalization of Myhill-Nerodetheorem for fuzzy languages, having finite or infinite rank, is introduced in[16] for fuzzy languages based on continuous triangular norms (t-norms)[24],([0 , , ⊗ , factorization , in par-ticular maximal factorization .In essence, a factorization is a form of common factor extraction. Specif-ically, a factorization is a pair of functions ( g, f ) such that they satisfy ℓ = g ( ℓ ) ⊗ f ( ℓ ) for any fuzzy language ℓ . This notion was initially intro-duced in weighted automata by Kirsten and M¨aurer [23] and applied to thearea of fuzzy automata in order to develop efficient constructions for de-terminization and minimization of fuzzy automata. In fact, factorizationshave allowed researchers to provide minimization algorithms [28][9] and de-terminization methods for weighted automata [23]. In the context of fuzzyautomata, factorizations have been studied to obtain determinization meth-ods [14][15][40], and minimization algorithms for fuzzy automata [17][18][41].Although [16] deals with characterizing fuzzy languages with infiniterank, that work is somewhat restrictive because it assumes zero-divisor-free t-norm based monoids and maximal factorizations. In general, thatproperty on the monoid is a necessary condition for the existence of maximalfactorizations as it has been recently proved in [13].3he motivation to write this paper is to study the recognizability offunctions ℓ : Σ ∗ → M defined for any arbitrary monoid ( M, · , ℓ : Σ ∗ → M is atotal function. We call these functions M -languages. As in previous works,recognizability is based on deterministic finite automata, but in this case,their components take values on M ( M -DFAs). Let L (Σ ∗ , M ) be the setof all M -languages. Our main objective is to provide a general Myhill-Nerode theorem valid for any L (Σ ∗ , M ). We use factorizations on L (Σ ∗ , M )in order to get a characterization of M -languages. The characterization ofan M -language ℓ ∈ L (Σ ∗ , M ) is based on a right congruence on Σ ∗ inducedby a factorization on L (Σ ∗ , M ), ( g, f ), and ℓ . That congruence is denotedby ≡ ( g,f ) ℓ in the paper.The general Myhill-Nerode theorem presented in this paper establishesthat ℓ ∈ L is recognized by an M -DFA if and only if there exists a factor-ization on L , ( g, f ) , such that the right congruence on Σ ∗ induced by thefactorization ( g, f ) and f ( ℓ ) ∈ L , has finite index .The proof is made by construction. In this way, we prove that if ≡ ( g,f ) f ( ℓ ) has finite index then there exists an M -DFA that recognizes the M -language f ( ℓ ), and, as ( g, f ) is a factorization, g ( ℓ ) · f ( ℓ ) = ℓ , which implies that ℓ is also recognized by such an M -DFA. In the other direction, if an M -language ℓ is recognized by some M -DFA, A , then there exists a naturalfactorization on L (Σ ∗ , M ) induced by A , ( g A , f A ), such that ≡ ( g A ,f A ) f A ( ℓ ) hasfinite index. The construction of a factorization induced by an M -DFA is nottrivial and requires the notion of transition-equalized automata. However,it is not necessary to construct explicitly such kind of M -DFAs to get theirfactorizations [19].Each possible factorization on L (Σ ∗ , M ) has its own recognition capa-bility . Thus, we study the recognition capability of three particular cases:trivial factorization, maximal factorizations and natural factorizations. Theformulation of factorizations on L (Σ ∗ , M ) allows us to define the compo-sition of factorizations to form new factorizations. In this way, we provethat the recognition capability of the composition of natural factorizationspreserves the recognition capability of each individual natural factorization.The rest of the paper is organized as follows. Section 2 and section 3present a formal framework to define the operations and main propertiesof factorizations. Section 4 is a short introduction to M -DFAs. In section4, we introduce the sufficient condition for recognizability of M -languagesbased on factorizations and the definition of right congruence based on fac-torizations. This section presents the main properties of the M -DFA basedon this kind of congruences. Section 6 is devoted to natural factorizations,i.e., the factorizations induced via M -DFAs. The properties of the M -DFAconstructed under a natural factorization are also included in this section.The general Myhill-Nerode theorem is proved in section 7. The recognitioncapability of a factorization is defined in section 8 and three cases of studyare considered: trivial factorization, maximal factorizations and composi-tion of natural factorizations. Finally, some concluding remarks end thepaper.
2. Preliminaries
Let f : A → B and f : B → C be two well defined functions. In thispaper, the composition of the functions f and f is denoted by f ◦ f anddefined by ( f ◦ f )( a ) = f ( f ( a )) for any a ∈ A . We will assume that everyfunction is a total function.Let ( M, · ,
1) be an arbitrary monoid where M , · and 1 represent thesupport set, the multiplication operation, and the identity element of themonoid respectively. In general, we identify each monoid with the name ofits support set.Let Σ be a finite alphabet of symbols. The set Σ ∗ denotes the set of allfinite words over Σ. We use ε to represent the empty word. Then, Σ ∗ isthe free monoid generated by Σ under the operation of concatenation. Letus consider functions from Σ ∗ to M , i.e., ℓ : Σ ∗ → M . The set L (Σ ∗ , M ) = { ℓ | ℓ : Σ ∗ → M } is the set containing all those kind of functions. Wedo not write (Σ ∗ , M ) when it is clear in the context of discussion. Thus, L (Σ ∗ , M ) is simply denoted by L . We consider that L represents the set ofall possible M -languages on Σ. In fact, if the monoid M is the elementalBoolean monoid then, any ℓ ∈ L may be interpreted as the characteristicfunction of an ordinary language, i.e., a subset of Σ ∗ .In the following, we extend the multiplication operation · of the monoid M to different contexts. The context for the symbol · will clarify its interpre-tation. Given ℓ ∈ L and m ∈ M , m · ℓ ∈ L is defined as ( m · ℓ )( γ ) = m · ℓ ( γ )for every γ ∈ Σ ∗ . Obviously, ( n · m ) · ℓ = n · ( m · ℓ ) for any n, m ∈ M . For anyfunction r : Σ ∗ → Σ ∗ (a word-transformation), any ℓ ∈ L , and any m ∈ M ,the next property holds: m · ( ℓ ◦ r ) = ( m · ℓ ) ◦ r (1)5et us consider functions of the form f : L → L and g : L → M . We definethe sets F ( L ) = { f | f : L → L } and G ( L, M ) = { g | g : L → M } whichare simply denoted by F and G respectively. In F , f e denotes the identityfunction, i.e., f e ( ℓ ) = ℓ for any ℓ ∈ L . In G , g e denotes the constant function g e ( ℓ ) = 1 for every ℓ ∈ L . Since composition of functions is associative then( F, ◦ , f e ) is a monoid. As the image of any function in G is a subset of M ,we also extend · in the following way: for any g, g ′ ∈ G , g · g ′ is defined as( g · g ′ )( ℓ ) = g ( ℓ ) · g ′ ( ℓ ) for any ℓ ∈ L . Thus, g · g ′ is a well defined functionof G . Clearly, ( G, · , g e ) is a monoid.Given the monoids G and F introduced above, we put our attention tothe cartesian product G × F and pairs of functions ( g, f ) ∈ G × F . We definethe binary operation • : G × F → F as follows: given g ∈ G and f ∈ F ,( g • f )( ℓ ) = g ( ℓ ) · f ( ℓ ) for any ℓ ∈ L . Thus, ( g • f ) is a well defined functionof F . Furthermore, for any g, g ′ ∈ G and f, f ′ ∈ F , it is simple to provethat, ( g · g ′ ) • f = g • ( g ′ • f ) (2)( g ◦ f ′ ) • ( f ◦ f ′ ) = ( g • f ) ◦ f ′ (3)We provide a product operation for elements of G × F . The binaryoperation ∗ : ( G × F ) → G × F is defined as follows:( g , f ) ∗ ( g , f ) = ( g · ( g ◦ f ) , f ◦ f ) (4)for any ( g , f ), ( g , f ) ∈ G × F . Obviously, ( g , f ) ∗ ( g , f ) ∈ G × F . Itis not difficult to prove that ∗ is associative. In addition, the pair ( g e , f e ) isthe identity element for ∗ . Therefore, ( G × F, ∗ , ( g e , f e )) is a monoid. Weabuse of the confidence of the reader by using the term composition insteadof ∗ when this operation is evident from its context of application.Given a finite family { ( g i , f i ) ∈ G × F } i :1 ..n with n ≥
1, the composition( g , f ) ∗ ( g , f ) ∗ ... ∗ ( g n , f n ), is denoted by ∗| ni =1 ( g i , f i ). By using (4)successively, the result can be expressed in the form: ∗| ni =1 ( g i , f i ) = ( n Y i =1 g i ◦ ( ◦| j = i − f j ) , ◦| i = n f i ) (5)where ◦| and Q represent the quantifiers of ◦ and · respectively. By conven-tion, ◦| ∅ ( . ) = f e and Q ∅ ( . ) = g e . Thus, (5) is also applicable when n = 0.In that case, ∗| ∅ ( f i , g i ) = ( f e , g e ). 6 xample 1. As an example of (5), let us consider n = 3 : ∗| i =1 ( g i , f i ) = ( g , f ) ∗ ( g , f ) ∗ ( g , f ) =(( g ◦ f e ) · ( g ◦ f ) · ( g ◦ ( f ◦ f )) , f ◦ f ◦ f ) =( g · ( g ◦ f ) · ( g ◦ ( f ◦ f )) , f ◦ f ◦ f ) (cid:3) In order to obtain a more compact notation, we will write equation (5)in the form [( g i , f i )] n = ([ g i ◦ [ f j ] i − ] n , [ f i ] n ) (6)where [( g i , f i )] n = ∗| ni =1 ( g i , f i ), [ f i ] n = ◦| i = n f i , and [ g i ◦ [ f j ] i − ] n = Q ni =1 g i ◦ ( ◦| j = i − f j ). By using this notation, we obtain, by (4), that[( g i , f i )] n +11 = [( g i , f i )] n ∗ ( g n +1 , f n +1 ) =([ g i ◦ [ f j ] i − ] n · ( g n +1 ◦ [ f j ] n ) , f n +1 ◦ [ f i ] n ) (7)for the composition of a family of n + 1 pairs of functions in G × F , with n ≥ Remark 1.
Let us observe that, by (6), [( g i , f i )] n +11 = ([ g i ◦ [ f j ] i − ] n +11 , [ f i ] n +1 ) .In some proofs, we are interested in the operation [ g i ◦ [ f j ] i − ] n +11 • [ f i ] n +1 . [ g i ◦ [ f j ] i − ] n +11 • [ f i ] n +1 = (by (7)) =([ g i ◦ [ f j ] i − ] n · ( g n +1 ◦ [ f j ] n )) • ( f n +1 ◦ [ f i ] n ) = (by (2)) =([ g i ◦ [ f j ] i − ] n ) • (( g n +1 ◦ [ f j ] n ) • ( f n +1 ◦ [ f i ] n )) = (by (3)) =([ g i ◦ [ f j ] i − ] n ) • (( g n +1 • f n +1 ) ◦ [ f i ] n ) In conclusion, [ g i ◦ [ f j ] i − ] n +11 • [ f i ] n +1 = [ g i ◦ [ f j ] i − ] n • (( g n +1 • f n +1 ) ◦ [ f i ] n ) (8) (cid:3) For each word-transformation r : Σ ∗ → Σ ∗ , we define the function ∂ r ∈ F , ∂ r : L → L , as ∂ r ( ℓ ) = ℓ ◦ r for any ℓ ∈ L . We say that ∂ r ( ℓ ) is the derivative of ℓ by the word-transformation r . Example 2.
Let us consider, for each α ∈ Σ ∗ , the transformation α : Σ ∗ → Σ ∗ defined by α ( γ ) = αγ for any γ ∈ Σ ∗ . Thus, the derivative ∂ α ( ℓ ) ,satisfies that ∂ α ( ℓ ) = ℓ ◦ α by the definition given above. Then, ∂ α ( ℓ )( γ ) = ℓ ( α ( γ )) = ℓ ( αγ ) for each word γ . Let us observe that ∂ α ( ℓ ) may be viewed s a generalization of the Brzozowski derivative of an ordinary language bya word [3]. In addition, ∂ β ◦ ∂ α = ∂ α ◦ β = ∂ αβ holds for any words α and β .Clearly, ∂ ε = f e . In the rest of this paper, ∂ α is simply denoted by ∂ α forany word α . (cid:3) Given a word-transformation r : Σ ∗ → Σ ∗ and a pair ( g, f ) ∈ G × F , thenext equation holds: g • ( ∂ r ◦ f ) = ∂ r ◦ ( g • f ) (9) Proof : For any ℓ ∈ L ,( g • ( ∂ r ◦ f ))( ℓ ) = g ( ℓ ) · ( ∂ r ( f ( ℓ ))) = g ( ℓ ) · ( f ( ℓ ) ◦ r ) = ( by (1) ) =( g ( ℓ ) · f ( ℓ )) ◦ r = ( g • f )( ℓ ) ◦ r = ∂ r (( g • f )( ℓ )) = ( ∂ r ◦ ( g • f ))( ℓ ) (cid:3)
3. Factorizations on L (Σ ∗ , M ) Let us observe that the identity element ( g e , f e ) ∈ G × F satisfies that g e • f e = f e since, for any ℓ ∈ L , ( g e • f e )( ℓ ) = g e ( ℓ ) · f e ( ℓ ) = 1 · ℓ = ℓ . Weassume the hypothesis that there exist other pairs ( g, f ) ∈ G × F with thesame property, i.e., ( g • f )( ℓ ) = g ( ℓ ) · f ( ℓ ) = ℓ for any ℓ ∈ L . In that case, g ( ℓ ) ∈ M divides each value ℓ ( α ) ∈ M for any word α , i.e., it is a commonfactor for ℓ . Thus, we may say that the pair ( g, f ) factorizes L . In general, Definition 1.
A pair of functions ( g, f ) ∈ G × F is a factorization on L if ( g, f ) satisfies that g • f = f e (10)The identity element ( g e , f e ) is called the trivial factorization on L . We willstudy the properties obtained from the definition of factorization withoutconsidering additional properties about the monoid or the functions involvedin the factorization. If ( g, f ) is a factorization on L then ∂ r = ∂ r ◦ ( g • f ) = g • ( ∂ r ◦ f ) (11) ∂ r = ( g • f ) ◦ ∂ r = ( g ◦ ∂ r ) • ( f ◦ ∂ r ) (12)for any word-transformation r : Σ ∗ → Σ ∗ . Those results are consequence of(9), (3), and Definition 1. Lemma 1.
Let { ( g i , f i ) ∈ G × F } i :1 ..n be an arbitrary finite family of n ≥ factorizations on L . The composition [( g i , f i )] n is a factorization on L . roof : By (6), [( g i , f i )] n = ([ g i ◦ [ f j ] i − ] n , [ f i ] n ). By induction on n :- Basis. if n = 0 then [( g i , f i )] ∅ = ( g e , f e ), the trivial factorization.- Hypothesis. Let us assume that [( g i , f i )] n is a factorization on L for somearbitrary n ≥
0, i.e., [ g i ◦ [ f j ] i − ] n • [ f i ] n = f e (Definition 1).- Induction Step. Let us consider the composition [( g i , f i )] n +11 where ( g n +1 , f n +1 )is a factorization on L . By (8), the fact that g n +1 • f n +1 = f e , and induc-tion Hypothesis, [ g i ◦ [ f j ] i − ] n +11 • [ f i ] n +1 = f e . Therefore, [( g i , f i )] n +11 is afactorization on L . (cid:3) Lemma 2.
Let { ( g ′ i , f ′ i ) ∈ G × F } i :1 ..n be a finite family of n ≥ factor-izations on L . Let { r i : Σ ∗ → Σ ∗ } i :1 ..n be a finite family of n ≥ word-transformations. For each i : 1 ..n , the pair ( g i , f i ) ∈ G × F is defined as ( g i , f i ) = ( g ′ i ◦ ∂ r i , f ′ i ◦ ∂ r i ) where ∂ r i is the derivative operator by the word-transformation r i . The composition of the family { ( g i , f i ) ∈ G × F } i :1 ..n , [( g i , f i )] n = ([ g i ◦ [ f j ] i − ] n , [ f i ] n ) , satisfies that [ g i ◦ [ f j ] i − ] n • [ f i ] n = ∂ ◦| ni =1 r i (13) Proof : Let us recall that ∂ r i ∈ F and ∂ r i ( ℓ ) = ℓ ◦ r i for any ℓ ∈ L . Byinduction on n :- Basis. if n = 0 then [( g i , f i )] ∅ = ( g e , f e ). By convention ◦| ∅ r i is the identityword-transformation. Thus, ∂ ◦| ∅ r i = f e . Therefore, g e • f e = f e since ( g e , f e )is the trivial factorization on L .- Hypothesis. Let us assume that (13) holds for an arbitrary n ≥ g i , f i )] n +11 where each( g i , f i ) is defined in Lemma 2. The new pair ( g n +1 , f n +1 ) is also definedas ( g n +1 , f n +1 ) = ( g ′ n +1 ◦ ∂ r n +1 , f ′ n +1 ◦ ∂ r n +1 ) for a given factorization on L ,( g ′ n +1 , f ′ n +1 ), and a word-transformation r n +1 .By definition of ( g n +1 , f n +1 ), g n +1 • f n +1 = ( g ′ n +1 ◦ ∂ r n +1 ) • ( f ′ n +1 ◦ ∂ r n +1 ).Then, by (12), g n +1 • f n +1 = ∂ r n +1 .By (8), and substitution,[ g i ◦ [ f j ] i − ] n +11 • [ f i ] n +1 = [ g i ◦ [ f j ] i − ] n • ( ∂ r n +1 ◦ [ f i ] n ).By (9), [ g i ◦ [ f j ] i − ] n +11 • [ f i ] n +1 = ∂ r n +1 ◦ ([ g i ◦ [ f j ] i − ] n • [ f i ] n ).By Hypothesis and definition of ∂ r n +1 ,[ g i ◦ [ f j ] i − ] n +11 • [ f i ] n +1 = ∂ r n +1 ◦ ∂ ◦| ni =1 r i = ∂ ◦| n +1 i =1 r i . Therefore, the Lemmaholds. (cid:3) Previous results have been provided for arbitrary factorizations, word-transformations and their derivatives. The importance of factorizations on9 and the equation (13) is that they provide a reasonable theoretical basisfor recognizability of M -languages via deterministic finite automata withvalues on a monoid M as we will show in the next sections.
4. Deterministic finite automata with values on a monoid
We present a short introduction to deterministic finite automata withvalues on a monoid.
Definition 2.
Let ( M, · , be a monoid. A Deterministic Finite Automa-ton with values on the monoid M , ( M -DFA in short), is a tuple A =( Q, Σ , u, i u , δ, w, ρ ) where • Q is a finite nonempty set of states; • Σ is a finite alphabet; • u ∈ Q is the unique initial state ; • i u ∈ M is the initial value assigned to the initial state u ∈ Q ; • δ : Q × Σ → Q is the state-transition function ; • w : Q × Σ → M is the monoid-transition function that assigns valuesfrom M to each transition; and • ρ : Q → M is the final-function that assigns final values from M toeach state in Q . The state-transition function δ is extended to Σ ∗ . The extended function δ ∗ : Q × Σ ∗ → Q is defined as(i) δ ∗ ( q, ε ) = q ; and, (ii) δ ∗ ( q, ασ ) = δ ( δ ∗ ( q, α ) , σ ) (14)for any q ∈ Q , α ∈ Σ ∗ , and σ ∈ Σ. As δ is a total function then δ ∗ ( q, α ) ∈ Q for every word α . Due this fact, it is common to say that A is complete . Inthe rest of this paper, δ ∗ ( q, α ) is simply denoted qα for each state q of A ,and word α . As A is a deterministic automaton, qα is the unique reachablestate from q by the word α . In other words, qα is an accessible state from q (by the word α ). An M -DFA A is accessible if Q = { uα | α ∈ Σ ∗ } , i.e., anystate in Q is accessible from the initial state.The monoid-transition function w is also extended to Σ ∗ . The extendedfunction w ∗ : Q × Σ ∗ → M is defined as(i) w ∗ ( q, ε ) = 1; and, (ii) w ∗ ( q, ασ ) = w ∗ ( q, α ) · w ( qα, σ ) (15)10or any q ∈ Q , α ∈ Σ ∗ , and σ ∈ Σ. Let us observe that the definition of w ∗ uses the extended function δ ∗ .Let α ∈ Σ ∗ be a word. The length of α is denoted by | α | . The k -thprefix of α is α [ k ] where 0 ≤ k ≤ | α | . By convention, α [0] = ε . In addition,the k -th symbol in α is denoted by α ( k ) where, by convention, α ( k ) = ε when k < k > | α | . Taken into account such a notation, by (15), we canexpand w ∗ ( q, α ) as follows: w ∗ ( q, α ) = w ( q, α (1)) · ...w ( qα [ i − , α ( i )) ... · w ( qα [ n − , α ( n ))= n Y i =1 w ( qα [ i − , α ( i )) (16)for any word α of length n ≥ q ∈ Q . It is also clear that, for any twowords α and β , and q ∈ Q , w ∗ ( q, αβ ) = w ∗ ( q, α ) · w ∗ ( qα, β ) (17)Given those previous definitions and extended functions, it is possible todefine the M -language recognized (or generated) by an M -DFA. Let A =( Q, Σ , u, i u , δ, w, ρ ) be an M -DFA. The M -language recognized by A , denoted A , A ∈ L , is defined by A ( α ) = i u · w ∗ ( u, α ) · ρ ( uα ) (18)for any α ∈ Σ ∗ . In addition, for each state q of A , the M -language A q ∈ L is defined by A q ( α ) = w ∗ ( q, α ) · ρ ( qα ) (19)for any α ∈ Σ ∗ . For each word α ∈ Σ ∗ and its derivative ∂ α ∈ F (seeExample 2), it is simple to prove that ∂ α ( A q ) = w ∗ ( q, α ) · A qα (20)for any q ∈ Q . Finally, ∂ α ( A ) = i u · ∂ α ( A u ). Definition 3. An M -language ℓ ∈ L , is a recognizable M -language, orsimply recognizable , if ℓ is recognized by some M -DFA. Two trivial consequences of Definition 3 are:(
Rcg1 ) If ℓ ∈ L is recognizable then, for any m ∈ M , m · ℓ isrecognizable; and( Rcg2 ) if ℓ ∈ L is recognizable then, there exists ℓ ′ ∈ L suchthat ℓ ′ is recognizable and ℓ = m · ℓ ′ for some m ∈ M .11f ℓ ∈ L is recognized by A = ( Q, Σ , u, i u , δ, w, ρ ), i.e., A = ℓ , then, by (18),( Rcg1 ) m · ℓ is recognized by the automaton A ′ = ( Q, Σ , u, m · i u , δ, w, ρ ).For ( Rcg2 ), let us consider A u = ( Q, Σ , u, , δ, w, ρ ). Then, by (18), ℓ ′ = A u and m = i u .Given two M -DFAs, A and B , we say that A is (language) equivalent to B when they recognize the same M -language, i.e., A = B .Among the equivalent automata to A , we may find a minimal one. Theminimal automaton has the minimal number of states for recognizing thesame M -language. Property 1.
Let A = ( Q, Σ , u, i u , δ, w, ρ ) be an M -DFA. If A is minimalthen A satisfies the next conditions: A is an accessible M -DFA. ( NcndS ) For all states p, q ∈ Q ,( ∃ m ∈ M : m · A p = A q ∨ A p = m · A q ) ⇒ p = q ( NcndW ) For all states p, q ∈ Q , A p = A q ⇒ p = q Proof : Let A = ( Q, Σ , u, i u , δ, w, ρ ) be a minimal M -DFA.1. If A is not accessible then there is an equivalent M -DFA A ′ with a lessernumber of states than A . A ′ is built by removing the unaccessible states of A . This is in contradiction with the initial hypothesis.2. Let us consider that for two states p = q , m · A p = A q for some m ∈ M .Let us define the M -DFA A ′ = ( Q, Σ , u, i u , δ ′ , w ′ , ρ ). A ′ is exactly equal to A excepting that, for any s ∈ Q and σ ∈ Σ, if δ ( s, σ ) = q then δ ′ ( s, σ ) = p and w ′ ( s, σ ) = w ( s, σ ) · m . That is, every transition ( s, σ ) ending in q in A ends in p in A ′ . It is easily to prove that A ′ is equivalent to A . However, in A ′ , q is not an accessible state. Removing q in A ′ , constructs an equivalentautomaton to A but with less states. Again, this is a contradiction.3. NcndW is consequence of
NcndS when m = 1. (cid:3) Those previous conditions are necessary conditions for an M -DFA to bea minimal one. Let us observe that ( NcndS ) is stronger than (
NcndW ).It is also possible to provide a sufficient condition for minimality.
Property 2.
Let A = ( Q, Σ , u, i u , δ, w, ρ ) be an accessible M -DFA, then ¬ ( ∃ α, β ∈ Σ ∗ , ℓ ∈ L, m, m ′ ∈ M : uα = uβ ∧ ∂ α ( A ) = m · ℓ ∧ ∂ β ( A ) = m ′ · ℓ ) ⇒ A is minimal (21)12 roof : As A is accessible, Q = { uα | α ∈ Σ ∗ } . Assume that k Q k = n .As A is a deterministic automaton, there exists n different words, α ... α n ,such that Q = { uα i | i : 1 ..n } . Suppose that A is not minimal. Then,there exists a minimal M -DFA A ′ = ( Q ′ , Σ , u ′ , i u ′ , δ ′ , w ′ , ρ ′ ) equivalent to A with k Q ′ k < k Q k . As A ′ is complete u ′ α i ∈ Q ′ for each i : 1 ..n . Bythe Pigeonhole principle, there are at least two different words α k and α k ′ ,with 1 ≤ k < k ′ ≤ n , such that u α k = u α k ′ and u ′ α k = u ′ α k ′ . Call α k and α k ′ by α and β respectively. Thus, ∂ α ( A ) = ∂ α ( A ′ ) and ∂ β ( A ) = ∂ β ( A ′ )since A is equivalent to A ′ . By (20), ∂ α ( A ′ ) = i u ′ · w ′∗ ( u ′ , α ) · A ′ u ′ α and ∂ β ( A ′ ) = i u ′ · w ′∗ ( u ′ , β ) · A ′ u ′ β . As u ′ α = u ′ β , then A ′ u ′ α = A ′ u ′ β = ℓ . Inconclusion, uα = uβ ∧ ∂ α ( A ) = m · ℓ ∧ ∂ β ( A ) = m ′ · ℓ . A contradictionhappens and the property holds. (cid:3) In general, we say that an M -DFA A = ( Q, Σ , u, i u , δ, ω, ρ ) is transition-equalized if for any p , q ∈ Q and σ , τ ∈ Σ, ∂ σ ( A q ) = ∂ τ ( A p ) ⇒ δ ( q, σ ) = δ ( p, τ ) ∧ w ( q, σ ) = w ( p, τ ) (22)This property has an important relation with the construction of factor-izations induced by an M -DFA (see section 6).
5. Recognizability via factorizations on L (Σ ∗ , M ) In this section, our main result states a sufficient condition for the rec-ognizability of M -languages via factorizations on L . We recall that, for aword α ∈ Σ ∗ , α ( i ) denotes the i -th symbol in α and α [ i ] denotes the i -thprefix of α . Given a factorization on L , ( g, f ), and a word α , let us definethe pairs ( g ◦ ∂ α ( i ) , f ◦ ∂ α ( i ) ) ∈ G × F with i : 1 .. | α | , and the composition[( g ◦ ∂ α ( i ) , f ◦ ∂ α ( i ) )] | α | . By (5) and the notation given in (6), that compositionreturns the pair ( W ( g,f ) α , S ( g,f ) α ) ∈ G × F , where S ( g,f ) α = [ f ◦ ∂ α ( i ) ] | α | (23) W ( g,f ) α = [ g ◦ ∂ α ( i ) ◦ S ( g,f ) α [ i − ] | α | (24)We write S α ( ℓ ) and W α ( ℓ ) instead of S ( g,f ) α and W ( g,f ) α in many part ofthe text when the factorization ( g, f ) is clear in the context of discussion.We recognize that it is not simple to provide an interpretations of thesefunctions. This is the reason for providing their main properties: S ε = f e and W ε = g e (25)13 σ = f ◦ ∂ σ and W σ = g ◦ ∂ σ (26) S β ◦ S α = S αβ (27) W α · ( W β ◦ S α ) = W αβ (28) W α • S α = ∂ α (29)( W β ◦ S α ) • ( S β ◦ S α ) = ∂ β ◦ S α (30)for any σ ∈ Σ, and α , β ∈ Σ ∗ .Equations (25), (26), (27) and (28) are directly obtained from the defi-nition of S α (23) and W α (24). Equation (29) is a consequence of the factthat ( W α , S α ) is an element of G × F that satisfies (13) in Lemma 2. Thelast equation (30) is obtained by (3) and the previous one (29).The main justification for introducing such a pair ( W α , S α ) is that, forany ℓ ∈ L and α ∈ Σ ∗ , ( W α • S α )( ℓ ) = ∂ α ( ℓ ) W α ( ℓ ) · S α ( ℓ ) = ℓ ◦ α ( W α ( ℓ ) · S α ( ℓ ))( ε ) = ( ℓ ◦ α )( ε ) W α ( ℓ ) · ( S α ( ℓ ))( ε ) = ℓ ( α )In conclusion, for any factorization on L , ( g, f ), ℓ ∈ L and α ∈ Σ ∗ : W ( g,f ) α ( ℓ ) · ( S ( g,f ) α ( ℓ ))( ε ) = ℓ ( α ) (31)This last equation suggests us a way for constructing the M -language ℓ takeninto account that, by (28), W ( g,f ) α ( ℓ ), with | α | = n , can be expanded as W α ( ℓ ) = W α (1) ( ℓ ) · ( W α (2) ◦ S α [1] )( ℓ ) · ... · ( W α ( n ) ◦ S α [ n − )( ℓ ) W α ( ℓ ) = Q | α | i =1 ( W α ( i ) ◦ S α [ i − )( ℓ ) (32)By (31) and the considered expansion, each value ℓ ( α ) ∈ M , is the productof | α | + 1 values from M for any factorization. The reader may compare thelanguage accepted by an M -DFA (18) with (31), and (16) with (32). Themain question is how to ensure that every ℓ ( α ) is finitely generated by someautomaton given a factorization. This question is solved in the followinglemma for recognizability. Lemma 3.
Let ℓ ∈ L be an M -language. If there exists a factorization on L , ( g, f ) , such that the set { S ( g,f ) α ( ℓ ) | α ∈ Σ ∗ } is finite, then ℓ is a recognizable M -language. roof : We construct an M -DFA that recognizes ℓ ∈ L . By assumption { S ( g,f ) α ( ℓ ) | α ∈ Σ ∗ } is finite. It is a nonempty set since ℓ is in that set.In the rest of the proof, we omit the superscript ( g, f ) excepting for somedefinitions. Let us consider the following automaton: Definition 4. N ( g,f ) ℓ = ( Q, Σ , S ε ( ℓ ) , , δ, w, ρ ) : • Q = { S α ( ℓ ) | α ∈ Σ ∗ } is the set of sates, each state is an M -language; • the initial state is S ε ( ℓ ) ∈ Q , by (25), S ε ( ℓ ) = ℓ ; • the initial value of S ε ( ℓ ) is ; • the state-transition function δ is defined as δ ( S α ( ℓ ) , σ ) = S ασ ( ℓ ) for any α ∈ Σ ∗ and σ ∈ Σ ; • the monoid-transition function w is defined as w ( S α ( ℓ ) , σ ) = ( W σ ◦ S α )( ℓ ) for any α ∈ Σ ∗ and σ ∈ Σ ; and • for each state, its final value is ρ ( S α ( ℓ )) = ( S α ( ℓ ))( ε ) for any α ∈ Σ ∗ . Both δ : Q × Σ → Q and w : Q × Σ → M are well defined functions.Given two words α and β , if S α ( ℓ ) = S β ( ℓ ) then S ασ ( ℓ ) = ( S σ ◦ S α )( ℓ ) =( S σ ◦ S β )( ℓ ) = S βσ ( ℓ ); and, trivially, ( W σ ◦ S α )( ℓ ) = ( W σ ◦ S β )( ℓ ). Therefore,the automaton N ( g,f ) ℓ is a well defined M -DFA since Q is a nonempty finiteset by the conditions of the lemma.It is simple to show that δ ∗ ( S α ( ℓ ) , β ) = S αβ ( ℓ ) for any words α and β .By notation, δ ∗ ( S α ( ℓ ) , β ) is represented as S α ( ℓ ) β (as in section 4). Thus, S α ( ℓ ) β = S αβ ( ℓ ).In the following, we prove that the M -language recognized by N ( g,f ) ℓ , innotation N ( g,f ) ℓ , satisfies N ( g,f ) ℓ = ℓ .For any α ∈ Σ ∗ :By (18), N ( g,f ) ℓ ( α ) = 1 · w ∗ ( S ε ( ℓ ) , α ) · ρ ( S ε ( ℓ ) α ).Since, S ε ( ℓ ) α = S α ( ℓ ), then N ( g,f ) ℓ ( α ) = w ∗ ( S ε ( ℓ ) , α ) · ρ ( S α ( ℓ )),By (16), N ( g,f ) ℓ ( α ) = ( Q | α | i =1 w ( S ε ( ℓ ) α [ i − , α ( i ))) · ρ ( S α ( ℓ )), and again, S ε ( ℓ ) α [ i −
1] = S α [ i − ( ℓ ), then, N ( g,f ) ℓ ( α ) = ( Q | α | i =1 w ( S α [ i − ( ℓ ) , α ( i ))) · ρ ( S α ( ℓ ))By definition of w () and ρ () given for the automaton N ( g,f ) ℓ ,15 ( g,f ) ℓ ( α ) = ( Q | α | i =1 ( W α ( i ) ◦ S α [ i − )( ℓ )) · ( S α ( ℓ ))( ε )By the considered expansion of W α ( ℓ ) given in (32), N ( g,f ) ℓ ( α ) = W α ( ℓ ) · ( S α ( ℓ ))( ε ).Finally, by (31), N ( g,f ) ℓ ( α ) = ℓ ( α ).Therefore, N ( g,f ) ℓ = ℓ . This fact concludes that ℓ is a recognizable M -language via the factorization ( g, f ). (cid:3) As Lemma 3 indicates, ℓ ∈ L is recognizable if the automaton N ( g,f ) ℓ isan M -DFA for some factorization on L , ( g, f ). In the construction providedin Definition 4, N ( g,f ) ℓ depends on ℓ and the factorization ( g, f ). The initialvalue of this automaton may be also changed without modifying the rest ofcomponents. In that case, we write N ( g,f ) ( ℓ,
1) for the original automatonand N ( g,f ) ( ℓ, m ) for the same automaton but with initial value m ∈ M .Let us observe that the finiteness property of the set of states Q ( g,f ) ℓ = { S α ( ℓ ) | α ∈ Σ ∗ } of N ( g,f ) ( ℓ, S α ( ℓ ) = S β ( ℓ ). This argument allows us to propose a right congruence on Σ ∗ . Definition 5.
Let ( g, f ) be a factorization on L and let ℓ be an M -language.The binary relation on Σ ∗ denoted by ≡ ( g,f ) ℓ is defined as follows: α ≡ ( g,f ) ℓ β ⇔ S ( g,f ) α ( ℓ ) = S ( g,f ) β ( ℓ ) (33) for any α , β ∈ Σ ∗ . Clearly, ≡ ( g,f ) ℓ is a congruence on Σ ∗ . In addition, ≡ ( g,f ) ℓ satisfies α ≡ ( g,f ) ℓ β ⇒ αγ ≡ ( g,f ) ℓ βγ (34)for any α , β , γ ∈ Σ ∗ . Proof : By definition of the congruence S ( g,f ) α ( ℓ ) = S ( g,f ) β ( ℓ ). Then, by (27), S αγ ( ℓ ) = ( S γ ◦ S α )( ℓ ) = ( S γ ◦ S β )( ℓ ) = S βγ ( ℓ ). Thus, αγ ≡ ( g,f ) ℓ βγ for every γ ∈ Σ ∗ . (cid:3) It concludes that ≡ ( g,f ) ℓ is a right congruence on Σ ∗ . Furthermore, ifthe quotient set Σ ∗ / ≡ ( g,f ) ℓ is finite, i.e., ≡ ( g,f ) ℓ has finite index , then the set Q ( g,f ) ℓ is also finite (and vice versa ): ≡ ( g,f ) ℓ has finite index ⇔ Q ( g,f ) ℓ = { S α ( ℓ ) | α ∈ Σ ∗ } is finite (35)16 orollary 1. Let ℓ ∈ L be an M -language. If there exists a factorization on L , ( g, f ) , such that the right congruence on Σ ∗ induced by the factorization, ≡ ( g,f ) ℓ , has finite index, then ℓ is a recognizable M -language.Proof : By (35), Q ( g,f ) ℓ is a finite set. Therefore, by Lemma 3, ℓ is recognizedby the M -DFA N ( g,f ) ( ℓ, (cid:3) We end this section by providing the main properties of the automaton N ( g,f ) ( ℓ, m ). Property 3.
Let ℓ ∈ L be an M -language. Let ( g, f ) be a factorization on L . If the automaton N ( g,f ) ( ℓ, is an M -DFA then, for any m ∈ M , the M -DFA N ( g,f ) ( ℓ, m ) satisfies the properties: N ( g,f ) ( ℓ, m ) = m · ℓ N ( g,f ) ( ℓ, m ) is an accessible M -DFA. Each state S α ( ℓ ) ∈ Q ( g,f ) ℓ is a recognizable M -language. For each state S α ( ℓ ) ∈ Q ( g,f ) ℓ , ( N ( g,f ) ( ℓ, m )) S α ( ℓ ) = S α ( ℓ ) . N ( g,f ) ( ℓ, m ) satisfies the necessary condition of minimality ( NcdW ).Proof :1. We recall that m ∈ M is the initial value in the automaton N ( g,f ) ( ℓ, m )that replaces the original initial value 1 in N ( g,f ) ( ℓ, Rcg1 ) (seesection 4), as N ( g,f ) ( ℓ,
1) is an M -DFA that recognizes ℓ (proof of Lemma3) then, N ( g,f ) ( ℓ, m ) is an M -DFA that recognizes m · ℓ ∈ L for any m ∈ M .2. By Definition 4 in Lemma 3, the set Q ( g,f ) ℓ = { S α ( ℓ ) | α ∈ Σ ∗ } containsall accessible states from the initial state S ε ( ℓ ) = ℓ (see the proof belowDefinition 4 in Lemma 3).3. As Q ( g,f ) ℓ is finite then, for any S α ( ℓ ) ∈ Q ( g,f ) ℓ , the set Q ( g,f ) S α ( ℓ ) , that isobtained by replacing ℓ by S α ( ℓ ), is also finite and; thus, each state S α ( ℓ ) isa recognizable M -language by Lemma 3. In fact, Q ( g,f ) S α ( ℓ ) ⊆ Q ( g,f ) ℓ because,by (27), for any word β , S αβ ( ℓ ) = ( S β ◦ S α )( ℓ ) = S β ( S α ( ℓ )) ∈ Q ( g,f ) ℓ .4. As S α ( ℓ ) is a state of N ( g,f ) ( ℓ, m ) then, by (19), ( N ( g,f ) ( ℓ, m )) S α ( ℓ ) isgiven by ( N ( g,f ) ( ℓ, m )) S α ( ℓ ) ( β ) = w ∗ ( S α ( ℓ ) , β ) · ρ ( S α ( ℓ ) β ) for any β ∈ Σ ∗ ,where w ∗ () and ρ () are the extended monoid-transition function and final17unction provided in (15) and Definition 4. Following a similar reasoning asin the proof of Lemma 3, we have that w ∗ ( S α ( ℓ ) , β ) = Q | β | i =1 ( W β ( i ) ◦ S β [ i − )( S α ( ℓ )) = W β ( S α ( ℓ )) = ( W β ◦ S α )( ℓ ),and ρ ( S α ( ℓ ) β ) = ρ ( S αβ ( ℓ )) = ( S αβ ( ℓ ))( ε ) = (( S β ◦ S α )( ℓ ))( ε )Thus, for any word β ,( N ( g,f ) ( ℓ, m )) S α ( ℓ ) ( β ) = ( W β ◦ S α )( ℓ ) · (( S β ◦ S α )( ℓ ))( ε ) =((( W β ◦ S α ) • ( S β ◦ S α ))( ℓ ))( ε ) = by (30)(( ∂ β ◦ S α )( ℓ ))( ε ) = ( ∂ β ( S α ( ℓ )))( ε ) = S α ( β )In conclusion, ( N ( g,f ) ( ℓ, m )) S α ( ℓ ) = S α ( ℓ ).5. The necessary condition for minimality ( NcndW ) (see Property 1.3) isfulfilled in the M -DFA N ( g,f ) ( ℓ, m ) because for all states S α ( ℓ ) and S β ( ℓ )of that automaton, if ( N ( g,f ) ( ℓ, m )) S α ( ℓ ) = ( N ( g,f ) ( ℓ, m )) S β ( ℓ ) then, by theprevious Property 3.4, S α ( ℓ ) = S β ( ℓ ). (cid:3)
6. Natural factorizations on L (Σ ∗ , M ) Recognizability of M -languages provided in section 5 has been based onthe hypothesis that, for arbitrary monoids, there exist general factorizationson L (Σ ∗ , M ). In this section, we show that each M -DFA is able to define anatural factorization on L . The construction of a factorization on L inducedby an M -DFA is based on equalization of transitions (see (22)).Let A = ( Q, Σ , u, i u , δ, w, ρ ) be an M -DFA (Definition 2). Recall that the M -language recognized by A is denoted by A (see (18)), and that A q (see19) is the M -language given for each state q of A .We define the set P A = { (( σ, q ) , ∂ σ ( A q )) , (( ε, u ) , A ) | σ ∈ Σ , q ∈ Q } and thefollowing relation on P A , denoted by ≈ A :(( σ, q ) , ℓ ) ≈ A (( τ, p ) , ℓ ′ ) ⇔ ℓ = ℓ ′ (36)for every (( σ, q ) , ℓ ), (( τ, p ) , ℓ ′ ) ∈ P A .By the given definition, ≈ A is an equivalence relation on P A . This equiv-alence relation has been constructed taken into account the antecedent ofthe implication given in (22), and it will allow us to define a factorization on L . Clearly, the quotient set P A / ≈ A contains a finite number of equivalenceclasses. Let π be a function for selecting a unique representative element foreach class in P A / ≈ A . Under a given selection π , each class C A ∈ P A / ≈ A ,18s denoted by C πA (( σ, q ) , ∂ σ ( A q )). This notation indicates explicitly the rep-resentative element of the class. The given selection π is defined is such away that it always provides the class C πA (( ε, u ) , A ). Definition 6.
Let A = ( Q, Σ , u, i u , δ, w, ρ ) be an M -DFA, let P A / ≈ A bethe quotient set of the equivalence relation ≈ A (36), and let π be a selectionfunction of representative elements for the classes in P A / ≈ A . The functions f πA : L → L and g πA : L → M are defined as follows:For each ℓ ∈ Lf πA ( ℓ ) = A u if ℓ = A ∧ C πA (( ε, u ) , A ) ∈ P A / ≈ A f πA ( ℓ ) = A qσ if ℓ = ∂ σ ( A q ) ∧ C πA (( σ, q ) , ∂ σ ( A q )) ∈ P A / ≈ A f πA ( ℓ ) = ℓ otherwise (37) g πA ( ℓ ) = i u if ℓ = A ∧ C πA (( ε, u ) , A ) ∈ P A / ≈ A g πA ( ℓ ) = w ( q, σ ) if ℓ = ∂ σ ( A q ) ∧ C πA (( σ, q ) , ∂ σ ( A q )) ∈ P A / ≈ A g πA ( ℓ ) = 1 otherwise (38)By the given definition, f πA and g πA are well defined functions of F and G respectively. We prove that such pair of functions is a factorization on L . Lemma 4.
Let A = ( Q, Σ , u, i u , δ, w, ρ ) be an M -DFA. The pair ( g πA , f πA ) ∈ G × F (Definition 6) is a factorization on L induced by the automaton A .Proof : For any ℓ ∈ L . By Definition 6, • ℓ = A . Then, g πA ( ℓ ) · f πA ( ℓ ) = i u · A u = A (by (19) and (18)). • ℓ = ∂ τ ( A p ) for some τ ∈ Σ and p ∈ Q , then there exists some class C πA ∈ P A / ≈ A such that ℓ = ∂ σ ( A q ) or ℓ = A . This last caseis the same as the given above. For the former case, assume that(( σ, q ) , ∂ σ ( A q )) is the given representative by π . Then, g πA ( ℓ ) · f πA ( ℓ ) = w ( q, σ ) · A qσ = ∂ σ ( A q ) = ℓ (by (20)). • Otherwise, g πA ( ℓ ) · f πA ( ℓ ) = 1 · ℓ .Therefore, g πA • f πA = f e . By Definition 1, the pair ( g πA , f πA ) ∈ G × F is afactorization on L . (cid:3) Let us observe that the pair ( g πA , f πA ) depends on the selection function π , and that Lemma 4 holds for any selection π . Given an M -DFA A =19 Q, Σ , u, i u , δ, w, ρ ), define the sets b P A = {A , ∂ σ ( A q ) | σ ∈ Σ , q ∈ Q } and b Q A = {A q | q ∈ Q } . Any factorization on L induced by A , ( g πA , f πA ), satisfiesthat f πA ( ℓ ) ∈ b Q A if ℓ ∈ b P A f πA ( ℓ ) = ℓ otherwise (39)for any ℓ ∈ L . Lemma 5.
Let A = ( Q, Σ , u, i u , δ, w, ρ ) be an M -DFA. For any factoriza-tion on L induced by A , ( g πA , f πA ) , the right congruence ≡ ( g πA ,f πA ) f πA ( A ) has finiteindex.Proof : Recall that A is the M -language recognized by A . As A is an M -DFA, the set of states Q is finite, and the sets of M -languages b Q A = {A q | q ∈ Q } and b P A = {A , ∂ σ ( A q ) | σ ∈ Σ , q ∈ Q } are also finite. We prove that forany α ∈ Σ ∗ , S ( g πA ,f πA ) α ( f πA ( A )) ∈ b Q A (40)For convenience, we omit the superscript ( g πA , f πA ) in S ( g πA ,f πA ) α . By induc-tion on the prefixes of α :- Basis. By (25), S ε ( f πA ( A )) = f πA ( A ). As A ∈ b P A , then, by (39), S ε ( f πA ( A )) ∈ b Q A .- Hypothesis. Let us assume that S α ′ ( f πA ( A )) ∈ b Q A where α ′ is a prefix of α .- Induction step. Let us consider α ′ σ be a prefix of α with σ ∈ Σ. By(27) and (26), S α ′ σ ( f πA ( A )) = ( S σ ◦ S α ′ )( f πA ( A )) = f πA ( ∂ σ ( S α ′ ( f πA ( A )))). Byinduction hypothesis, S α ′ ( f πA ( A )) ∈ b Q A , i.e., there exists some q ∈ Q suchthat S α ′ ( f πA ( A )) = A q . By substitution, S α ′ σ ( f πA ( A )) = f πA ( ∂ σ ( A q )). As ∂ σ ( A q ) ∈ b P A , then, by (39), f πA ( ∂ σ ( A q )) ∈ b Q A . Therefore, S α ′ σ ( f πA ( A )) ∈ b Q A .This fact proves that the set Q ( g πA ,f πA ) f πA ( A ) = { S α ( f πA ( A )) | α ∈ Σ ∗ } is finite because Q ( g πA ,f πA ) f πA ( A ) ⊆ b Q A . Therefore, by (35), ≡ ( g πA ,f πA ) f πA ( A ) has finite index. (cid:3) Remark 2.
Given an M -DFA A = ( Q, Σ , u, i u , δ, w, ρ ) , Lemma 5 statesthat ≡ ( g πA ,f πA ) f πA ( A ) has finite index. By Corollary 1 and Lemma 3, the M -DFA N ( g πA ,f πA ) ( f πA ( A ) , recognizes the M -language f πA ( A ) . y Property 3.1, N ( g πA ,f πA ) ( f πA ( A ) , g πA ( A )) recognizes g πA ( A ) · f πA ( A ) . By Def-inition 6 (or Lemma 4), g πA ( A ) · f πA ( A ) = i u · A u = A .Therefore, N ( g πA ,f πA ) ( f πA ( A ) , g πA ( A )) is equivalent to A . Previous Remark shows that N ( g πA ,f πA ) ( f πA ( A ) , g πA ( A )) is equivalent to A .Those automata could be very different because the former one satisfies allthe properties given in Property 3 but, however, these properties may beabsent in the automaton A . The M -DFA N ( g πA ,f πA ) ( f πA ( A ) , g πA ( A )) has theirtransitions equalized. This is derived by the way factorization ( g πA , f πA ) isconstructed. The next property collects the main properties involved anyfactorization on L induced by A . Property 4.
Let A = ( Q, Σ , u, i u , δ, w, ρ ) be an M -DFA. For any factoriza-tion on L induced by A , ( g πA , f πA ) , the following properties hold: N ( g πA ,f πA ) ( f πA ( A ) , g πA ( A )) is equivalent to A . N ( g πA ,f πA ) ( f πA ( A ) , g πA ( A )) is transition-equalized. If A is minimal then N ( g πA ,f πA ) ( f πA ( A ) , g πA ( A )) is minimal. There is a minimal and transition-equalized M -DFA equivalent to A . If A is minimal then f πA is idempotent, i.e., f πA ◦ f πA = f πA If A is minimal and transition-equalized then f π i A = f π j A for any selec-tion functions π i and π j .Proof :1. The equivalence is shown in Remark 2.2. Let N ( g πA ,f πA ) ( f πA ( A ) , g πA ( A )) = ( Q ( g πA ,f πA ) f πA ( A ) , Σ , f πA ( A ) , g πA ( A ) , δ, w, ρ ) be thestructure of such automaton provided in Definition 4. Let us consider twostates S α ( f πA ( A )) and S β ( f πA ( A )) and two symbols σ , τ ∈ Σ. By (40), S α ( f πA ( A )), S β ( f πA ( A )) ∈ b Q A , i.e., there are two states q , p ∈ Q , such that S α ( f πA ( A )) = A q and S β ( f πA ( A )) = A p . Now consider (22): ∂ σ ( N ( g πA ,f πA ) ( f πA ( A ) , g πA ( A )) S α ( f πA ( A )) ) = ∂ τ ( N ( g πA ,f πA ) ( f πA ( A ) , g πA ( A )) S β ( f πA ( A )) ) , by Property 3.4 ∂ σ ( S α ( f πA ( A ))) = ∂ τ ( S β ( f πA ( A )))then, ∂ σ ( A q ) = ∂ τ ( A p )This fact implies that (( σ, q ) , ∂ σ ( A q )) and (( τ, p ) , ∂ τ ( A p )) ∈ C πA , i.e, theyare in the same class of P A / ≈ A . Then, by Definition 6, f πA ( ∂ σ ( A q )) =21 πA ( ∂ τ ( A p )) and g πA ( ∂ σ ( A q )) = g πA ( ∂ τ ( A p )).Thus, δ ( S α ( f πA ( A )) , σ ) = S ασ ( f πA ( A )) = ( S σ ◦ S α )( f πA ( A )) = f πA ( ∂ σ ( A q )) = f πA ( ∂ τ ( A p )) = δ ( S β ( f πA ( A )) , τ ). In a similar way, w ( S α ( f πA ( A )) , σ ) = ( W σ ◦ S α )( f πA ( A )) = g πA ( ∂ σ ( A q )) = g πA ( ∂ τ ( A p )) = w ( S β ( f πA ( A )) , τ ). In conclusion, N ( g πA ,f πA ) ( f πA ( A ) , g πA ( A )) is transition-equalized.3. By the proof in Lemma 5, Q ( g πA ,f πA ) f πA ( A ) ⊆ b Q A . By definition k b Q A k ≤ k Q k where Q is the set of states of the minimal M -DFA A . Then, k Q ( g πA ,f πA ) f πA ( A ) k ≤k Q k . Therefore, N ( g πA ,f πA ) ( f πA ( A ) , g πA ( A )) is a minimal M -DFA.4. It is a consequence of the previous properties.5. This property is proved by exhaustive case analysis and the applicationof the condition NcndS (Property 1.2) for a minimal M -DFA. We omit theproof by brevity.6. By Definition 6 and (39):- If ℓ / ∈ b P A , then f π i A ( ℓ ) = f π j A ( ℓ ) = ℓ .- If ℓ ∈ b P A and ℓ = A , then f π i A ( ℓ ) = f π j A ( ℓ ) = A u .- If ℓ ∈ b P A and ℓ = ∂ σ ( A q ) and C π i A (( σ, q ) , ∂ σ ( A q )) ∈ P A / ≈ A ,then C π i A (( σ, q ) , ∂ σ ( A q )) = C π j A (( τ, p ) , ∂ τ ( A p )) with ∂ σ ( A q ) = ∂ τ ( A p ). As A is transition-equalized, by (22), qσ = pτ . There-fore, f π i A ( ℓ ) = A qσ = A pτ = f π j A ( ℓ ). (cid:3)
7. General Myhill-Nerode Theorem
Results presented in previous sections allow us to enunciate a generaltheorem for recognizability of M -languages. Theorem 1. ( General Myhill-Nerode Theorem ) Let ℓ ∈ L be an M -language. The following two conditions are equivalent:(i) ℓ is a recognizable M -language;(ii) there exists a factorization on L , ( g, f ) , such that the rightcongruence ≡ ( g,f ) f ( ℓ ) has finite index. roof :(i) ⇒ (ii). If ℓ is a recognizable M -language then there is an M -DFA A =( Q, Σ , u, i u , δ, w, ρ ) such that A = ℓ . By Lemma 4, the pair ( g πA , f πA ) ∈ G × F (Definition 6) is a factorization on L induced by the automaton A . ByLemma 5, the right congruence ≡ ( g πA ,f πA ) f πA ( A ) has finite index. In addition, byRemark 2, the M -DFA N ( g πA ,f πA ) ( f πA ( A ) , g πA ( A )) recognizes A = ℓ .(ii) ⇒ (i). If there exists a factorization on L , ( g, f ), such that the rightcongruence ≡ ( g,f ) f ( ℓ ) has finite index then, by Lemma 3, f ( ℓ ) is recognized bythe M -DFA N ( g,f ) ( f ( ℓ ) , M -DFA N ( g,f ) ( f ( ℓ ) , g ( ℓ ))recognizes the M -language g ( ℓ ) · f ( ℓ ) = ℓ since ( g, f ) is a factorization on L . (cid:3)
8. Recognition Capability of factorizations on L (Σ ∗ , M ) For a monoid M and an alphabet Σ, the set of all recognizable M -languages in L (Σ ∗ , M ) is denoted by R L . By following the previous generalMyhill-Nerode Theorem, we can define the recognition capability of a factor-ization on L , ( g, f ), as the setRcgCap ( g,f ) L = { ℓ ∈ L | N ( g,f ) ( f ( ℓ ) ,
1) is an M -DFA } (41)This set, RcgCap ( g,f ) L , represents the set of all recognizable M -languages fora given factorization on L , i.e, the languages recognized by N ( g,f ) ( f ( ℓ ) , m )for some m ∈ M . Obviously, RcgCap ( g,f ) L ⊆ R L . In this section, we studythe recognition capability of three types of factorizations: (a) trivial factor-ization; (b) maximal factorizations; and (c) composition of natural factor-izations. For every monoid M , L (Σ ∗ , M ) admits the trivial factorization ( g e , f e ),which is the identity element of G × F under composition of factoriza-tions. By (23) and (24), for any ℓ ∈ L and α ∈ Σ ∗ , S ( g e ,f e ) α ( ℓ ) = ∂ α ( ℓ )and W ( g e ,f e ) α ( ℓ ) = 1. In addition, f e ( ℓ ) = ℓ .Let us consider that N ( g e ,f e ) ( ℓ,
1) is an M -DFA for a language ℓ ∈ L . ByDefinition 4, Q ( g e ,f e ) ℓ = { ∂ α ( ℓ ) | α ∈ Σ ∗ } is finite and its monoid-transitionfunction satisfies that w ( ∂ α , σ ) = 1 for any state and symbol.The M -DFA N ( g e ,f e ) ( ℓ, ℓ , is merely and ordinary DFA23quipped with a final-state function ρ . Furthermore, by (18), ( N ( g e ,f e ) ( ℓ, β ) = ρ ( ∂ β ( ℓ )) = ( ∂ β ( ℓ ))( ε ) = ℓ ( β ). Therefore, as Q ( g e ,f e ) ℓ is finite, ℓ has finite rank.In conclusion, only M -languages of finite rank may be recognized by usingthe trivial factorization.Previous discussion allows us to introduce the notion of a ρ -DFA, i.e., anordinary DFA with a final-state function ρ . We can represent a ρ -DFA by A = ( Q, Σ , u, δ, ρ ) where ( Q, Σ , u, δ ) is a DF A and ρ : Q → M . A ρ -DFA A ,recognizes the M -language A ( α ) = ρ ( uα ) for any word α . One can definethe notion of minimal ρ -DFA, and by Property 1.3 and Property 2, it is con-cluded that the sufficient and necessary condition for an accessible ρ -DFAto be minimal is just the condition NcndW , i.e., A is a minimal ρ -DFA ifand only if for every p , q ∈ Q , A p = A q ⇒ p = q . Therefore, N ( g e ,f e ) ( ℓ, ρ -DFA that recognizes ℓ since N ( g e ,f e ) ( ℓ, NcdW ). However, this fact does not prevent to findanother factorization on L that provides an M -DFA with lesser states thana minimal ρ -DFA (see fig.2 and fig. 3 in [17]).Let us observe that N ( g e ,f e ) ( ℓ, m ) = N ( g e ,f e ) ( m · ℓ,
1) since ℓ is of finite rankand N ( g e ,f e ) ( ℓ, m ) is a ρ -DFA. Therefore, for any monoid M , R L ⊇ RcgCap ( g e ,f e ) L = { ℓ ∈ L | ℓ is recognized by a ρ -DFA } In the context of fuzzy automata, ρ -DFAs are called crisp deterministicfuzzy automata. This kind of automata has been studied by Ignatovi´c et al.[22] from the perspective of the Myhill-Nerode Theorem and minimizationalgorithms for crisp deterministic fuzzy automata. Let us consider that L (Σ ∗ , M ) admits a maximal factorization. By theresults in [13], if M contains a zero element then ( M, · ,
1) is zero-divisor-free .In addition, ( M \ { } , · ,
1) is also a monoid. In particular, [13] studies mge -monoids and their conditions to obtain maximal factorizations. In order tosimplify this case of study, we consider that ( M, · ,
1) is a monoid without azero element.A maximal factorization on L , ( g h , h ) ∈ G × F , is a factorization that satis-fies h ( m · ℓ ) = h ( ℓ ) for any ℓ ∈ L and m ∈ M . This strong property impliesthat h is idempotent: h ( ℓ ) = h ( g h ( ℓ ) · h ( ℓ )) = h ( h ( ℓ )) for any ℓ ∈ L .Let us consider that N ( g h ,h ) ( h ( ℓ ) ,
1) is an M -DFA for a language ℓ ∈ L . An mge -monoid satisfies left and right cancellation axioms and the right most generalequalizer axiom [13].
24y (23) for any ℓ ∈ L and α ∈ Σ ∗ , S ( g h ,h ) α ( h ( ℓ )) = h ( ∂ α ( h ( ℓ ))). The readermay prove the given expression by induction on | α | by using that h is idem-potent and a maximal factorization. The hint is to apply equation (11)when proving the induction step. This expression simplifies the definitionof W ( g h ,h ) α ( h ( ℓ )) in (24). Then, the set of states of N ( g h ,h ) ( h ( ℓ ) ,
1) is the set Q ( g h ,h ) h ( ℓ ) = { h ( ∂ α ( h ( ℓ ))) | α ∈ Σ ∗ } .Let us recall that by Property 3.1, N ( g h ,h ) ( h ( ℓ ) , g ( ℓ )) recognizes ℓ ∈ L . Theinteresting aspect of this automata is that it satisfies the sufficient conditionof minimality (Property 2). Property 5.
Let ( g h , h ) be a maximal factorization on L . If N ( g h ,h ) ( h ( ℓ ) , g ( ℓ )) is an M -DFA then it is a minimal M -DFA that recognizes ℓ ∈ L .Proof : Let us consider an M -DFA A = ( Q, Σ , u, i u , δ, w, ρ ) such that it isaccessible, but it does not satisfy the sufficient condition for minimality (seeProperty 2): for two words α , β ∈ Σ ∗ , two values m , m ′ ∈ M , and an M -language ℓ ′ ∈ L , uα = uβ ∧ ∂ α ( A ) = m · ℓ ′ ∧ ∂ β ( A ) = m ′ · ℓ ′ .As ( g h , h ) is a maximal factorization, h ( ∂ α ( A )) = h ( ∂ β ( A )) because h ( m · ℓ ′ ) = h ( m ′ · ℓ ′ ) = h ( ℓ ′ ). By (20), ∂ α ( A ) = i u · w ∗ ( u, α ) · A uα . Then, h ( ∂ α ( A )) = h ( A uα ). Similarly, h ( ∂ β ( A )) = h ( A uβ ). Therefore, h ( A uα ) = h ( A uβ ). By identifying, A with N ( g h ,h ) ( h ( ℓ ) , g ( ℓ )), which is accessible, then uα = S α ( h ( ℓ )) = h ( ∂ α ( h ( ℓ ))) and uβ = h ( ∂ β ( h ( ℓ ))). By Property 3.4, as A uα = uα and A uβ = uβ , then h ( h ( ∂ α ( h ( ℓ )))) = h ( h ( ∂ β ( h ( ℓ )))). As h () isidempotent, then uα = uβ following our identification. This is a contradic-tion with the initial hypothesis. Therefore, N ( g h ,h ) ( h ( ℓ ) , g ( ℓ )) satisfies thesufficient condition for minimality. (cid:3) A maximal factorization on L (Σ ∗ , M ) achieves the maximal recognitioncapability. Lemma 6.
Let ( g h , h ) be a maximal factorization on L . Then, R L = RcgCap ( g h ,h ) L Proof : Let ℓ ∈ R L be recognized by an M -DFA A = ( Q, Σ , u, i u , δ, w, ρ ), i.e., A = ℓ . Without loss of generality, A is accessible, i.e, Q = { uα | α ∈ Σ ∗ } .The set b Q A = {A uα | α ∈ Σ ∗ } is finite. Thus, h ( b Q A ) = { h ( A uα ) | α ∈ Σ ∗ } is also finite. By (20), ∂ α ( A ) = i u · w ∗ ( u, α ) · A uα . Then, as ( g h , h ) is amaximal factorization, h ( ∂ α ( A )) = h ( A uα ). As A = g h ( A ) · h ( A ); by (11), h ( ∂ α ( A )) = h ( ∂ α ( g h ( A ) · h ( A ))) = h ( g h ( A ) · ∂ α ( h ( A ))) = h ( ∂ α ( h ( A ))).Therefore, the set { h ( ∂ α ( h ( ℓ ))) | α ∈ Σ ∗ } is finite. This set is the set of statesof the M -DFA N ( g h ,h ) ( h ( ℓ ) , g ( ℓ )) which recognizes ℓ . By the definitions given25t the beginning of this section, ℓ ∈ RcgCap ( g h ,h ) L , i.e., R L ⊆ RcgCap ( g h ,h ) L ;thus, R L = RcgCap ( g h ,h ) L . (cid:3) When the maximal factorization has an explicit formulae, it is possible toconstruct determinization and minimization algorithms for automata. Max-imal factorizations produce very efficient constructions. Kirsten and M¨aurer[23] show that their determinization algorithm of weighted automata is opti-mal using maximal factorizations and the zero-divisor-free condition (Theo-rem 3.3 in [23]). This behaviour has been corroborated in some determiniza-tion methods for fuzzy automata [15][40]. The original Mohri’s minimizationalgorithm for weighted automata over tropical semiring applies a maximalfactorization [28]. Other examples of applications of maximal factorizationsare in [9][17][41][16].
Let us consider an arbitrary monoid M . Let A be an M -DFA. By Prop-erty 4.4, there exists a minimal and transition-equalized M -DFA equiva-lent to A . Thus, we consider that A is a minimal and transition-equalized M -DFA. By Property 4.6, the factorization on L induced by A is unique.That factorization is simply denoted ( g A , f A ). By Lemma 5 and Remark 2,RcgCap ( g A ,f A ) L is not empty since f A ( A ) is in this set. Let us recall that, byLemma 1, the composition of factorizations on L is again a factorization on L . We study the composition of natural factorizations to provide the resultthat the composition preserves the recognition capability of each individualnatural factorization. Lemma 7.
Let { Ak } k =1 ..n be a finite family of n minimal and transition-equalized M -DFAs; and, let { ( g Ak , f Ak ) } k =1 ..n be the famility of the natu-ral factorizations on L induced by those automata. The factorization on L , ( g, f ) = [( g Ak , f Ak )] n , obtained by the composition of the family { ( g Ak , f Ak ) } k =1 ..n ,satisfies the next property,RcgCap ( g,f ) L ⊇ n [ k =1 RcgCap ( g Ak ,f Ak ) L (42) Proof : For each M -DFA Ak = ( Q Ak , Σ , uk, i uk , δ k , w k , ρ k ), with k : 1 ..n : Q Ak is the set of states; b Q Ak = {A k q | q ∈ Q Ak } ; and b P Ak = {A k, ∂ σ ( A k q ) | q ∈ Q Ak , σ ∈ Σ } . By definition of composition of factorizations on L , f = f An ◦ ... ◦ f A . By (39), it is simple to show that, f ( ℓ ) ∈ S nk =1 b Q Ak if ℓ ∈ S nk =1 b P Ak f ( ℓ ) = ℓ otherwise (43)26or any ℓ ∈ L . Let us observe that if ℓ / ∈ S nk =1 b P Ak then f ( ℓ ) = ℓ . Let j , n ≥ j ≥
1, be the first index such that ℓ ∈ b P Aj , then f ( ℓ ) ∈ S nk = j b Q Ak . Thisis so because, by (39), ( f Aj − ◦ ... ◦ f A )( ℓ ) = ℓ , f Aj ( ℓ ) ∈ b Q Aj and f Aj ( ℓ )may belong (or not) to any b P Ak with k : j + 1 ..n .Let us consider an arbitrary ℓ ∈ L such that, for some arbitrary Aj , with1 ≤ j ≤ n , ℓ ∈ RcgCap ( g Aj ,f Aj ) L . That is, the M -DFA N ( g Aj ,f Aj ) ( f Aj ( ℓ ) , f Aj ( ℓ ). The finite set of states of this automaton is Q ( g Aj ,f Aj ) f Aj ( ℓ ) = { S ( g Aj ,f Aj ) α ( f Aj ( ℓ )) | α ∈ Σ ∗ } . By (23) and (39), it is simple to show that S ( g Aj ,f Aj ) α ( f Aj ( ℓ )) ∈ b Q Aj or S ( g Aj ,f Aj ) α ( f Aj ( ℓ )) = ∂ α ( ℓ ). This last case hap-pens when each language in the composition of S ( g Aj ,f Aj ) α ( f Aj ( ℓ )) does notbelong to b P Aj and f Aj behaves like the identity f e . This structure of Q ( g Aj ,f Aj ) f Aj ( ℓ ) is important in the next step of the proof.We claim that the following property holds for the composition and thelanguage ℓ , S ( g,f ) α ( f ( ℓ )) ∈ ( n [ k =1 b Q Ak ) ∪ Q ( g Aj ,f Aj ) f Aj ( ℓ ) (44)for any α ∈ Σ ∗ .By induction on the length of α ∈ Σ ∗ :- Basis. Let α = ε . By (25), S ( g,f ) ε ( f ( ℓ )) = f ( ℓ ). By (43), f ( ℓ ) ∈ S nk =1 b Q Ak or f ( ℓ ) = ℓ . In this last case, ℓ / ∈ b P Aj . This implies that, by (25) and (39), S ( g Aj ,f Aj ) ε ( f Aj ( ℓ )) = S ( g Aj ,f Aj ) ε ( ℓ ) = ℓ ∈ Q ( g Aj ,f Aj ) f Aj ( ℓ ) . The property holds.- Hypothesis. Let us assume that (44) is valid for α ∈ Σ ∗ .- Induction Step. Let α ′ = ασ with α ∈ Σ ∗ and σ ∈ Σ. By (26) and (27), S ( g,f ) ασ ( f ( ℓ )) = ( S ( g,f ) σ ◦ S ( g,f ) α )( f ( ℓ )) = f ( ∂ σ ( S ( g,f ) α ( f ( ℓ )))). By Hypothesisand (44), we have two main cases:Case (a). S ( g,f ) α ( f ( ℓ ))) ∈ b Q Ar for some 1 ≤ r ≤ n . Then, S ( g,f ) α ( f ( ℓ ))) = A r q for some q ∈ Q Ar . In that case, ∂ σ ( A r q ) ∈ b P Ar . By (43), f ( ∂ σ ( A r q )) ∈ S nk =1 b Q Ak . Thus, S ( g,f ) ασ ( f ( ℓ )) ∈ S nk =1 b Q Ak , and the property holds.Case (b). S ( g,f ) α ( f ( ℓ ))) ∈ Q ( g Aj ,f Aj ) f Aj ( ℓ ) . By the structure of this set of statesgiven above, we have two subcases:Case (b1). S ( g,f ) α ( f ( ℓ ))) ∈ b Q Aj . The proof is the same as in Case(a). 27ase (b2). S ( g,f ) α ( f ( ℓ ))) = ∂ β ( ℓ ) for some β ∈ Σ ∗ . Thus, f ( ∂ σ ( ∂ β ( ℓ ))) = f ( ∂ βσ ( ℓ )). Then, by (43), if ∂ βσ ( ℓ ) / ∈ S nk =1 b P Ak ,then f ( ∂ βσ ( ℓ )) = ∂ βσ ( ℓ ). As ∂ βσ ( ℓ ) / ∈ b P Aj , then ∂ βσ ( ℓ ) = f Aj ( ∂ βσ ( ℓ )) = f Aj ( ∂ σ ( ∂ β ( ℓ ))) ∈ Q ( g Aj ,f Aj ) f Aj ( ℓ ) . Then, S ( g,f ) ασ ( f ( ℓ )) ∈ Q ( g Aj ,f Aj ) f Aj ( ℓ ) and the property holds. Finally, by (43), if ∂ βσ ( ℓ ) ∈ S nk =1 b P Ak , then f ( ∂ βσ ( ℓ )) ∈ S nk =1 b Q Ak . In this case, S ( g,f ) ασ ( f ( ℓ )) ∈ S nk =1 b Q Ak and the property holds again.Therefore, { S ( g,f ) α ( f ( ℓ )) | α ∈ Σ ∗ } ⊆ ( S nk =1 b Q Ak ) ∪ Q ( g Aj ,f Aj ) f Aj ( ℓ ) . This impliesthat it is finite. By Lemma 3, f ( ℓ ) ∈ L is recognized by the M -DFA N ( g,f ) ( f ( ℓ ) , ℓ ∈ RcgCap ( g Aj ,f Aj ) L implies ℓ ∈ RcgCap ( g,f ) L ;and, as ℓ and Aj has been selected in an arbitrary way, then (42) holds. (cid:3)
9. Conclusions
The Myhill-Nerode theory studies formal languages and deterministicautomata through right congruences and congruences on a free monoid. Let( M, · ,
1) be an arbitrary monoid. In this paper, we provide a general Myhill-Nerode theorem for M -languages, i.e., functions of the form ℓ : Σ ∗ → M . M -languages are studied from the aspect of their recognition by determinis-tic finite automata whose components take values on M ( M -DFAs). Unlikeother previous papers in the literature that deal with the problem of recog-nizability of languages on different algebraic structures, we do not assumeany additional property on the monoid. We characterize an M -language ℓ by a right congruence on Σ ∗ that is defined through the language ℓ and a fac-torization ( f, g ) on the set of all M -languages, denoted ≡ ( g,f ) ℓ . As ( g, f ) is afactorization then ℓ = g ( ℓ ) · f ( ℓ ) where g ( ℓ ) ∈ M and f ( ℓ ) is an M -language.Then, ℓ is characterized equivalently by both ≡ ( g,f ) ℓ or ≡ ( g,f ) f ( ℓ ) congruences.The main result of the paper (Theorem 1) states the equivalence of theconditions:(i) ℓ is a recognizable M -language;(ii) there exists a factorization on L , ( g, f ), such that the rightcongruence ≡ ( g,f ) f ( ℓ ) has finite index.The proof of the implication (ii) ⇒ (i) requires the explicit construction ofan M -DFA that recognizes f ( ℓ ). The properties of this automata based onthe congruence ≡ ( g,f ) f ( ℓ ) state that it is accessible and satisfies a weak nec-essary condition of minimality (Property 3). The proof of the implication28i) ⇒ (ii) requires that any M -DFA induces a factorization on the set of all M -languages. That factorization is called natural factorization induced byan M -DFA. The construction and composition of this kind of factorizationsis also studied in the paper. The provided formalism to study factorizationsand their composition establishes that the composition of natural factoriza-tions is also a factorization that preserves the recognition capability of eachindividual natural factorization (Lemma 7). In the particular case of the ex-istence of a maximal factorization on the set of all M -languages, we obtainthat the automata based on a maximal factorization is minimal (Property5) and that the recognition capability is maximal (Lemma 6). References [1] R. Bˇelohl´avek. Determinism and fuzzy automata. Information Sciences143 (2002) 205–209.[2] S. Bozapalidis, O. Louscou-Bozapalidou. On the recognizability of fuzzylanguages II. Fuzzy Sets and Systems 159 (2008) 107–113.[3] J. A. Brzozowski. Derivatives of Regular Expressions. Journal of ACM.11:4 (1964) 481-–494.[4] M. ´Ciri´c, M. Droste, J. Ignjatovi´c, H. Vogler. Determinization ofweighted finite automata over strong bimonoids. Information Sciences180 (2010) 3497–3520.[5] M. Droste, W. Kuich, and H. Vogler, editors. Handbook of WeightedAutomata. Springer-Verlag, Berlin, 2009.[6] M. Droste,W. Kuich, Weighted finite automata over hemirings, Theo-retical Computer Science 485 (2013) 38–48.[7] M. Droste, T. St¨uber, H. Vogler. Weighted finite automata over strongbimonoids. Information Sciences 180 (2010) 156–166.[8] S. Eilenberg. Automata, Languages and Machines. Academic Press.New York and London 1974.[9] J. Eisner. Simpler and more general minimization for weighted finite-state automata. In Proceedings of HLT-NAACL2003 conference, pp.64–71, 2003. 2910] S. Gerdjikov, S. Mihov. Myhill-Nerode Relation for Sequentiable Struc-tures. ArXiv e-prints. https://arxiv.org/abs/1706.02910, (2017).[11] S. Gerdjikov. A general class of monoids supporting canonisation andminimisation of (sub)sequential transducers. Klein, S.T., Mart´ın-Vide,C., Shapira, D. (eds.) Language and Automata Theory and Application,12th International Conference LATA2018 (2018).[12] S. Gerdjikov. Characterisation of (sub)sequential rational function overa general class monoids. CoRR, abs/1801.10063, (2018).[13] S. Gerdjikov, J. R. Gonz´alez de Mendivil. Conditions for the exis-tence of maximal factorizations. Fuzzy Sets and Systems. (on line)https://doi.org/10.1016/j.fss.2019.07.006 (2019).[14] J. R. Gonzalez de Mendivil, J. R. Garitagoitia. Fuzzy languages ofinfinite range: pumping lemmas and determinization procedure. FuzzySets and Systems 249 (2014) 1–26.[15] J. R. Gonzalez de Mendivil, J. R. Garitagoitia. Determinization of fuzzyautomata via factorization of fuzzy states. Information Sciences 283(2014) 165–179.[16] J. R. Gonz´alez de Mendivil. A generalization of Myhill-Nerode theoremfor fuzzy languages. Fuzzy Sets and Systems 301 (2016) 103–115.[17] J. R. Gonz´alez de Mendivil. Conditions for Minimal Fuzzy Determin-istic Finite Automata via Brzozowski’s Procedure. IEEE Transactionson Fuzzy Systems. 26 (4) (2018) 2409–2420.[18] J. R. Gonz´alez de Mendivil, F. Fari˜na. Canonization of max-min fuzzyautomata. Fuzzy Sets and Systems. 376 (2019) 152–168.[19] J. R. Gonz´alez de Mendivil, F. Fari˜na. Recognizability of languageswith values on a monoid. Report number: M02-2019-gsd. UniversidadP´ublica de Navarra. 2019.[20] J. E. Hopcroft, R. Motwani, J. D. Ullman.
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