finite automata theory with membership values in lattices

Information Sciences 181 (2011) 1003–1017

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Information Sciences

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Finite automata theory with membership values in lattices q

Yongming Li ⇑College of Computer Science, Shaanxi Normal University, Xi’an 710062, China

a r t i c l e i n f o

Article history:Received 9 August 2008Received in revised form 7 August 2010Accepted 8 November 2010

Keywords:Finite automataLanguageLatticeSubset construction

0020-0255/$ - see front matter � 2010 Elsevier Incdoi:10.1016/j.ins.2010.11.006

q This work is supported by National Science FounEducation of China (Grant No. 200807180005).⇑ Tel.: +86 2985310166; fax: +86 2985310161.

E-mail address: [email protected]

a b s t r a c t

In this paper, we study finite automata with membership values in a lattice, which arecalled lattice-valued finite automata. The extended subset construction of lattice-valuedfinite automata is introduced, then the equivalences between lattice-valued finite auto-mata, lattice-valued deterministic finite automata and lattice-valued finite automata withe-moves are proved. A simple characterization of lattice-valued languages recognized bylattice-valued finite automata is given, then it is proved that the Kleene theorem holdsin the frame of lattice-setting. A minimization algorithm of lattice-valued deterministicfinite automata is presented. In particular, the role of the distributive law for the truth val-ued domain of finite automata is analyzed: the distributive law is not necessary to manyconstructions of lattice-valued finite automata, but it indeed provides some conveniencein simply processing lattice-valued finite automata.

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1. Introduction

The concept of fuzzy automata was introduced in the very early age of fuzzy set theory [7,16,23,32]. Since finite automataconstitute a mathematical model of computation, fuzzy finite automata may be considered as an extended model which in-cludes notions like ‘‘vagueness’’ and ‘‘imprecision’’, i.e., notions frequently encountered in the study of natural languages. Soinvestigating fuzzy finite-state automata might reduce the gap between formal languages as studied in classical automatatheory [8] on the one hand and natural languages on the other hand. Usually, fuzzy automata took values in the unit interval[0,1]. To enhance the processing ability of fuzzy automata, the membership grades were extended to much general algebraicstructures. For example, automata theory based on complete residual lattices-valued logic has been primarily established in[24–26], and automata theory based on lattice-ordered monoids has been established in [19]. In fact, fuzzy finite automatacould be considered as a special instance of weighted automata considered recently [4,13,31,9], where weighted automatatake values in semirings, more general structures than lattice-ordered monoids when a lattice-ordered monoid is consideredas an idempotent semiring [13]. Of course, there are some distinct differences between fuzzy automata and weighted auto-mata. In fuzzy automata, it is stressed the role of order (or hierarchy) and fuzzy logic in automata theory, while, for weightedautomata, it is emphasized the weight of source used in automata theory. We can use fuzzy rules to represent fuzzy auto-mata [6], and fuzzy automata can be seen as a special kind of discrete fuzzy systems [18]. While weighted automata can beseen as the algebraic treatment of automata using semiring and formal power series [5,14].

When we use a lattice as the truth value domain of fuzzy automata or as a semiring involved in weighted automata, thislattice should satisfy distributive law. On the other hand, there are many kinds of lattices that are not distributive lattices.The lattices M3 (the diamond) and N5 (the pentagon) in Fig. 1 are two simple examples. The free lattices generated by more

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dation of China (GrantNo. 60873119) and the Higher School Doctoral Subject Foundation of Ministry of

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Fig. 1. Some lattices.

1004 Y. Li / Information Sciences 181 (2011) 1003–1017

than two elements are the other instances. There are some deep reasons to study such automata. First, this work may providesome useful ideas and more general framework to study automata theory based on quantum logic [33,27–29]. As we know,automata based on quantum logic used orthomodular lattices [11] as the truth values of the automata considered. In general,an orthomodular lattice as a lattice needs not to be a distributive lattice. Indeed, automata theory based on quantum logic,which maybe thought of as a logic foundation for quantum computing and quantum information, has some essential differ-ence with classical automata and forms an important direction in the study of quantum computation, see [33,27,28] for thedetailed explanation. In an orthomodular lattice, orthomodular laws hold instead of distributive laws, the Chinese lanternMO2 in Fig. 1 is one of such examples. For any Hilbert space H, the closed subspaces of H under the order of subset inclusionforms the standard orthomodular lattice which can be considered as the logic of quantum mechanics, see [2,11] for the de-tailed explanation. However, there are many (ortho-) lattices which do not satisfy the orthomodular laws. The simplest oneis the lattice O6 as in Fig. 1. We want to know how about the behaviors of finite automata with membership values in a gen-eral lattice. In this paper, we will use lattices as the structures of truth values domain. We do not require the used lattice tobe distributive, so the used lattice need not to be a semiring. We would obtain much more general results than fuzzy finiteautomata and weighted automata. Second, these results may be useful in multi-valued model checking. Some related workhas been done using multi-valued automata in Ref. [3,15], where the multi-valued domain is chosen as a De Morgan distrib-utive lattice. We shall extend multi-valued domain to a general lattice.

The contribution of this study contains at least three aspects. First, as we just said, lattice-valued finite automata in thisstudy is a common generalization of fuzzy automata and weighted automata. In this respective, the role of the distributivelaw for the truth valued domain of finite automata is analyzed. It is demonstrated that the distributive law is not necessary tomany constructions of lattice-valued finite automata, but it indeed provides some convenience in simply processing lattice-valued finite automata. Second, the technique of extended subset construction is introduced, using this technique, the equiv-alence between lattice-valued finite automata and lattice-valued deterministic finite automata is proved. Some results in[33] were strength, especially, the Kleene theorem in lattice-setting is presented. Third, we give a minimal algorithm of lat-tice-valued deterministic finite automata.

The content of this paper is arranged as follows. In Section 2, the definition of lattice-valued finite automata is introduced,then the relationship between the extendability of the state transition relation d and the distributive law of the lattice l isexploited. Section 3 introduces the extended subset construction of l-VFA, the determinization of l-VFA is shown. The rela-tionship between extended subset construction and lattice-valued subset construction and the distributive law of l is dis-cussed. Some results on determinization of lattice-valued finite automata in [1,15,33] are strengthened. In Section 4,some simple characterizations of l-valued regular languages are introduced, the operations on the l-valued regular languagesare discussed and Kleene Theorem in lattice setting is established. Section 5 discusses the minimization of l-VDFA, the min-imal algorithm of l-VDFA is presented. Some remarks are included in the conclusion part.

2. Finite automata with membership values in a lattice l

A lattice is a 6-tuple l = (L,6,^,_,0,1), 0 and 1 are the least and largest elements of L, respectively, 6 is the partial orderingin L; and for any a,b 2 L, a ^ b and a _ b stand for the greatest lower bound (or meet) and the least upper bound (or join) of aand b, respectively. We need lattice-valued logic (called l-valued logic in the paper, and it is indeed the geometric logic asstudied in [2], which is to provide algebraic semantics for observable events) to represent automata in this paper. We defineit in the following way. Since there is no negation operation and implication operation in a general lattice, we do not mentionthe negation connective and implication connective in lattice-valued logic. Similar to that of classical first-order logic, thesyntax of l-valued logic has two primitive connectives ^ (conjunction), _ (disjunction), and two primitive quantifier $(existential quantifier) and " (universal quantifier). In addition, we need to use some set-theoretical formulas. Let 2 (mem-bership) be a binary (primitive) predicate symbol. Then # and � (equality) can be defined with 2 as usual. The semantics ofl-valued logic is given by interpreting the connectives ^ and _ as the operations ^ and _ on l, respectively, and interpretingthe quantifier $ and " as the least upper bound and the greatest lower bound in l. Moreover, the truth value of set-theoreticalformula x 2 A is [x 2 A] = A(x). In the l-valued logic, 1 is the unique designated truth value; a formula u is valid iff [u] = 1, anddenoted by �lu.

In order to distinguish the symbols representing languages and the symbols representing lattices, we use symbol l torepresent a lattice, and use L to represent language. We use the symbols a, b, c, d, k to represent the elements of l.

Y. Li / Information Sciences 181 (2011) 1003–1017 1005

Definition 2.1. An l-valued finite automaton (l-VFA for short) is a 5-tuple A ¼ ðQ ;R; d; I; FÞ, where Q denotes a finite set ofstates, R a finite input alphabet, and d an l-valued subset of Q � R � Q; that is, a mapping from Q � R � Q into l, and it iscalled the l-valued transition relation. Intuitively, d is an l-valued (ternary) predicate over Q, R and Q, and for any p,q 2 Qand r 2 R, d(p,r,q) stands for the truth value of the proposition that input r causes state p to become q. I and F are l-valuedsubsets of Q; that is, a mapping from Q into l, which represent the initial state and final state, respectively. For each q 2 Q, I(q)indicates the truth value (in the underlying lattice-valued logic) of the proposition that q is an initial state, F(q) expresses thetruth value of the proposition that q is a finial state.

The propositions of the form

‘‘q is an initial state’’, written ‘‘q 2 I’’.‘‘q is a final state’’, written ‘‘q 2 F’’.‘‘input r causes state q to become p, according to the specification given by d’’, written ‘‘(q,r,p) 2 d’’.

denote the atomic propositions in our logical languages designated for describing l-valued automaton A. The truth values ofthe above three propositions are respectively I(q), F(q) and d(q,r,p). We use the symbols r,s to represent the elements in R,use the symbols x,h to denote the strings over R, and use e to represent the empty string over R. Let R* denote the set of allstrings over R. We use the symbols A;B to denote the l-valued finite automata.

For an l-VFA A, the l-valued unary recognizability predicate recA over R* is defined as a mapping from R* into l: foreach x 2 R*, let x = r1� � �rn for some n P 0,

x 2 recA¼def ð9q0 2 QÞ � � � ð9qn 2 QÞ � ðq0 2 I ^ qn 2 F ^ ðq0;r1; q1Þ 2 d ^ � � � ^ ðqn�1;rn; qnÞ 2 dÞ:

In other words, the truth value of the proposition that x is recognizable by A is given by

recAðxÞ ¼_fIðq0Þ ^ dðq0;r1; q1Þ ^ � � � ^ dðqn�1;rn; qnÞ ^ FðqnÞ : q0; . . . ; qn 2 Qg:

We call recA the l-valued language recognized or accepted by l-VFA A. We use l(R*) to denote the set of all l-valued lan-guages over R*, which is an l-valued subset of R*; that is, a mapping from R* to l. For an A 2 l(R*), if there is an l-VFA A suchthat A ¼ recA, then we call A an l-valued regular language or l-regular language on R, which is also called lattice-valued regularlanguage without mentioning the truth-valued lattice.

We give an example to illustrate the definition of l-VFA.

Example 2.1. Let l be the lattice N5 as shown in Fig. 1. Write the elements of l as 0, 1, a, b, c, where b < a and c cannot becomparable with a and b. An l-VFA A ¼ ðQ ;R; d; I; FÞ is defined as follows, where Q = {q0,q1,q2,q3,q4}, R = {0,1}, d(q0,0,q1) = b,d(q0,0,q2) = c, d(q1,1,q1) = 1, d(q2,0,q2) = 1, d(q1,0,q3) = 1, d(q2,0,q3) = 1, d(q3,0,q4) = a, d(q4,0,q4) = 1, d(q4,1,q4) = 1, andI ¼ 1

q0; F ¼ 1

q4. This l-VFA is represented as in Fig. 2, where we use the notation

Pt2T

atxt

to denote the l-valued subset withnonzero value at at the element xt for t 2 T and T is an index set, and if d(q,r,p) = r, then there is an edge label r

r from the nodewith name q to the node with name p in the graphic representation of the l-VFA A. By the simple calculation, we can see thatrecAðxÞ ¼ ða ^ bÞ _ ða ^ cÞ ¼ b if x 2 01*00(0 + 1)*, and recAðxÞ ¼ 0 in the other cases.

The extension of d, denoted d*, is defined as follows,

ðiÞ 8p 2 Q ; if p ¼ q; then d�ðq; e;pÞ ¼ 1; otherwise d�ðq; e;pÞ ¼ 0;

ðiiÞ 8h ¼ r1 � � �rn 2 R�; d�ðq;r1 � � �rn; pÞ ¼_fdðq;r1; q1Þ ^ � � � ^ dðqn�1;rn;pÞ : q1; . . . ; qn�1 2 Qg: ð1Þ

As far as the extension d* is concerned, for any h 2 R*, if h = h1h2, then it wish to satisfy the following equation

d�ðq; h1h2; pÞ ¼_r2Q

½d�ðq; h1; rÞ ^ d�ðr; h2; pÞ�: ð2Þ

Fig. 2. The automaton in Example 2.1.


To ensure Eq. (2) holds, we need l to be a distributive lattice, where l being a distributive lattice means the following twoequivalent distributive laws hold for any a,b,c 2 l,

a ^ ðb _ cÞ ¼ ða ^ bÞ _ ða ^ cÞ; a _ ðb ^ cÞ ¼ ða _ bÞ ^ ða _ cÞ: ð3Þ

Proposition 2.1. The following conditions are equivalent:

(i) l is a distributive lattice.(ii) For any l-VFA, A ¼ ðQ ;R; d; I; FÞ, and for any p,q 2 Q, h1,h2 2R*, the Eq. (2) holds, i.e.,

d�ðq; h1h2; pÞ ¼_r2Q

½d�ðq; h1; rÞ ^ d�ðr; h2; qÞ�:

The related result similar to Proposition 2.1 is also presented in [19,24], and the proofs are very similar, we omit the proofhere.

Let us define the operations of l-valued languages (c.f. [19,21,33]): for A,B 2 l(R*) and r 2 l, the union A _ B, the intersec-tion A ^ B, the scalar product rA, the concatenation AB, the Kleene closure A* are defined as follows: for any x 2R *,A _ B(x) = A(x) _ B(x), A ^ B(x) = A(x) ^ B(x), rA(x) = r ^ A(x), AB(x) =

W{A(x1) ^ B(x2) : x1x2 = x}, A*(x) =

W{A(x1)

^� � �^ A(xn) : n P 0, x1� � �xn = x}.The following proposition claims the relationship between the associativity of concatenation operation and the distrib-

utivity of lattice l.

Proposition 2.2. The following statements are equivalent.

(i) l is a distributive lattice.(ii) For any A,B,C 2 l(R*), (AB)C = A(BC).

Proof. (i)) (ii) is obvious. Conversely, for any a,b,c 2 l, we want to prove the distributive law a ^ (b _ c) = (a ^ b) _ (a ^ c)holds. For r 2 R, we take A,B,C 2 l(R*) as follows,

AðxÞ ¼b; if x ¼ r1; if x ¼ e;0; otherwise:

8><>: BðxÞ ¼

c; if x ¼ r;1; if x ¼ e;0; otherwise:

8><>: CðxÞ ¼

a; if x ¼ r;0; otherwise:

�

That is, A ¼ 1e þ b

r ; B ¼ 1e þ c

r and C ¼ ar. Then (AB)C(rr) = AB(r) ^ C(r) = AB(r) ^ a = (A(r) ^ B(e)) _ (A(e) ^ B(r)) ^ a =

((b ^ 1) _ (1 ^ c)) ^ a = a ^ (b _ c). Since A(BC)(rr) = (A(r) ^ BC(r)) _ (A(e) ^ BC(rr)) = (b ^ a) _ (1 ^ B(r) ^ C(r)) = (b ^ a) _(c ^ a) = (a ^ b) _ (a ^ c), by associativity law, we have, (AB)C = A(BC). It follows that the distributive law a ^ (b _ c) =(a ^ b) _ (a ^ c) holds. h

3. Determinization of l-valued finite automata and extended subset constructions

First, we show that the image set of each lattice-valued regular langauge is always a finite set of l.

Lemma 3.1 [17]. Let l be a lattice, and X a finite subset of l. Then the ^-semilattice of l generated by X, written as X^, is finite, the_-semilattice of l generated by X, denoted X_, is also finite, where X^ = {x1 ^� � �^ xk:k P 1,x1, . . ., xk 2 X} [ {1}, and X_ ={x1 _� � �_ xk:k P 1, x1, . . ., xk 2 X} [ {0}.

Proposition 3.1. Let A ¼ ðQ ;R; d; I; FÞ be an l-VFA. Then the image set of the l-valued language recA, as a mapping from R* to l, isfinite; that is, the subset ImðrecAÞ ¼ fr 2 l : 9x 2 R�; recAðxÞ ¼ rg of l is finite.

Proof. For any x = r1� � �rk 2R*, observing that recAðxÞ ¼WfIðq0Þ ^ dðq0;r1; q1Þ ^ � � � ^dðqk�1;rk; qkÞ ^ FðqkÞ : q0; . . . ; qk 2 Qg.

On input x = r1� � �rk 2 R*, there are only finite accepting paths, assumed as m, causing an initial state q0 2 I to become a finalstate qk 2 F. For the i-th accepting path, we let ai0 = I(q0), ai1 = d(q0,r1,q1), . . .,aik = d(qk�1,rk,qk) and ai,k+1 = F(qk). Then thetruth value of recAðxÞ can be calculated as, recAðxÞ ¼ ða10 ^ � � � ^ a1k ^ a1;kþ1Þ _ � � � _ ðam0 ^ � � � ^ amk ^ am;kþ1Þ. LetX = Im(d) [ Im(I) [ Im(F), then X is obviously a finite subset of l and aij 2 X for any 1 6 i 6m and 0 6 j 6 k + 1. For anyx 2 R*, by the above observation, it follows that recAðxÞ 2 ðX^Þ_, so ImðrecAÞ# ðX^Þ_. By Lemma 3.1, (X^)_ is a finite subset ofl, and thus ImðrecAÞ, as a subset of (X^)_, is also a finite subset of l. h

Due to Proposition 3.1, for any l-VFA, the image set of its recognizable lattice-valued language is always finite. Then wehave the following observation: the lattice l may be infinite as a set, but for a given l-VFA A, only a finite subset of l is em-ployed in the operating of A. This observation is the core in the introducing of extended subset construction in this section.


The notion of nondeterminism plays a central role in the theory of computation. The nondeterministic mechanism en-ables a device to change its states in a way that is only partially determined by the current state and the input symbol.The concept of l-VFA is obviously a generalization of nondeterministic finite automaton (NFA for short). In classical theoryof automata, each nondeterministic finite automaton is equivalent to a deterministic one; more precisely, there exists adeterministic finite automaton (DFA for short) which accepts the same language as the originally given nondeterministicone does. The construction of DFA from an NFA is the well-known subset construction introduced by Rabin and Scott[30]. With respect to the case of l-VFA, the situation is more complex. In fact, as shown in [33], the subset construction doesnot work well for l-VFA even for an orthomodular lattice l. That is, for an l-VFAA, one can construct an l-valued deterministicfinite automaton B, as defined in [33] using the modified subset construction. However, B is not necessarily equivalent to A,i.e., the equality recA ¼ recB does not hold in general. Some conditions that guarantee the equivalence between A and B aregiven in [33]. Therefore, it is an open problem whether an l-VFA can always be determinizable. We shall show that the an-swer is affirmative for a general lattice l. We shall introduce subset construction in lattice setting which we call it the ex-tended subset construction. First, we define the notion of deterministic l-VFA.

Definition 3.1 (c.f. [19]). An l-valued deterministic finite automaton (l-VDFA for short) is a 5-tuple A ¼ ðQ ;R; d; q0; FÞ, whereQ, R and F are the same as those in an l-valued automaton, q0 2 Q is the initial state, and the lattice-valued transition relationd is crisp and deterministic; that is, d is a mapping from Q � R into Q.

Note that our definition differs from the usual definition of a deterministic finite automaton only in that the final statesform an l-valued subset of Q. This, however, makes it possible to accept words to certain truth degrees (in the lattice setting),and thus to recognize lattice-valued languages.

For an l-VDFA, A ¼ ðQ ;R; d; q0; FÞ, its corresponding l-valued recognizability predicate recA 2 lðR�Þ is defined as: for allx = r1� � �rn 2R*,

x 2 recA¼def ð9q1 2 QÞ � � � ð9qn 2 QÞ:ðqn 2 F ^ dðq0;r1Þ ¼ q1 ^ � � � ^ dðqn�1;rnÞ ¼ qnÞ: ð4Þ

Write d* the extension of transition relation d by putting d*(q,e) = q and d*(q,xr) = d(d*(q,x),r) for any q 2 Q and x 2 R*and r 2 R, then the truth value of the proposition x 2 recA is given by,

recAðxÞ ¼ Fðd�ðq0;xÞÞ: ð5Þ

For any l-VFA, A ¼ ðQ ;R; d; I; FÞ, we now introduce the extended subset construction to construct an equivalent l-VDFAAd ¼ ðQ d;R;g; S; EÞ from A.

Let X = Im(d) [ Im(I) [ Im(F), then X is obvious a finite subset of l. Let l1 = X^. By Lemma 3.1, l1 is a ^-semilattice of l gen-erated by X and is also finite subset of l. Choose

Q d ¼ 2Q�ðl1�f0gÞ; ð6Þ

where 2Q�ðl1�f0gÞ denotes the set of all subsets of Q � (l1 � {0}). Then Qd is obvious a finite set. Take

S ¼ fðq; IðqÞÞ : q 2 Q and IðqÞ – 0g; ð7Þ

then S 2 Qd. The state transition relation g:Qd � R ? Qd is defined as, for any (q,r) 2 Q � (l1 � {0}) and r 2 R,

gðfðq; rÞg;rÞ ¼ fðp; dðq;r; pÞ ^ rÞ : p 2 Q and dðq;r;pÞ ^ r–0g; ð8Þ

and for Z 2 Qd,

gðZ;rÞ ¼[fgðfðq; rÞg;rÞ : ðq; rÞ 2 Zg: ð9Þ

By the definition of l1, l1 is closed under finite meet operation, i.e., for any a,b 2 l1, a ^ b 2 l1, it follows that, for any r 2 l1 andfor any (p,r,q) 2 Q � R � Q, r ^ d(p,r,q) 2 l1, and thus g({(q,r)},r) 2 Qd for any (q,r) 2 Q � (l1 � {0}). Then the mapping g iswell defined. The l-valued final state E : Qd ? l is defined by, for any Z 2 Qd,

EðZÞ ¼_fr ^ FðqÞ : ðq; rÞ 2 Zg: ð10Þ

Then Ad is an l-VDFA.

Theorem 3.1. For any l-VFA, A ¼ ðQ ;R; d; I; FÞ, the l-VDFA Ad ¼ ðQd;R;g; S; EÞ constructed above is equivalent to A, i.e.,recA ¼ recAd .

Proof. We wish to show by induction on the length jxj of input string x that g*(S,x) = {(qn, I(q0) ^ d(q0,r1,q1) ^� � �^d(qn�1,rn,qn)) : q0, . . .,qn 2 Q and rn = I(q0) ^ d(q0,r1,q1) ^� � �^ d(qn�1,rn,qn) – 0}, where x = r1� � �rn for n P 0. The result istrivial for jxj = 0, since x = e and g*(S,x) = {(q0, I(q0)) : q0 2 Q and I(q0) – 0}. Suppose that the hypothesis is true for inputsof length n or less. Let x = r1� � �rn+1 be a string of length n + 1, write x = r1� � �rn, then x = xrn+1. Then

g�ðS; xrnþ1Þ ¼ gðg�ðS; xÞ;rnþ1Þ:


By the inductive hypothesis,

g�ðS; xÞ ¼ fðqn; Iðq0Þ ^ dðq0;r1; q1Þ ^ � � � ^ dðqn�1;rn; qnÞÞ : q0; . . . ; qn 2 Q and rn

¼ Iðq0Þ ^ dðq0;r1; q1Þ ^ � � � ^ dðqn�1;rn; qnÞ – 0g:

By the definition of g,

gðg�ðS; xÞ;rnþ1Þ ¼[

ðqn ;rnÞ2g�ðS;xÞgðfðqn; rnÞg;rnþ1Þ ¼

[ðqn ;rnÞ2g�ðS;xÞ

fðqnþ1; rn ^ dðqn;rnþ1; qnþ1ÞÞ : qnþ1

2 Q and rn ^ dðqn;rnþ1; qnþ1Þ– 0g¼ fðqnþ1; Iðq0Þ ^ dðq0;r1; q1Þ ^ � � � ^ dðqn�1;rn; qnÞ ^ dðqn;rnþ1; qnþ1ÞÞ : q0; . . . ; qnþ1 2 Q and rnþ1

¼ Iðq0Þ ^ dðq0;r1; q1Þ ^ � � � ^ dðqn;rnþ1; qnþ1Þ– 0g;

which establishes the inductive hypothesis.By the definition of l-valued final state E, for any input x = r1� � �rn 2R* (n P 0), we have

recAd ðxÞ ¼ Eðg�ðS;xÞÞ ¼_frn ^ FðqnÞ : ðqn; rnÞ 2 g�ðS;xÞg ¼

_fIðq0Þ ^ dðq0;r1; q1Þ ^ � � � ^ dðqn�1;rn; qnÞ ^ FðqnÞ

: q0; . . . ; qn 2 Q and Iðq0Þ ^ dðq0;r1; q1Þ ^ � � � ^ dðqn�1;rn; qnÞ – 0g

¼_fIðq0Þ ^ dðq0;r1; q1Þ ^ � � � ^ dðqn�1;rn; qnÞ ^ FðqnÞ : q0; . . . ; qn 2 Qg ¼ recAðxÞ:

Thus recAd ¼ recA; A and Ad are equivalent. h

Theorem 3.1 gives the subset construction of finite automaton in lattice setting (under lattice-valued logic). In fact, in thecase of l = B2 = {0,1} (in Fig. 1), the underlying logic is the classical logic, the extended subset construction is just the ordinarysubset construction.

We give an example to illustrate the technique of the extended subset construction introduced above.

Example 3.1. Let l be the free lattice generated by four elements a, b, c and d. Then l is an infinite lattice, which is not adistributive lattice and is not locally finite generated. Here a lattice l is called locally finite generated, which means that, forany finite subset X of l, the sublattice of l generated by X is also finite.

An l-VFAA ¼ ðQ ;R; d; I; FÞ is defined as follows, Q = {p,q}, R = {r}, I(p) = 1 and I(q) = c, F(p) = c and F(q) = 1, and d(p,r,q) = a,d(p,r,p) = b, d(q,r,p) = c and d(q,r,q) = d. See Fig. 3 for the representation of A.

Using the extended subset construction, the determinization of A is induced as follows. In this example, l is an infinitelattice, and l1 = {a,b,c,d,0,1}^ = {a,b,c,d,0,1,a ^ b,a ^ c,a ^ d,b ^ c,b ^ d,c ^ d,a ^ b ^ c,a ^ b ^ d,a ^ c ^ d,b ^ c ^ d} has 16elements. In the construction of Ad, the state set Qd is 2Q�ðl1�f0gÞ; Ad will have 230 states. To give a full construction of Ad

is a tedious work. Fortunately, it is sufficient to give those states which are useful in generating the l-valued languagerecognized by Ad from the initial state S.

The initial state is S = {(p,1), (q,c)}. By the simple calculation, we have g(p0,r) = p1, g(p1,r) = p2, g(p2, r) = p3, g(p3,r) = p3,where p0 = S, p1 = {(p,b), (p,c), (q,a), (q,c ^ d)}, p2 = {(p,b), (p,a ^ c), (p,b ^ c), (p,c ^ d), (q,a ^ b), (q,a ^ c), (q,a ^ d), (q,c ^ d)}, p3 ={(p,b), (p,a ^ c), (p,b ^ c), (p,c ^ d), (p,a ^ b ^ c), (p,a ^ c ^ d), (p,b ^ c ^ d), (q,a ^ b), (q,a ^ c), (q,a ^ d), (q,c ^ d)}, (q,a ^ b ^ c),(p,a ^ b ^ d), (p,a ^ c ^ d)}. Let P = {p0,p1,p2,p3}, then the state transition function g is defined as, g(p0,r) = p1, g(p1,r) = p2

and g(p2,r) = p3, g(p3,r) = p3. The l-valued final state E is defined as, E(p0) = c, E(p1) = a _ c and E(p2) = E(p3) = (a ^ b) _(a ^ c) _ (a ^ d) _ (b ^ c) _ (c ^ d). This complete the construction of Ad ¼ ðP;R;g; p0; EÞ (see Fig. 4). Then recAð¼ recAd Þ can besimply calculated as follows,

recAðxÞ ¼c; if x ¼ e;a _ c; if x ¼ r;ða ^ bÞ _ ða ^ cÞ _ ða ^ dÞ _ ðb ^ cÞ _ ðc ^ dÞ; otherwise:

8><>:

Fig. 3. The automaton A in Example 3.1.

Fig. 4. The automaton Ad in Example 3.1.


In [1,15,33], the authors used another approach, which is called lattice-valued subset construction here, to construct thedeterminization of an l-VFA. However, their constructions were not universally valid for any lattice. In fact, in [1,15,33], theauthors assumed that the lattice satisfies the locally finite generated condition, that is, for any finite subset X of a lattice, thesublattice generated by X is also finite. In general, locally finite generated condition is not satisfied by any lattice. For exam-ple, the free lattice generated by three elements is itself infinite, which does not satisfy the locally finite generated condition.Even if a lattice l satisfies the locally finite generated condition, lattice-valued subset construction does not bring the determ-inization of an l-VFA. We give here the related explicit construction as follows.

For a lattice l satisfying locally finite generated condition, let A ¼ ðQ ;R; d; I; FÞ be an l-VFA. Take X = Im(d) [ Im(I) [ Im(F)and let l1 be the sublattice generated by X. Then l1 is finite. Construct an l-VDFA A0 from A as follows, write A0 ¼ðQ 0;R; d0; q00; F

0Þ, where Q 0 ¼ lQ1 ; q00 ¼ I 2 Q 0; d0 : Q 0 � R! Q 0 is defined as d0(A,r)(q) =

Wp2Q(A(p) ^ d(p,r,q)) for any A 2 Q0,

r 2 R and q 2 Q, and F0 : Q0 ? l is defined as F0(A) =W

q2Q(A(q) ^ F(q)) for any A 2 Q0. We call A0 the lattice-valued determini-zation of A, which was introduced in [1,15,33] as the determinization of an l-VFA. We have the following observation.

Proposition 3.2. The following statements are equivalent.

(i) l is a distributive lattice.(ii) For any l-VFA A; A is equivalent to A0, i.e., recA ¼ recA0 .

Proof(i)) (ii) is easy.(ii)) (i) It suffices to show that the meet ^ is distributive over the union _; that is,

a ^ ðb _ cÞ ¼ ða ^ bÞ _ ða ^ cÞ

for all a,,b,c 2 l. Let a,b,c 2 l.We construct an l-VFA, A ¼ ðQ ;R; d; I; FÞ, as follows, Q = {q0,q1,q2,q3,q4}, let r 2R, then d(q0,r,q1) = b, d(q0,r,q2) = c,

d(q1,r,q3) = 1, d(q2,r,q3) = 1, d(q3,r,q4) = a, and d(q,r,p) = 0 in other case. Let I = {q0}, F = {q4}. This l-VFA is represented inFig. 5.

By a simple calculation, we have d*(q0,rrr,q4) = (a ^ b) _ (a ^ c). Hence, recAðrrrÞ ¼ ða ^ bÞ _ ða ^ cÞ.Consider the l-valued subset construction A0 from A. Then

d0ðI;rÞðpÞ ¼_r2Q

IðrÞ ^ dðr;r;pÞ ¼ dðq0;r;pÞ:

d0(I,rr)(p) = d0(d0(I,r),r)(p) =W

r2Qd0(I,r)(r) ^ d(r,r, p) =W

r2Qd(q0,r, r) ^ d(r,r, p) = (d(q0,r, q1) ^ d(q1,r, p)) _ (d(q0,r, q2) ^d(q2,r,p)) = (b ^ d(q1,r,p)) _ (c ^ d(q2,r,p)), which is equal to b _ c if p = q3 and 0 in other cases.

d0(I,rrr)(p) =W

r2Qd0(I,rr)(r) ^ d(r,r,p) = d0(I,rr)(q3) ^ d(q3,r,p) = (b _ c) ^ d(q3,r,p), which is equal to a ^ (b _ c) if p = q4

and 0 in other cases.Therefore, recA0 ðrrrÞ ¼ F 0ðd0ðI;rrrÞÞ ¼

Wp2Q d0ðI;rrrÞðpÞ ^ FðpÞ ¼ a ^ ðb _ cÞ. Finally, from assumption (ii), we assert that

a ^ (b _ c) = (a ^ b) ^ (a ^ c). h

Theorem 3.1 and Proposition 3.2 show that the distributive law is not necessary to the determinization of an l-VFA, and itindeed provides some convenience in the simple construction of the determinization of an l-VFA. This view may be seen asone of the mail contribution of the paper.

In fact, we have generalized the lattice-valued determinization of an l-VFA to fuzzy finite automata with membershipvalues in lattice-order monoids in [19]. Further discussion of determinization of an l-VFA was done in [10].

We continue to study the relationship between l-VFA and l-VFA with e-moves. Let us first represent the definition of l-VFAwith e-moves.

Definition 3.2 (c.f. [22,33,26]). An l-valued automaton with e-moves (l-VFAe for short) is a five-tuple A ¼ ðQ ;R; d; I; FÞ inwhich all components are the same as in an l-valued automaton (without e-moves), but the domain of the l-valued transitionrelation d is changed to Q � (R [ {e}) � Q; that is, d is a mapping from Q � (R [ {e}) � Q into l, where e stands for the emptystring of input symbols.

Fig. 5. The automaton constructed in the proof of (ii)) (i) in Proposition 3.2.


Now let A ¼ ðQ ;R; d; I; FÞ be an l-valued automaton with e-moves. Then the recognizability recA is also defined as an l-valued unary predicate over R*, and it is given by

x 2 recA¼def ð9n P 0Þð9s1 2 R [ fegÞ � � � ð9sn 2 R [ fegÞ � ð9q0 2 QÞ � � � ð9qn 2 QÞ � ðq0 2 I ^ qn 2 F ^ ðq0; s1; q1Þ2 d ^ � � � ^ ðqn�1; sn; qnÞ 2 d ^ s1 � � � sn ¼ xÞ

for all x 2 R*. The defining equation of recA may be rewritten in terms of truth value as follows:

recAðxÞ ¼_fIðq0Þ ^ dðq0; s1; q1Þ ^ � � � ^ dðqn�1; sn; qnÞ ^ FðqnÞ : n P 0; s1; . . . ; sn 2 R [ feg

satisfying s1 � � � sn ¼ x; and q0; . . . ; qn 2 Qg:

We shall show that l-VFA and l-VFAe are equivalent in the sequel. First, we study a special kind of l-VFAe in which tran-sition is crisp, that is, d is a crisp subset of Q � (R [ {e}) � Q. In this case, d can be seen as a mapping from Q � (R [ {e}) to 2Q.

Let A ¼ ðQ ;R; d; q0; FÞ be an l-VFAe with crisp transition and with a unique initial state q0 2 Q, the explicit expression ofrecA can be induced as follows. First, we give the extension d* : 2Q � R* ? 2Q using the notion of e-closure. For q 2 Q, the e-closure of q, denoted EC(q), is defined as,

ECðqÞ ¼ fp 2 Q : there exists n P 0 and q0; . . . ; qn satisfying qi 2 dðqi�1; eÞ for any i ¼ 1; . . . ;n;

in which q0 ¼ q and qn ¼ pg:

For any subset X of Q, the e-closure of X, denoted EC(X), is defined as

ECðXÞ ¼[q2X

ECðqÞ:

In particular, EC({q}) = EC(q). Then d* is defined inductively as,

d�ðq; eÞ ¼ ECðqÞ;d�ðq;xrÞ ¼ ECðdðd�ðq;xÞ;rÞÞ for any q 2 Q ; x 2 R� and r 2 R:

Then

d�ðX;xÞ ¼[q2X

d�ðq;xÞ:

It follows that

d�ðq;xrÞ ¼ d�ðd�ðq;xÞ;rÞ

for any q 2 Q, x 2R* and r 2 R. By the definition of unitary predicate rec over R*, the truth valued recA for an l-VFAe withcrisp transition is defined as follows: for any x 2 R*,

recAðxÞ ¼_fFðqÞ : q 2 d�ðq0;xÞg:

We construct an equivalent l-VFA B from the above A as follows, where B ¼ ðQ ;R;g; q0; EÞ. The l-valued transition g isdefined as: for any q 2 Q and r 2 R,

gðq;rÞ ¼ d�ðq;rÞ:

If q – q0, then

EðqÞ ¼ FðqÞ;
and _
Eðq0Þ ¼ fFðqÞ : q 2 ECðq0Þg:

Note that B has no e-transitions.

Lemma 3.2. For any l-VFAe with crisp transition A, the l-VFA B constructed as above is equivalent to A, i.e., recA ¼ recB .

Proof. We wish to show by induction on jxj that g*(q,x) = d*(q,x). However, this statement may not be true for x = e, sinceg*(q,e) = {q}, while d*(q,e) = EC(q). We therefore begin our induction at 1.

Let jxj = 1. Then x is a symbol r, and g(q,r) = d*(q,r) by definition of g. Suppose that the hypothesis holds for inputs oflength n or less. Let x = xr be a string of length of n + 1 with symbol r in R. Then

g�ðq; xrÞ ¼ gðg�ðq; xÞ;rÞ:

By the inductive hypothesis, g*(q,x) = d*(q,x). Let d*(q,x) = X, we must show that g(X,r) = d*(q,xr). But

gðX;rÞ ¼[q2Q

gðq;rÞ ¼[q2X

d�ðq;rÞ:


Then as X = d*(q,x) we have
[q2X
d�ðq;rÞ ¼ d�ðq; xrÞ:

Thus
g�ðq; xrÞ ¼ d�ðq; xrÞ:
To complete the proof we shall show that recBðxÞ ¼WfFðqÞ : q 2 d�ðq0;xÞg.

If x = e, this statement is immediate from the definition of E. That is, g*(q0,e) = {q0}, then recBðeÞ ¼WfEðqÞ : q 2

g�ðq0; eÞg ¼ Eðq0Þ ¼WfFðqÞ : q 2 d�ðq0; eÞg.

If x – e, then x = xr for some symbol r. We have two cases to discuss.

Case I: q0 R g*(q0,xr). By the definition of E and the equality g*(q0,xr) = d*(q0,xr), it follows that

recBðxrÞ ¼_fEðqÞ : q 2 g�ðq0; xrÞg ¼

_fFðqÞ : q 2 d�ðq0; xrÞg:

Case II: q0 2 g*(q0,xr). Then EC(q0) # d*(q0,xr) = g*(q0,xr). Thus,

recBðxrÞ ¼_fEðqÞ : q 2 g�ðq0; xrÞg ¼

_fEðqÞ : q 2 d�ðq0; xrÞg ¼

_fEðqÞ : q 2 d�ðq0; xrÞ � fq0gg _ Eðq0Þ ¼

_fFðqÞ : q

2 d�ðq0; xrÞ � fq0gg __fFðqÞ : q 2 ECðq0Þg ¼

_fFðqÞ : q 2 d�ðq0; xrÞg:

Hence, for any x 2 R�; recBðxÞ ¼WfFðqÞ : q 2 d�ðq0; xrÞg ¼ recAðxÞ. This shows that recA ¼ recB , and thus A and B are

equivalent. h

Let A ¼ ðQ ;R; d; I; FÞ be an l-VFAe. We construct an equivalent l-VFAe B ¼ ðP;R;g; S; EÞ with crisp transition from A asfollows.

Let X = Im(d) [ Im(I) [ Im(F), and l1 = X^. Choose P ¼ 2Q�ðl1�f0gÞ, and S = {(q, I(q)) : q 2 Q and I(q) – 0}, then P is a finite setand S 2 P. The state transition g : P � (R [ {e}) ? P is defined by,

gðfðq; rÞg; sÞ ¼ fðp; r ^ dðq; s; pÞÞ : p 2 Q and r ^ dðq; s;pÞ–0g

for any (q,r) 2 Q � (l1 � {0}) and s 2 R [ {e}. We define

gðZ; sÞ ¼[ðq;rÞ2Z

gðfðq; rÞg; sÞ

for any Z 2 P and s 2 R [ {e}. Then g is well defined as discussed in the extended subset construction from an l-VFA to anl-VDFA. The lattice-valued final state E : P ? l is defined as,

EðZÞ ¼_fr ^ FðqÞ : ðq; rÞ 2 Zg:

Lemma 3.3. For any l-VFAe A ¼ ðQ ;R; d; I; FÞ, the l-VFAe with crisp transition B constructed as above is equivalent to A, i.e.,
recA ¼ recB .
Proof. The proof is very similar to that of Theorem 3.1, we omit it here. h

Combining the above two lemmas, we can conclude the following theorem which shows the equivalence between l-VFAeand l-VFA.

Theorem 3.2. For any l-VFAe A, there is an l-VFA B such that A and B are equivalent, i.e., recA ¼ recB .

Combining Theorems 3.1 and 3.2, we can see the equivalence between l-VFAe, l-VFA and l-VDFA.

Corollary 3.1. For any l-VFAe A, there is an l-VDFA B such that A and B are equivalent, i.e., recA ¼ recB .

As an application of Theorem 3.1, we present pumping lemma in lattice-valued setting as follows, which is independent ofthe distributive law of the used truth-valued lattice. Qiu has presented the pumping lemma under automata theory based oncomplete residuated lattice-valued logic recently, and see [26] for the details.

Proposition 3.3 (Pumping lemma in lattice setting). For an l-regular language A : R* ? l, there exists positive integer n, for anyinput string z 2 R*, if jzjP n, then there are u,v,w 2R* such that juvj 6 n, v – e, z = uvw, and for any non-negative integer k, theequality A(uvkw) = A(uvw) holds.

Proof. Since A is l-regular, it is accepted by an l-VDFA A ¼ ðQ ;R; d; q0; FÞ with some particular number of states, say n. Con-sider an input of n or more symbols z = r1� � �rm, m P n, and for i = 1, . . .,m, let d*(q0,r1� � �ri) = qi. It is not possible for each ofthe n + 1 states q0, . . .,qn be different, since there are only n different states. Thus there are two integers j1 and j2,


0 6 j1 < j2 6 n, such that qj1¼ qj2

. Let u ¼ r1 � � �rj1 ; v ¼ rj1þ1 � � �rj2 ; w ¼ rj2þ1 � � �rm, then juvj = j2 6 n, v – e and z = uvw.Observing that d�ðq0;r1 � � �rj1rj2þ1 � � �rmÞ ¼ d�ðd�ðq0;r1 � � �rj1 Þ;rj2þ1 � � �rmÞ ¼ d�ðqj1

;rj2þ1 � � �rmÞ ¼ d�ðqj2;rj2þ1 � � �rmÞ ¼ qm,

and for any k P 1; d�ðq0;r1 � � �rj1 ðrj1þ1 � � �rj2 Þkrj2þ1 � � �rmÞ ¼ d�ðd�ðd�ðq0;r1 � � �rj1 Þ; ðrj1þ1 � � �rj2 Þ

kÞ;rj2þ1 � � �rmÞ ¼ d�ðd�ðqj1;

ðrj1þ1 � � �rj2 ÞkÞ;rj2þ1 � � �rmÞ ¼ d�ðqj2

;rj2þ1 � � �rmÞ ¼ qm. Therefore, for any k P 0; AðuvkwÞ ¼ recAðuvkwÞ ¼ Fðd�ðq0;uvkwÞÞ ¼FðqmÞ ¼ Fðd�ðq0;uvwÞÞ ¼ recAðuvwÞ ¼ AðuvwÞ. h

4. Kleene Theorem for l-valued finite automata

We use lR(R) to denote the set of l-regular languages over R. In this section, we will study the regular operations on lR(R),and show that the Kleene theorem (c.f. [12]) holds for l-VFA. As a corollary, we shall show that Kleene theorem holds underquantum logic, which strengthens the results obtained in [33]. Indeed, in [33], the author declared that Kleene theorem inquantum logic was relied on the distributive law of orthomodular lattice, we shall show that this restriction is not necessary.

Let us first give a simple characterization of l-valued regular languages.

Theorem 4.1. Let A : R* ? l be an l-valued language over R. Then the following statements are equivalent.

(i) A is an l-regular language.(ii) There exist k1, . . ., km 2 l � {0}, and regular languages L1, . . ., Lm such that A ¼

Wmi¼1ki1Li

, where 1Lidenotes the characteristic

function of Li.(iii) There exist k1, . . ., km 2 l � {0}, and pairwise disjoint regular languages L1, . . ., Lm satisfying the equality A ¼

Wmi¼1ki1Li

.

Proof(i)) (iii) Since A is an l-valued regular language, there is an l-VDFA A ¼ ðQ ;R; d; q0; FÞ to recognize A. That is, for allx 2 R�; AðxÞ ¼ recAðxÞ ¼ Fðd�ðq0;xÞÞ. Write Im(F) � {0} = {k1, . . .,km}, and let Fi = {q 2 Q : F(q) = ki}. For this Fi, we constructa DFA, Ai ¼ ðQ ;R; d; q0; FiÞ. Let the language recognized by Ai be Li, then Li is a regular language, and evidently, the family{L1, . . .,Lm} is pairwise disjoint. Moreover, A(x) = r iff F(d*(q0,x)) = r, iff there is i such that r = ki and x 2 Li, which shows thatA ¼

Wmi¼1ki1Li

.(iii)) (ii) is obvious.(ii)) (i) Since each Li is regular, there is a DFA Ai ¼ ðQi;R; d; q0i; FiÞ to recognize Li. We can assume that Qi \ Qj = ;

whenever i – j. Define an l-VFA, A ¼ ðQ ;R; d; q0; FÞ as follows, Q ¼Sm

i¼1Qi [ fq0g, where q0 RSm

i¼1Qi, and d : Q � R ? 2Q is,d(q0,r) = {d1(q01,r), . . .,dm(q0m, r)}, for q 2 Qi, d(q,r) = {di(q,r)}; F(q0) =

W{ki : q0i 2 Fi}, and when q – q0,

FðqÞ ¼ki; if q 2 Fi;

0; otherwise:

�

Then it can be easily verified that A ¼ recA ¼Wm

i¼1ki1Li. Hence A is an l-valued regular language. h

We call the l-valued language satisfying the condition (ii) or (iii) in the above theorem the l-valued recognizable step lan-guage, and write the set of all l-valued recognizable languages on R as step(R), which is equal to lR(R).

The following proposition gives the stratified characterization of l-valued recognizable step languages.

Corollary 4.1. Let A : R* ? l be an l-valued language over R. Then the following statements are equivalent.

(i) A is an l-regular language.(ii) The image set Im(A) is finite, and for any r 2 Im(A) � {0}, the r-cut of A, Ar = {x 2 R* : A(x) P r} is a regular language over

R and A ¼W

r2Im�f0gr1Ar .(iii) The image set Im(A) is finite, and for any r 2 Im(A) � {0}, the r-level of A, A[r] = {x 2 R* : A(x) = r} is a regular language over

R and A ¼W

r2Im�f0gr1A½r� .

Theorem 4.2. The family step(R) or lR(R) is closed under the operations of union, intersection, scalar product, concatenation andKleene closure.

Proof. Let A,B 2 step(R). By Theorem 4.1, we can assume A ¼Wm

i¼1ki1Li; B ¼

Wnj¼1dj1Mj

, where, all Li and Mj are regular lan-guages and fLigm

i¼1 are pairwise disjoint, fMjgmi¼1 are also pairwise disjoint.

With respect to the union, we have A _ B ¼Wm

i¼1ki1Li_Wn

j¼1dj1Mj. By Theorem 4.1, it follows that A _ B 2 step(R).

With respect to the intersection, we have A ^ B ¼Wm

i¼1Wn

j¼1ðki ^ djÞ1Li\Mj. By Theorem 4.1, it follows that A ^ B 2 step(R).

With respect to the scalar product, for each r 2 l, we have rA(x) = r ^ A(x), then rA ¼Wm

i¼1ðr ^ kiÞ1Li. Therefore,

rA 2 step(R).For the operation of concatenation, since AB(x) =

W{A(x1) ^ B(x2) : x = x1x2}, it follows that AB ¼

Wmi¼1Wn

j¼1ðki ^ djÞ1LiMj.

This shows that AB 2 step(R).


For the Kleene closure, A* is defined by, A*(x) =W

{A(x1) ^ � � �A(xk) : k P 0, x = x1� � �xk} for any x 2R *. SinceA ¼

Wmi¼1ki1Li

, and L1, . . .,Lm are pairwise disjoint regular languages and ki – 0 for each i, it follows that Im(A) � {0} ={k1, . . .,km}, and Li = {x 2 R* : A(x) = ki} (i = 1, . . .,m). For any nonempty subset K of the set {1,2, . . .,m}, we can assume thatK = {i1, . . ., is}. Let rK ¼ ki1 ^ � � � ^ kis ; LðKÞ ¼

Sp1 ��ps

Lþp1Lþp2

L�p1Lþp3ðLp1[ Lp2

Þ� � � � Lþps�1ðLp1[ � � � [ Lps�2

Þ�LþpsðLp1[ � � � [ Lps

Þ�, wherep1� � �ps is a permutation of {i1, . . ., is}, and L(K) is taken unions under all permutations of {i1, . . ., is}. Hence L(K) is a regularlanguage. It is easily verified that A* =

W;–K # {1,2,. . .,m}rK 1L(K) _ 1{e}. By Theorem 4.1, it follows that A* 2 step(R). h

Remark 4.1. Recall that a negation on a lattice l is a mapping h from l into l such that, a 6 b implies h(a) P h(b) and hh(a) = afor any a,b 2 l. Furthermore, if the negation h also satisfies the conditions: a ^ h(a) = 0 and a _ h(a) = 1 for any a 2 l, then h iscalled the complement over l. In this case, l is an orthocompletment lattice. An orthomodular lattice is an orthocompletmentlattice satisfying the orthomodular laws as presented as follows:

a 6 b implies a _ ðhðaÞ ^ bÞ ¼ b:

If l is a lattice with a negation h, then for an l-language A : R* ? l, we can define the negation of A on l(R*) as, h(A)(x) = -h(A(x)). Then for any A 2 lR(R*), h(A) is also l-regular. In fact, if A ¼

Wmi¼1ki1Li

for some ki 2 l and regular language Li overR, then hðAÞ ¼

Wmi¼1hðkiÞ1R��Li

_ 1R��ðL1[��[LmÞ.

Definition 4.1 ([19,33]). The language of l-valued regular expressions over alphabet R has the alphabet(R [ {e,;}) [ (l [ {+, �,�}). The symbols in R [ {e,;} will be used to denote atomic expressions, and the symbols in l [ {+, �,�}will be used to stand for operators for building up compound expressions: � and all r 2 l are the unary operators, and +, �are binary ones. We use a, b to act as meta-symbols for regular expressions and L(a) for the language denoted by expressiona. More explicitly, L(a) will be used to denote an l-valued subset of R*; that is, L(a) 2 l(R*). The l-valued regular expressionsand the l-valued languages denoted by them are formally defined as follows:

(i) For each r 2 R, r is a regular expression, and L(r) = {r}; e and ; are regular expressions, and L(e) = {e}, L(;) = ;.(ii) If both a and b are regular expressions, then for each r 2 l, ra, a + b, a � b, a* are all regular expressions, and

L(ra) = rL(a), L(a + b) = L(a) _ L(b), L(a � b) = L(a)L(b), L(a*) = L(a)*.

Theorem 4.3 (Kleene Theorem in lattice setting). For an l-valued language A 2 l(R*), A can be recognized by an l-VFA iff thereexists an l-valued regular expression a over R such that A = L(a).

Proof. If A can be recognized by an l-VFA, then by Theorem 4.1, there exist k1, . . .,kn 2 l � {0}, and regular languages L1, . . .,Ln

such that A ¼Wn

i¼1ki1Li. Since each Li is a regular language, by classical Kleene Theorem, there exists a regular expression ai

over R such that L(ai) = Li. Let a = k1a1 + � � � + knan, then a is an l-valued regular expression, and LðaÞ ¼Wn

i¼1kiLðaiÞ ¼Wni¼1ki1Li

¼ A.Conversely, assume that there exists an l-valued regular expression a such that A = L(a). We show that A can be

recognized by an l-VFA inductively on the number of operation symbols occurring in a. If there is no operation symbol in a,then a = r 2R, e or ;. In this case, L(a) = {r}, {e} or ;, and L(a) can be recognized by a classical DFA. The classical DFA isevidently an l-VDFA, so L(a) can be recognized by an l-VDFA in this case. Inductively, since the family of recognizablelanguages by l-VDFA is closed under union, intersection, scalar product, concatenation and Kleene closure (by Theorem 4.2),it follows that L(a) can be recognized by an l-VDFA for any l-valued regular expression a. h

Remark 4.2. As a corollary, when l is an orthomodular lattice, Theorem 4.3 is exactly the Kleene theorem in quantum logic.Therefore, the strictly restrictions on Kleene theorem under quantum logic as presented in [33] are not necessary. It alsoshows that Kleene theorem holds in any lattice-valued finite automata, the distributive law is not necessary to the validityof Kleene theorem.

5. Minimization of lattice-valued finite automata

Finite automata are used to design complex system. Finding a minimum representation of finite automata is a critical is-sue arising in such design. In this part, we shall extend the minimal algorithm of deterministic finite automata (DFA) in clas-sical logic [8] to those of lattice setting. To make the meaning of minimization of an l-VDFA clear, we first introduce somenecessary notions, which are slight modifications of related notions in [20,23].

Given an l-VDFA A ¼ ðQ ;R; d; q0; FÞ, let Qa = {d*(q0,h) : h 2R*}. If Q = Qa, then A is called accessible. The elements of Qa arecalled accessible states, and the elements of Q � Qa are called inaccessible elements. Consider da ¼ djQa�R, the restriction map-ping of d on Qa � R, and Fa ¼ FjQa , the restriction mapping on Qa. Then Aa ¼ ðQa;R; da; q0; F

aÞ is called the accessible part of A.It can be easily shown that these two l-VDFAs are equivalent, that is, they accept the same l-language, recA ¼ recAa . And wesay that an l-VDFA is accessible iff A ¼ Aa.


Obviously, Aa is constructed from A by removing all inaccessible states.Next, we give some notions necessary to compare two or more l-VDFAs. Given two l-VDFAs A ¼ ðQ ;R; d; q0; FÞ and

B ¼ ðP;R;g; p0; EÞ, a homomorphism from A to B, denoted u : A ! B, is a mapping u : Q ? P such that u(d(q,r)) = g(u(q),r),u(q0) = p0 and E(u(q)) 6 F(q) for any q 2 Q and r 2R. It can be easily deduced that u(d*(q,h)) = g*(u(q),h) for any h 2 R*. Fur-thermore, if E(p) =

W{F(q) : u(q) = p}, then u is called a strong homomorphism. A strong homomorphism u : A ! B is called an

isomorphism if u is one-to-one and onto.

Lemma 5.1. For two l-VDFAs A and B, a homomorphism u : A ! B is an isomorphism iff there is another homomorphismw : B ! A such that u � w ¼ 1B and w �u ¼ 1A.

Proof. ‘‘If part’’: In this case, u is a bijection, i.e., u is one-to-one and onto. What is left is to show that u is a strong homo-morphism. In fact, for any q 2 Q, E(u(q)) 6 F(q) = F(wu(q)) 6 E(u(q)), thus E(u(q)) = F(q)‘‘Only if part’’: Since u is an isomor-phism, u is a bijection from Q to P, then there exists an inverse of u, which is denoted as w:P ? Q; we show that w is also ahomomorphism. First, since u(q0) = p0, w(p0) = q0. Second, if w(p) = q, then u(d*(q,h)) = g*(u(q),h) = g*(p,h), thus w(g*(p,h)) =d*(q,h) = d*(w(p),h). Finally, since u is strong and bijection, E(p) = F(w(p)). h

Lemma 5.2. If u : A ! B is a homomorphism between two l-VDFAs, then recB 6 recA, i.e., recBðhÞ 6 recAðhÞ for any h 2R*. Fur-thermore, if u is strong, then recA ¼ recB .

Proof. For any h 2 R�; recBðhÞ ¼ Eðg�ðp0; hÞÞ ¼ Eðg�ðuðq0Þ; hÞÞ ¼ Eðuðd�ðq0; hÞÞÞ 6 Fðd�ðq0; hÞÞ ¼ recAðhÞ, that is, recB 6 recA. Ifu is strong, then recBðhÞ ¼ Eðg�ðp0; hÞÞ ¼ Eðg�ðuðq0Þ; hÞÞ ¼ Eðuðd�ðq0; hÞÞÞ ¼

WfFðqÞ : uðqÞ ¼ uðd�ðq0; hÞÞgP Fðd�ðq0; hÞÞ ¼

recAðhÞ, that is, recB P recA, thus recB ¼ recA. h

For two equivalent l-VDFAs A and B, given a homomorphism u : A ! B, and consider the restriction of u on Aa. For anyq 2 Qa, there exists h 2 R* such that q = d*(q0,h), in this case, u(q) = u(d*(q0,h)) = g*(p0, h). Let p = g*(p0,h), then p 2 Pa. Thisdemonstrates that the restriction of u on Aa is just the homomorphism ua : Aa ! Ba. Furthermore, we show that ua is alsostrong and surjective. For any p 2 Pa, there is h 2 R* such that p = g*(p0,h). We select q = d*(q0,h). Then q 2 Qa and ua(q) = p,thus ua is surjective.

Since ua is surjective and Fðd�ðq0; hÞÞ ¼ recAðhÞ ¼ recBðhÞ ¼ Eðg�ðp0; hÞÞ ¼ Eðuðd�ðq0; hÞÞÞ; ua is also a strong homomor-phism. We thus obtain the following results.

Theorem 5.1. Let u : A ! B be a homomorphism and assume recA ¼ recB . We can obtain the restriction mappingua : Aa ! Ba; ua is a strong homomorphism and is a surjective mapping.

Corollary 5.1. Let A and B be two accessible l-VDFAs such that recA ¼ recB . There is then at most one homomorphism u : A ! B.This homomorphism is strong and is surjective mapping.

Proof. We need to prove that there is at most one homomorphism fromA to B. If there are two homomorphisms u1,u2 fromA to B, then for any q 2 Q = Qa, there is h 2 R*, u1(q) = u1(d*(q0,h)) = g*(u1(q0),h) = g*(p0,h) and similarly, u2(q) = g*(p0,h).Thus u1(q) = u2(q) for any q 2 Q. This shows that u1 = u2. h

As the above two results, there are preorders on the class, ACCE(A), of all the accessible l-VDFAs A such that recA ¼ A,where A is an l-regular language. If A and B are objects in ACCE(A), there exists at most one onto strong homomorphismu : A ! B. If we write B 6 A whenever such an onto strong homomorphism exists, we obtain a preorder on the class ofall accessible l-VDFAs accepting the same l-language A. We show that if A 6 B and B 6 A then A and B are isomorphic. In-deed, if u : A ! B and w : B ! A are strong homomorphism, then so is u � w : B ! B and w �u : A ! A. Thus, by Corollary5.1, u � w ¼ 1B; w �u ¼ 1A, where 1A is the identity mapping on Q. Thus u is an isomorphism with w as inverse as stated inLemma 5.1. The isomorphisic class of accessible l-VDFAs accepting l-language A thus forms a partially order set. The naturalquestion to be asked is whether this ordered set has a minimal element. An equivalent formulation concerns an issues ofexistence of an accessible l-VDFA A0 accepting A such that for any other such l-VDFA A a strong homomorphism A ! A0

exists. The l-VDFA A0 with this property is then determined uniquely up to an isomorphism, and it will be appropriate tocall it minimal. In the sequel, we study the structure of minimal l-VDFA.

We first proceed with the characterization of an l-regular language.

Theorem 5.2 (Myhill–Nerode theorem in lattice setting). The following three statements are equivalent.

(i) The l-language A 2 l(R*) is accepted by some l-VFA.(ii) There is a right invariant equivalence relation � of finite index on R* such that, for any x,y 2 R*, x � y implies A(x) = A(y),

where an equivalence relation � on R* being right invariant means that x � y implies xz � yz for any z 2 R*.(iii) Let equivalence relation �A on R* be defined as, for x,y 2 R*, x � Ay iff A(xz) = A(yz) for any z 2 R*. Then �A has finite index.


Proof. (i)) (ii) Assume that A is accepted by some l-VDFA A ¼ ðQ ;R; d; q0; FÞ. Let �A be the equivalence relation x�Ay iffd*(q0,x) = d*(q0,y).�A is right invariant since, for any z, if d*(q0,x) = d*(q0,y), then d*(q0,xz) = d*(q0,yz). The index of �A is finite,since the index is at most the number of states in Q. Furthermore, if x�Ay, then d*(q0,x) = d*(q0,y), and thusA(x) = F(d*(q0,x)) = F(d*(q0,y)) = A(y). (ii)) (iii) We show that any equivalence relation E satisfying (ii) is a refinement of�A; that is, every equivalence class is entirely contained in some equivalence class of � A. Thus the index of �A can notbe greater than the index of E and so is finite. Assume that xEy. Then since E is right invariant, for each z 2 R*, xz E yz,and thus A(xz) = A(yz). Thus x � Ay, and hence the equivalence class of x in E is contained in the equivalence class of x in� A. We conclude that each equivalence class of E is contained within some equivalence class of �A.

(iii)) (i) Suppose that the index of �A is finite. Let [h] denote the congruence class of h 2 R*. Let AA ¼ ðQ ;R; d; q0; FÞ be anl-VDFA, where Q = {[h] : h 2R*}, and d([h],r) = [hr] for any h 2 R*,r 2 R. Set q0 = [e], and F:Q ? L by F([h]) = A(h). Since thedefinition of �A, the l-VDFA AA is well defined. Also, recAA

ðhÞ ¼ Fðd�ðq0; hÞÞ ¼ Fð½h�Þ ¼ AðhÞ. Thus for anyh 2 R�; recAA ðhÞ ¼ AðhÞ. Hence A ¼ recAA and so A is an l-regular language. h

The l-VDFA constructed in the proof of (iii)) (i) of the above theorem will be denoted by AA in the following. The impor-tance of AA stems from the following.

Theorem 5.3. For any l-regular language A on R, there exists a minimal accessible l-VDFA accepting A, which is just the one AA

constructed in Theorem 5.2.

Proof. It is obvious that the l-VDFAAA constructed in Theorem 5.2 is accessible. Next, let B ¼ ðP;R;g; p0; EÞ be any accessiblel-VDFA accepting A, we show that there exists a (unique) homomorphism from B to AA. Since B is an accessible l-VDFA, forany p 2 P, there exists hp 2 R* such that g*(p0,hp) = p. Define u:P ? Q as, u(p) = [hp], then u is well defined. This is because, ifg*(p0,h) = g*(p0,hp) for another h 2R*, then h�Bhp, where�B is defined in Theorem 5.2. Therefore, since�B is an refinement of�A, h � Ahp, [h] = [hp].

Therefore, u is well defined. Next, for any h 2 R *, since u(g*(p0,h)) = [h], u is a surjective mapping. Moreover,since g*(p0,e) = p0, u(p0) = [e] = q0, and u(g(p,r)) = u(g*(p0,hp),r) = u(g*(p0,hpr)) = [hpr] = d([hp],r) = d(u(p),r). And EðpÞ ¼Eðg�ðp0; h0ÞÞ ¼ recBðhpÞ ¼ AðhpÞ ¼ Fðd�ð½e�; hpÞÞ ¼ Fð½hp�Þ ¼ FðuðpÞÞ. Therefore, u is a strong, surjective homomorphism. Thisshows that AA is the minimal accessible l-VDFA accepting A. h

We refer to AA as the minimal l-VDFA of A; any l-VDFA isomorphic with AA will also be ’a’ minimal l-VDFA for A. Further-more, for any l-VDFA B with recB ¼ A, the proof of Theorem 5.3 leads to a surjective mapping from B to A (possiblenonunique).

Remark 5.1. In fact, from the construction of accessible l-VDFA, we know that the number of states of the minimalaccessible l-VDFA is the least among all l-VDFAs accepting the same language, and this minimal l-VDFA is determined up toan isomorphism (i.e. renaming of the states).

To give an effective algorithm to minimize a given l-VDFA B ¼ ðP;R;g; p0; EÞ, we have to define an equivalence relation �s

on the states of B. For p1,p2 2 P, p1 �s p2 iff E(g*(p1,h)) = E(g*(p2,h)) for any h 2R*. We say that p1 is distinguishable from p2 ifthere exists a h such that E(g*(p1,h)) – E(g*(p2,h)). If this equivalence relation is the identity, we say that the l-VDFA B is re-duced. Thus, B is reduced if p1 � sp2 implies p1 = p2. B is reduced iff u defined in Theorem 5.3 is injective. Since u is strong andis a surjective mapping we obtain.

Corollary 5.2. An l-VDFA B accepting A is minimal iff B is accessible and reduced.

When we use an l-VFA as a model of fuzzy computation, it is necessary to use an effective given domain for lattice as truthvalues set. In the Algorithm 1 below, we assume that the lattice l is an effective given domain.

Given an l-VDFA B, the minimum l-VDFA A such that recA ¼ recB can be constructed as follows:

Algorithm 1. In the following, a lattice l, as an effective given domain, is fixed. For convenience, let states in P be numberedfrom p0 to pn�1.

Step 1. For pi,pj 2 P where i < j, if E(pi) – E(pj), mark (pi,pj) as pi being distinguishable from pj.Step 2. For each pair (pi,pj) where i < j and (pi,pj) is unmarked doIf for some input r, (g(pi,r),g(pj,r)) is marked thenStep 2.1. Mark (pi,pj).Step 2.2. Recursively marked all unmarked pairs on the list for (pi,pj) and on the lists of other pairs that are marked at thisstep.elseStep 2.3. For each input r doPut (pi,pj) on the list for (g(pi,r),g(pj,r)) unless g(pi,r) = g(pj,r).


Step 3. Equivalence classes of P are constructed as follows:For i = 0 to n � 1 doFor j = i + 1 to n � 1 doIf (pi,pj) is unmarked, pj is in [pi], the equivalence class containing pi.Step 4. Define minimum l-VDFA A ¼ ðQ ;R; d; q0; FÞ as follows:

Q ¼ f½pi� : pi 2 Pg; dð½pi�;rÞ ¼ ½gðpi;rÞ�; q0 ¼ ½p0� and Fð½pi�Þ ¼ EðpiÞ:

Theorem 5.4. The l-VDFA constructed by means of algorithm 1, with inaccessible states removed, is minimum.

Proof. By Corollary 5.2, it suffice to show that (pi,pj) is marked if pi,pj are distinguishable. Assume pi is distinguishable frompj, and let h 2R* be a shortest string distinguishing pi from pj. We prove by induction on the length of h that (pi,pj) is marked.If h = e, (pi,pj) will be marked at step 1. Assume that hypothesis is true for jhj < k,k > 1, and let jhj = k, write h = rh0. Lett = g(pi,r) and s = g(pj,r). Now h0 distinguishes t from s. Since jh0j = k � 1, by induction, (t,s) will be remarked. If (t,s) is con-sidered before (pi,pj), (pi,pj) will be marked at Step 2.1. Otherwise, (pi,pj) will be marked at Step 2.2, after (t,s) has been mar-ked. h

Considering the time complexity of Algorithm 1, let jRj = k and jQj = n. Then Step 1 in Algorithm takes O(n2) times. Thetotal time spent on the loop of Step 2 is O(kn2), and Step 3 takes O(n2) times, Step 4 uses O(kn) times. So, the total timeof Algorithm 1 is O(kn2). Of course, if we consider the transformation of an l-VFA into an l-VDFA, and then minimize thetransformed l-VDFA as in Algorithm 1, the time complexity is more high as seen from the construction of Qd in Theorem 3.1.

6. Conclusion

The notion of lattice-valued finite automata was introduced in this paper. It generalized the notion of fuzzy automatawith membership values in a distributive lattice. Some general results were obtained here. In particular, finite automata un-der quantum logic [33] could be considered as a special case of lattice-valued finite automata with orthomodular lattice astruth values domain. Some results in [33] were strengthened here. In particular, the extended subset construction of lattice-valued finite automata was introduced, then the equivalence between lattice-valued finite automata, lattice-valued deter-ministic finite automata and lattice-valued finite automata with e-moves was established. The extended subset constructionand the determinization method considered in [1,15,33] were compared. A simple characterization of lattice-valued lan-guages recognized by lattice-valued finite automata was presented, then it was proved that the Kleene theorem holds inthe frame of lattice-setting. Minimization algorithm of lattice-valued deterministic finite automata was also presented.The role of distributive law in a lattice was analyzed. It was shown that the distributive law is not necessary to many con-struction in l-VFA, it indeed provides some convenience to the simple construction in l-VFA. This view can be seen one of themain contribution of the paper. The next problem is to consider the potential application of lattice-valued finite automata tomodel checking in lattice-setting (c.f. [3,15]).

References

[1] R. Belohlávek, Determinism and fuzzy automata, Information Sciences 142 (2002) 205–209.[2] G. Birkhoff, Lattice Theory, third ed., Amer. Math. Soc., Providence, Rhode Island, USA, 1973.[3] M. Chechik, B. Devereux, A. Gurfinkel, S. Easterbrook, Multi-valued symbolic model-checking, ACM Transactions on Software Engineering and

Methodology 12 (4) (2004) 1–38.[4] M. Droste, P. Gastin, Weighted automata and weighted logics, Theoretical Computer Science 380 (2007) 69–86.[5] S. Eilenberg, Automata, Languages and Machines, vols. A, B, Academic Press, New York, 1974.[6] C.A. Gama, A.G. Evsukoff, P. Weber, N.F.F. Ebecken, Parameter identification of recurrent fuzzy systems with fuzzy finite-state automata representation,

IEEE Transactions on Fuzzy Systems 16 (1) (2008) 213–224.[7] M.M. Gupta, G.N. Saridis, Fuzzy Automata and Decision Processes, North-Holland, New York, 1977.[8] J.E. Hopcroft, J.D. Ullman, Introduction to Automata Theory, Languages and Computation, Addison-Wesley, New York, 1979.[9] J. Ignjatovic, M. Ciric, Formal power series and regular operations on fuzzy languages, Information Sciences 180 (7) (2010) 1104–1120.

[10] J. Ignjatovic, M. Ciric, S. Bogdanovic, Determinization of fuzzy automata with membership values in complete residuated lattices, Information Sciences178 (1) (2008) 164–180.

[11] G. Kalmbach, Orthomodular Lattices, Academic Press, London, 1983.[12] S.C. Kleene, Representation of events in nerve nets and finite automata, in: C.E. Shannon, J. McCarthy (Eds.), Automata Studies, Princeton University

Press, Princeton, NJ, 1956, pp. 3–42.[13] W. Kuich, G. Rahonis, Fuzzy regular languages over finite and infinite words, Fuzzy Sets and Systems 157 (2006) 1532–1549.[14] W. Kuich, A. Salomaa, Semirings, Automata Languages, EATCS Monographs on Theoretical Computer Science, vol. 5, Springer-Verlag, Berlin/Heidelberg/

New York/Tokyo, 1986.[15] O. Kupferman, Y. Lustig, Lattice automata, in: Proceedings of VWCAI2007, LNCS, vol. 4349, 2007, pp. 199–213.[16] E.T. Lee, L.A. Zadeh, Note on fuzzy languages, Information Sciences 1 (1969) 421–434.[17] Y.M. Li, Z.H. Li, Free semilattices and strongly free semilattices generated by partially ordered sets, Northeastern Mathematical Journal 9 (3) (1993)

359–366.[18] Y.M. Li, Analysis of Fuzzy Systems, Academic Press, Beijing, 2005 (in Chinese).[19] Y.M. Li, W. Pedrycz, Fuzzy finite automata and fuzzy regular expressions with membership values in lattice-ordered monoids, Fuzzy Sets and Systems

156 (2005) 68–92.


[20] Y.M. Li, W. Pedrycz, Minimization of lattice finite automata and its application to the decomposition of lattice languages, Fuzzy Sets and Systems 158(2007) 1423–1436.

[21] P. Li, Y.M. Li, Algebraic properties of LA-languages, Information Sciences 176 (2006) 3232–3255.[22] Z.H. Li, P. Li, Y.M. Li, The relationships among several types of fuzzy automata, Information Sciences 176 (2006) 2208–2226.[23] J.N. Mordeson, D.S. Malik, Fuzzy Automata and Languages: Theory and Applications, Chapman & Hall/CRC, Boca Raton, London, 2002.[24] D.W. Qiu, Automata theory based on complete residuated lattice-valued logic, Science in China Series F 44 (6) (2001) 419–429.[25] D.W. Qiu, Automata theory based on complete residuated lattice-valued logic (II), Science in China Series F 45 (6) (2002) 442–452.[26] D.W. Qiu, Pumping lemma in automata theory based on complete residuated lattice-valued logic, Fuzzy Sets and Systems 157 (2006) 2128–2138.[27] D.W. Qiu, Automata theory based on quantum logic: reversibilities and pushdown automata, Theoretical Computer Science 386 (2007) 38–56.[28] D.W. Qiu, Automata theory based on quantum logic: some characterizations, Information and Computation 190 (2007) 179–195.[29] D.W. Qiu, Notes on automata theory based on quantum logic, Science in China Series F: Information Sciences 50 (2) (2007) 154–169.[30] M.O. Rabin, D. Scott, Finite automata and their decision problems, IBM Journal of Research and Development 3 (1959) 114–125.[31] M.P. Schützenberger, On the definition of a family of automata, Information and Control 4 (1961) 245–270.[32] W. Wechler, The Concept of Fuzzyiness in Automata and Language Theory, Akademie-Verlag, Berlin, 1978.[33] M.S. Ying, A theory of computation based on quantum logic (I), Theoretical Computer Science 344 (2005) 134–207.

finite automata theory with membership values in lattices

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