Approximation of fuzzy context-free grammars

Information Sciences 179 (2009) 3920–3929

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Approximation of fuzzy context-free grammars q Yongbing Wang a,b, Yongming Li a,c,* a b c

College of Mathematic and Information Science, Shaanxi Normal University, Xi’an 710062, China School of Mathematics and Computation, Anqing Teachers College, Anqing 246133, China 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 25 August 2007 Received in revised form 21 February 2009 Accepted 22 June 2009

Keywords: Fuzzy context-free grammars Fuzzy context-free languages Approximation Equivalence Sensitivity

a b s t r a c t Fuzzy context-free max-H grammar (or FCFGH , for short), as a straightforward extension of context-free grammar, has been introduced to express uncertainty, imprecision, and vagueness in natural language fragments. Li recently proposed the approximation of fuzzy finite automata, which may effectively deal with the practical problems of fuzziness, impreciseness and vagueness. In this paper, we further develop the approximation of fuzzy context-free grammars. In particular, we show that a fuzzy context-free grammar under max-H compositional inference can be approximated by some fuzzy context-free grammar under max–min compositional inference with any given accuracy. In addition, some related properties of fuzzy context-free grammars and fuzzy languages generated by them are studied. Finally, the sensitivity of fuzzy context-free grammars is also discussed. Ó 2009 Elsevier Inc. All rights reserved.

1. Introduction In recent years, fuzzy system theory has been applied in various fields, such as the control systems [7,34]. In most of these applications, the main purpose is to construct a fuzzy system to approximate a desired control or decision system. In fact, we can use different fuzzy inference methods in designing a fuzzy system [6,14,32]. However, the most commonly used fuzzy inference method is known as the compositional rule of inference (or CRI) or max-H compositional inference for some tnorm H. It was shown that the systems under different t-norms are equivalent in the approximation sense [17,19]. That is to say, if a fuzzy system with max–min compositional inference approximately describes a given practical process, so does another fuzzy system under max-H compositional inference with the t-norm H distinct from min. Formal context-free grammars theory not only lays the foundations for complexity theory, but also contributes to many fields such as the lexical analysis [22], the description of natural languages [13,26,27] and the programming languages [11,23,25]. Notably, the precision of formal languages is much sharper than the imprecision of natural languages. To bridge the gap between them, some measures have been proposed. For instance, the concepts of stochastic [29] and fuzzy languages [4,15,20,21,24,28] were introduced. As one of the generators of fuzzy languages, fuzzy context-free grammars have also been used to solve interesting problems such as intelligent interface design [10,11], clinical monitoring [33], neural networks [8,35] and pattern recognition [5]. And the theory of fuzzy logic plays a fundamental role in the process of forming the fuzzy context-free grammars [9,30,36].

q This work is supported by National Science Foundation of China (Grant No. 10571112, 69873119) and the Higher School Doctoral Subject Foundation of Ministry of Education under Grant 200807180005. * Corresponding author. Address: College of Mathematic and Information Science, Shaanxi Normal University, Xi’an 710062, China. Tel.: +86 2985307628; fax: +86 2985310161. E-mail address: [email protected] (Y. Li).

0020-0255/$ - see front matter Ó 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.ins.2009.06.028

Y. Wang, Y. Li / Information Sciences 179 (2009) 3920–3929

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In [1,2], Asveld used fuzzy context-free grammars with max–min and max-product compositional inference to model a fuzzy system, respectively. In fact, there are many types of fuzzy context-free grammars that can be implemented in practical processes [10,31]. Inspired by this, we consider the following topics. A crucial theoretical problem is whether fuzzy context-free grammars under max-H compositional inference using different t-norm H are equivalent when modeling practical processes. In this paper, we first introduce the concepts of fuzzy context-free max-H grammars (or FCFGH s, for short) and fuzzy context-free languages (or FCFLs, for short). We show that a fuzzy context-free max–min grammar (or FCFG^ , for short) is not as powerful as an FCFGH in recognizing fuzzy languages. Fortunately, we prove that an FCFGH can be approximated by some FCFG^ with any given accuracy. Also the equivalent relations between FCFG^ and FCFGH are discussed. This forms the first topic of this study. Then we introduce the sensitivity of FCFGH s for some t-norm H, which is an attractive issue in fuzzy systems. Since fuzzy systems are the approximation description of real systems, some approximation errors may occur in the formulation of fuzzy models. Hence, it is necessary to require the models of fuzzy systems to be tolerant of approximation errors. For this purpose, as a model of real system, fuzzy context-free grammar needs to be sensitive for some minor approximation errors, which is another topic of this paper. The remainder of this paper is organized as follows: In Section 2, we first recall the definition and some examples of t-norm. Then we introduce the definitions of FCFGH s, FCFLs, and discuss some properties of FCFGH s. In Section 3, we show that an FCFGH can be approximated by some FCFG^ with any given accuracy for any t-norm satisfying weakly finite generated condition (or WFGC, in short), even if they are not equivalent in general. We investigate the sensitivity of FCFGH s in Section 4. Finally, some conclusions are presented. The reader can refer to [1,2,11,17] as useful reference materials in the context of this study.

2. Preliminaries In this section, we first briefly recall some basic concepts about t-norm, since the membership degree of a computation depends on a specified t-norm. Refer to [12] for more details. Definition 2.1. The term triangular norm (briefly t-norm) is a binary operation H on unit interval [0, 1] which is commutative, associative, monotone, and has 1 as neutral element, i.e., it is a function H : ½0; 12 ! ½0; 1 such that for any a; b; c 2 ½0; 1:

ðT1Þ aHb ¼ bHa; ðT2Þ ðaHbÞHc ¼ aHðbHcÞ; ðT3Þ a1 Hb1 6 a2 Hb2 whenever a1 6 a2 and b1 6 b2 ; ðT4Þ 1Ha ¼ a: Throughout this paper, to keep a unified notation established in the function theory, we shall consistently use the prefix notation for t-norms (and t-conorms). Example 2.1. We give some common used t-norms [9,12] as follows. (1) Minimum operation ‘‘^” on [0, 1] is a t-norm, that is, a ^ b ¼ minfa; bg for any a; b 2 ½0; 1. In particular, minimum operation satisfies the property a ^ a ¼ a for any a 2 ½0; 1. (2) Product operation on [0, 1] is a t-norm which is a strict Archimedean t-norm, i.e., aa < a and an > 0 for any a 2 ð0; 1Þ, where n is any positive integer. (3) Drastic product operation is a t-norm which is defined as aHb ¼ a ^ b when ða; bÞ 2 ½0; 12 and 0 otherwise. (4) Lukasiewicz t-norm is a t-norm which is defined as aHb ¼ ða þ b  1Þ _ 0 for any a; b 2 ½0; 1, where _ is the maximum operation. It is a nilpotent Archimedean t-norm which satisfies aHa < a for any a 2 ð0; 1Þ and there exists positive integer n such that an ¼ aHaH    Ha ðn timesÞ ¼ 0. (5) Nilpotent minimum is a t-norm defined as aHb ¼ a ^ b when a þ b > 1 and 0 otherwise. Note that the drastic product operation and the minimum operation are the smallest and the largest operation in the family of t-norms, respectively. In the following text, we suppose that T is an alphabet with 1 6 jTj < 1 and T  is the free monoid generated from T with the operation of concatenation. A string over T is a sequence of symbols from T; the empty string is denoted by . For any x 2 T  , jxj denotes the length of x; and a crisp formal language over T is a subset of T  : Inclusion and strict inclusion relations of sets are denoted by # and , respectively. Then we define the concept of a fuzzy context-free max-H grammar in the following way: Definition 2.2. A fuzzy context-free max-H grammar (or FCFGH , for short) is a system G ¼ ðT; N; P; g; h; HÞ such that the following conditions hold:

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(1) (2) (3) (4) (5)

Y. Wang, Y. Li / Information Sciences 179 (2009) 3920–3929

T and N are disjoint finite nonempty sets; P is a finite collection of productions of the form A ! x, where A 2 N; x 2 ðT [ NÞ ; g is a function from N into ½0; 1; h is a function from P into ð0; 1; H is a specified t-norm which is used to define the fuzzy language generated by an FCFGH .

In Definition 2.2, the elements of T are called terminals, the elements of N are called nonterminals, and gðAÞ is the grade of membership that A is the starft symbol of G. For p ¼ A ! x 2 P; hðpÞ is the grade of applying the production p 2 P, which means A will be replaced by x with the grade of membership hðpÞ. For the sake of simplicity, we represent a fuzzy production q in the form A ! x, where q ¼ hðA ! xÞ. We omit H in the definition of FCFGH if H is understood in the context. In particular, if aHb ¼ a ^ b, the fuzzy context-free max-H grammar G is called a fuzzy context-free max–min grammar ðFCFG^ Þ. G

G

Formally, to define the language generated by an FCFGH G ¼ ðT; N; P; g; hÞ, we first define two relations ) and ) between G

strings in ðT [ NÞ . If A ! x 2 P, then c1 Ac2 ) c1 xc2 for any c1 ; c2 2 ðT [ NÞ , and we call that c1 Ac2 derives directly c1 xc2 in G

G



G

G

grammar G. ) is the reflexive and transitive closure of ). For simplicity, we use ) and ) for ) and ), respectively, if there 

is no confusion. More specifically, if a1 ; a2 ; . . . ; am 2 ðT [ NÞ and ai ) aiþ1 ; i ¼ 1; 2; . . . ; m  1, then we say a1 ) am or a1 derives am in fuzzy grammar G. q  Here, binary relations ) and ) are extended to two fuzzy relations on ðT [ NÞ  ðT [ NÞ . We use c1 Ac2 ) c1 xc2 to exq press that c1 Ac2 directly derives c1 xc2 by applying a production A ! x. In general, a ) b represents that a directly derives b by applying all those possible productions in P, and the degree is thus defined as

lða ) bÞ ¼

_ fhðA ! xÞ : c1 Ac2 ¼ a; c1 xc2 ¼ b;

A ! x 2 P; c1 ; c2 2 ðT [ NÞ g:



Furthermore, a ) b means the reflexive and transitive closure of ), that is, a ) a1 ; a1 ) a2 ; . . . ; ak1 ) ak ; ak ) b, for some positive integer k and ai 2 ðT [ NÞ ; i ¼ 1; 2; . . . ; k. Naturally the degree of a deriving b is defined as 

lða ) bÞ ¼

_ flða ) a1 ÞHlða1 ) a2 ÞH    Hlðak1 ) ak ÞHlðak ) bÞ : ai 2 ðT [ NÞ ; i ¼ 1; 2; . . . ; kg

where H is a specified t-norm. q In the above definition, we note that a derives b with the grade of membership q if and only if a ) bðmod-Þ, where q

- ¼ ðr0 ; l0 Þðr1 ; l1 Þ    ðrk ; lk Þ; a; b 2 ðT [ NÞ ; ri ¼ Ai !i xi 2 P; i ¼ l0 ; l1 ; . . . ; lk ; li 2 N, and q ¼ q0 Hq1 H    Hqk , then there exist ai 2 ðT [ NÞ ; i ¼ 0; 1; . . . ; k such that a0 ¼ a; akþ1 ¼ b; and for each i; aiþ1 is obtained from ai by replacing the li th occurrence 0 of Ai in ai by xi ; otherwise, we write a ) bðmod-Þ. q 0 If for every i; li ¼ 1 and ai ¼ uAi v , where u 2 T  ; i ¼ 0; 1; . . . ; k; we write a ) bðmod-Þ; otherwise, we write a ) bðmod-Þ. L

L

Definition 2.3. The fuzzy language fG generated by an FCFGH G ¼ ðT; N; P; g; hÞ is a mapping from T  to ½0; 1 defined as

fG ðxÞ ¼

_



gðAÞHlðA ) xÞ

A2N 

for all x 2 T . Let f be a fuzzy language over T; f is called a fuzzy context-free language (FCFL) if f ¼ fG for some FCFGH G. Let G be an FCFGH , then fG is the fuzzy language generated by G and fGL is the fuzzy language generated by G using left-most derivations only. Definition 2.4. Two FCFGH s G1 and G2 are said to be equivalent if fG1 ¼ fG2 . Now we present the relationship between fG and fGL in the following theorem. Theorem 2.1. If fG is an FCFL, then fG ¼ fGL for some FCFGH G. Proof. Suppose that G ¼ ðT; N; P; g; hÞ. For each A 2 N; x 2 T  , if A ) xðmod-Þ for some q

- 2 ðP  NÞ ; then

q

A)L xðmod-0 Þ for some

-0 2 ðP  f1gÞ . This can be accomplished by induction on jxj. h

In the following theorem, we shall show that an FCFGH G with a crisp start nonterminal is equivalent to the one with fuzzy start nonterminal. Theorem 2.2. Let f be a fuzzy language over T, then f can be generated by an FCFGH G ¼ ðT; N; P; g; hÞ if and only if there exists an 0 FCFGH G0 ¼ ðT; N 0 ; P 0 ; A0 ; h Þ with a crisp start nonterminal A0 such that f ¼ fG0 .

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0

Proof. For the ‘‘only if” part. Let G ¼ ðT; N; P; g; hÞ be an FCFGH such that f ¼ fG . Construct an FCFGH G0 ¼ ðT; N 0 ; P 0 ; g0 ; h Þ as follows: let A0 R T [ N; N 0 ¼ N [ fA0 g;

8 W hðA ! xÞ; if A 2 N; x 2 ðT [ NÞ ; > > > A!x2P < W W 0 h ðA ! xÞ ¼ gðA0 ÞhðA0 ! xÞ; if A ¼ A0 ; x 2 ðT [ NÞ ; > 0 0 > A 2N A !x2P > : 0; otherwise;

and

g0 ðAÞ ¼



1; ifA ¼ A0 ; 0;

ifA – A0 : 0

It is clear that G0 is equivalent to G, i.e., fG0 ¼ fG . Obviously, in this case, we can write G0 ¼ ðT; N 0 ; P 0 ; A0 ; h Þ for G0 . Thus f ¼ fG0 . ‘‘If” part is clear. h We give an example to illustrate the proof of the above theorem. Henceforth, we need to make some notations. For each x in T and x in T  , let #x ðxÞ be the number of times that the symbol x occurs in the string x. Usually, all productions A ! x1 ; A ! x2 ; . . . ; A ! xn with the same left-hand side A are collected in a single expression of the form A ! x1 jx2 j    jxn . Moreover, we use X ¼ fx1 =q1 ; x2 =q2 ; . . . ; xn =qn g as a concise representation of the fuzzy set on a finite set X ¼ fx1 ; . . . ; xn g with membership function defined by lðxi Þ ¼ qi ð1 6 i 6 nÞ. Example 2.2. Let H be the usual product operation of numbers, and G ¼ ðT; N; P; g; hÞ be defined as follows, N ¼ fS; A; Bg; T ¼ fx1 ; x2 g; gðSÞ ¼ gðAÞ ¼ gðBÞ ¼ 1 and P with the grade of membership is defined by

S ! AB=1jBA=1jAA=0:1jBB=0:9; A ! AS=1jSA=1jAB=0:7jBA=0:3jx1 =1; B ! BS=1jSB=1jBA=0:8jAB=0:2jx2 =1: Then this FCFGH belongs to the class FCFGH s with fuzzy start symbol, and the language generated by this FCFGH can be calculated as follows:

8 0; > > > > < _ ð 9 Þð#x2 ðxÞ#x1 ðxÞÞ=2 ;  f G ð xÞ ¼ gðZÞlðZ ) xÞ ¼ 101 ð#x ðxÞ#x ðxÞÞ=2 >ð Þ 1 2 > ; Z2N > 10 > : 1;

if jxj ¼ 0; if #x2 ðxÞ P #x1 ðxÞ þ 2 and jxj is even; if #x1 ðxÞ P #x2 ðxÞ þ 2 and jxj is even; otherwise: 0

We deduce the equivalent FCFGH G0 as follows: G0 ¼ ðT; N 0 ; P0 ; A0 ; h Þ is defined as, N 0 ¼ N [ fA0 g; T ¼ fx1 ; x2 g and P0 ¼ P [ P, where P with the grade of membership is defined by

A0 ! AS=1jSA=1jBS=1jSB=1jAA=0:1jBB=0:9jx1 =1jx2 =1: 

For example, fG0 ðxÞ ¼ lðA0 ) xÞ ¼ 0:81, where jxj ¼ 6; #x2 ðxÞ ¼ 4 and #x1 ðxÞ ¼ 2 for any x 2 T  . Theorem 2.3. If f is an FCFL, then f ¼ fG for some FCFGH G ¼ ðT; N; P; A0 ; hÞ such that the following conditions hold: (1) for any A 2 N, there exist c1 ; c2 2 T  such that hðA0 ! c1 Ac2 Þ > 0; (2) for any A 2 N, there exists x 2 T  such that hðA ! xÞ > 0.

Proof (1) If for any c1 ; c2 2 T  we have hðA0 ! c1 Ac2 Þ ¼ 0, then eliminating symbol A from N will not change the generated language. (2) If there exists x 2 T  such that hðA ! xÞ ¼ 0, then we eliminate symbol A from N which would not change the generated language. h Further, an FCFGH satisfying (1) and (2) is said to be reduced, and only the reduced fuzzy context-free grammars are considered in this paper. 3. Approximation of fuzzy context-free max- grammars Some notations need to be introduced in this section. Let T be a set. For a fuzzy subset f : T  ! ½0; 1, let Rðf Þ ¼ ff ðxÞ : x 2 T  ; f ðxÞ > 0g. For any a 2 ½0; 1, two subsets fa and f½a of T  are defined as, fa ¼ fx : x 2 T  and f ðxÞ P ag and f½a ¼ fx : x 2 T  and f ðxÞ ¼ ag. The support of f is a subset of T  defined as suppðf Þ ¼ fx : x 2 T  ; f ðxÞ > 0g.

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Since the operations max and min applied to a specified set do not yield new, different values in the codomain, if we take H as the minimum operation ‘‘^”, then we have the following theorem. Lemma 3.1. Let G1 ¼ ðT; N 1 ; P1 ; A01 ; h01 Þ and G2 ¼ ðT; N 2 ; P 2 ; A02 ; h02 Þ be two FCFGH s such that f ¼ fG1 and g ¼ fG2 , then the union f _ g can be generated by an FCFGH . Proof. Without loss of generality, we assume that N 1 \ N 2 ¼ ;. Construct a new fuzzy grammar G ¼ ðT; N; P; A0 ; hÞ such that    Then, fG ðxÞ ¼ lðA0 ) xÞ ¼ lðA0 ) A01 ) xÞ _ lðA0 ) A02 ) xÞ N ¼ N 1 [ N 2 [ fA0 g; P ¼ P 1 [ P2 [ fA0 ! A01 =1jA02 =1g.   ¼ lðA01 ) xÞ _ lðA02 ) xÞ ¼ f ðxÞ _ gðxÞ. That is to say, fG ¼ f _ g; f _ g can be generated by an FCFGH G. h Theorem 3.1. Let f : T  ! ½0; 1 be a fuzzy language, then the following statements are equivalent. (1) f can be generated by an FCFG^ . (2) Rðf Þ is finite, and fa is a context-free language (or CFL) for any a 2 Rðf Þ. Proof. ð1Þ ) ð2Þ Suppose that f ¼ fG , where G ¼ ðT; N; P; A0 ; hÞ is an FCFG^ . Then for any generated string x 2 T  by G, we  have fG ðxÞ ¼ lðA0 ) xÞ. Since Rðf Þ # RðhÞ [ f1g and RðhÞ is finite, Rðf Þ is finite. For any a 2 Rðf Þ, we construct a context-free grammar (CFG) Ga ¼ ðT; N; P a ; A0 Þ, where P a ¼ fA ! x 2 P : hðA ! xÞ P ag. Then we have x 2 fa () f ðxÞ P a () x 2 LðGa Þ, that is, fa can be generated by CFG Ga , i.e., fa is a CFL. ð2Þ ) ð1Þ Since fa is a CFL for any a 2 Rðf Þ, then fa can be generated by a CFG G ¼ ðT; N; P; A0 Þ. Clearly, a ^ fa can be generated by an FCFG^ G0 ¼ ðT; N; P ^ a; A0 ; hÞ, and a ^ fa is a fuzzy subset of T  defined as a ^ fa ðxÞ ¼ a if x 2 fa and 0 a otherwise, P ^ a ¼ fA ! x : A ! x 2 Pg, where the t-norm H is the minimum operation. Notice that Rðf Þ is finite and the W family of fuzzy languages generated by FCFG^ is closed under finite union operation, thus f ¼ a2Rðf Þ ða ^ fa Þ can be generated by an FCFG^ . h Example 3.1. Let L be a CFL over T ¼ fx1 ; x2 ; x3 g such that the complement T   L is not a CFL. For example, L ¼ T   fxn1 xn2 xn3 jn P 1g is a CFL. In fact, we could divide L into three parts, i.e., L ¼ L1 [ L2 [ L3 , where L1 ¼ T   fxi1 xj2 xk3 ji; j; k P 1g; L2 ¼ fxi1 xj2 xk3 ji; j; k P 1 and i–jg

and

L3 ¼ fxi1 xj2 xk3 ji; j; k P 1 and j–kg.

We

know

that

fxi1 xj2 xk3 ji; j; k

P 1g is regular language, and the family of regular language is closed under the operation of complement, thus L1 is regular. Clearly, L1 is a CFL. Also, it is well known that L2 and L3 are CFLs and the family of CFLs is closed under finite union operation. Therefore, L is a CFL. Assume that L can be generated by a CFG G ¼ ðT; N; P; A0 Þ. Constructing an FCFG^ G1 ¼ ðT; N 1 ; P1 ; A0 ; hÞ such that P 1 ¼ P [ fA0 ! x1 A0 =0:5jx2 A0 =0:5jx3 A0 =0:5j=0:5g, where the grade of membership of all productions in P are 1. Then fG ðxÞ P 0:5 for any generated string x 2 T  . Note that f1 ¼ f½1 ¼ L and f½0:5 ¼ T   L, the latter is not a CFL. Hence, in general, for an FCFL f, the level language f½a is not a CFL. Moreover, this example shows that the family of FCFLs is not closed under the operations of complement and intersection. As shown in Theorem 3.1, a fuzzy context-free grammar under max–min compositional inference can be realized by a simple model, that is, an FCFG^ . Then we shall study the issue whether a fuzzy context-free grammar under max-H compositional inference can be approximated by some FCFG^ with any given accuracy in the following part. Now we turn to a few definitions [22] and lemmas [17] to establish the further properties of FCFGH . Definition 3.1. A substitution w over an alphabet T is a mapping w : T ! PðT  Þ which is extended to strings over T by w : T  ! PðT  Þ with wðÞ ¼ wðÞ and wðxxÞ ¼ wðxÞwðxÞ; x 2 T; x 2 T  , and languages over T by w : PðT  Þ ! PðT  Þ with S wðLÞ ¼ fwðxÞjx 2 Lg. Definition 3.2. An a-transducer is a six-tuple M ¼ ðX; Q ; Y; H; q0 ; FÞ; where X; Q and Y are finite nonempty sets (of input, state and output symbols, respectively), H is a finite subset of Q  X   Y   Q ; q0 2 Q is the initial state, and F # Q is the set of accepting states. Let M ¼ ðX; Q ; Y; H; q0 ; FÞ be an a-transducer. For each x 2 X  ; let MðxÞ ¼ fx0 2 Y  j there exist x1 ; x2 ; . . . ; xk 2 X  ; y1 ; y2 ; . . . ; yk 2 Y  and q1 ; q2 ; . . . ; qk 2 Q such that x ¼ x1 x2    xk ; x0 ¼ y1 y2    yk ; qk 2 F; and ðqi1 ; xi ; yi ; qi Þ 2 H for all i; 1 6 i 6 kg. For each L # X  ; let

MðLÞ ¼

[

MðxÞ:

x2L

It is well known that if L is a CFL and M is an a-transducer, then MðLÞ is a CFL. It is also well known that if L is a CFL, and w is a substitution such that wðxÞ is a CFL for all x 2 T, then wðLÞ is also a CFL. Let H be a t-norm. For any a 2 ½0; 1, we can inductively define the power of a as follows: a0 ¼ 1; a1 ¼ a, and anþ1 ¼ an Ha. Then for any subset D of [0, 1], the subalgebra of ð½0; 1; HÞ generated by D, denoted as SðDÞ, which is defined as follows: l

SðDÞ ¼ fal11 H    Hakk : a1 ; . . . ; ak 2 D and l1 ; . . . ; lk are nonnegative integersg:

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For any a 2 ð0; 1, we use Sa ðDÞ to represent a subset of SðDÞ consisting of elements of SðDÞ which is larger than or equal to a, i.e.

Sa ðDÞ ¼ fb : b 2 SðDÞ and b P ag: We have two conditions for t-norm H. Finite generated condition (or FGC, for short): For any finite subset D of [0, 1], SðDÞ is finite. Weakly finite generated condition (or WFGC, for short): For any finite subset D of [0, 1] and any a 2 ð0; 1; Sa ðDÞ is finite. Obviously, if H satisfies FGC, then H also satisfies WFGC. For example, minimum, Lukasiewicz and nilpotent minimum tnorms satisfy FGC, and product t-norm satisfies WFGC but not FGC. Lemma 3.2 ([17,18]). For a t-norm H; H does not satisfy FGC if and only if there exists a 2 ð0; 1Þ such that ai –aj whenever i–j. Lemma 3.3 ([17,18]). If H is an Archimedean t-norm, i.e., H is continuous and aHa < a for any a 2 ð0; 1Þ, then H satisfies WFGC. Furthermore, if H is nilpotent Archimedean t-norm, i.e., for any a < 1, there exists n P 1 such that an ¼ 0, then H satisfies FGC. Theorem 3.2. Let H be a t-norm satisfying WFGC, and f be a fuzzy language that can be generated by an FCFGH , then fa is a CFL for any a 2 ½0; 1. Proof. Let L ¼ fa , where f is generated by an FCFGH G ¼ ðT; N; P; A0 ; hÞ and a 2 ½0; 1. Without loss of generality, we assume 1

q

that N ¼ N 1 [ N 2 , where (1) N 1 \ N 2 ¼ £; (2) A0 2 N 1 ; (3) if A ! x 2 P, then A 2 N 1 ; (4) if A ! x 2 P, and q < 1, then A 2 N 2 . 1 Let P1 ¼ fA ! xjA 2 N 1 ; x 2 ðN [ TÞ g and P 2 ¼ P n P 1 . For all A 2 N 1 , let LðAÞ ¼ f0 , where GA ¼ ðT [ N 2 ; N 1 ; P1 ; A; hÞ. Clearly, LðAÞ is a CFL for all A 2 N 1 . Let w be a substitution such that wðAÞ ¼ LðAÞ if A 2 N 1 , and wðtÞ ¼ ftg if t 2 T [ N 2 . For all q where x 2 ðT [ N 1 [ N 2 Þ , let M r ¼ ðX; Q ; Y; H; q0 ; fq1 gÞ be an a-transducer, where r ¼ A ! x 2 P2 , X ¼ T [ N 2 ; Q ¼ fq0 ; q1 g; Y ¼ T [ N and H ¼ fðq; u; u; qÞjq 2 Q ; u 2 T [ N 2 g [ fðq0 ; A; x; q1 Þg. Note that if L0 # ðT [ N 2 Þ , then Mr ðL0 Þ is obtained from L by first omitting all words in L0 not containing any occurrence of A, and then replacing exactly one occurrence of A by x in the remaining words of L0 . q For all L1 ; L2 # ðT [ N 2 Þ and k 2 N, define L1 ! L2 if and only if there exists r 2 P 2 such that L02 ¼ ðMr ðL1 ÞÞ, if r ¼ A ! x 2 P 2 ,  0 where x 2 ðT [ N2 Þ , then there is nothing to do, i.e., L2 ¼ L2 ; otherwise, the similar construction of GA will be done by the q

same way for finite times, that is, there exists x 2 ðT [ N 1 [ N 2 Þ N 1 ðT [ N 1 [ N 2 Þ such that r ¼ A ! x 2 P 2 . It is clear that the numbers of the different nonterminals which is included in N 1 and occur in x is finite, since N 1 is a finite nonempty set, thus L2 can be obtained from L02 by applying finite times substitution. And L1 !k L2 if and only if there exist L00 ; L01 ; . . . ; L0k # ðT [ N 2 Þ , such that L1 ¼ L00 ; L2 ¼ L0k , and L0i1 ! L0i for i ¼ 1; 2; . . . ; k. Define L1 k L2 if and only if there exists L3 # ðT [ N 2 Þ such that L1 !k L3 and L2 ¼ L3 \ T  . Clearly, if L1 is a CFL and L1 )k L2 for some k, then L2 is also a CFL. To complete the proof, there should be two cases to be considered. Case 1: if a ¼ 0, we construct a CFG Ga ¼ ðT; N; P a ; A0 Þ, where Pa ¼ fA ! x 2 P : hðA ! xÞ P ag, then x 2 fa () f ðxÞ P a () x 2 LðGa Þ, that is to say, fa can be generated by a CFG, i.e., fa is a CFL. Case 2: if a > 0; then there exists a positive integer n such that x 2 L implies x 2 L2 for some L2 # T  , where wðA0 Þk L2 for some positive integer k 6 n. 

So for all x 2 L2 ; lðA0 ) xÞ P a, i.e., there exist a sequence of strings a0 ; a1 ; . . . ; ak , k 6 n and ai ) aiþ1 applying the Pk1 qi k1 production of Ai ! xi 2 P, where A0 ¼ a0 ; x ¼ ak ; i ¼ 0; 1; . . . ; k, then we have ql00 Hql11 H    Hqlk1 P a, and i¼0 li 6 n. Let S  L ¼ fL2 # T jwðA0 Þk L2 for some k 6 n and L2 \ L–£g, it follows that L2 # L. Thus L ¼ L2 2L L2 , therefore, L is a CFL, i.e., fa is a CFL for any a 2 ½0; 1. h Note that the converse of the above theorem is straightforward. From Theorem 3.2 and Lemma 3.3, we have the following corollary. Corollary 3.1. If H is an Archimedeam t-norm, and f is a fuzzy language generated by an FCFGH , then fa and supp(f) are CFL for any a 2 ð0; 1. Proof. It is clear that fa is a CFL for any a 2 Rðf Þ by Theorem 3.2 and Lemma 3.2. We will show that suppðf Þ is a CFL, this will be done by checking the two subsequent cases. Case 1: H is a strict Archimedeam t-norm, i.e., H is continuous, aHa < a and an > 0 for any a 2 ð0; 1Þ: In this case, suppose q f ¼ fG ; where G ¼ ðT; N; P; A0 ; hÞ is an FCFGH . Let G0 ¼ ðT; N; P 0 ; A0 Þ be a CFG, where P0 ¼ fA ! x : A ! x 2 Pg, then suppðf Þ 0 can be generated by G and suppðf Þ is a CFL. Case 2: H is a niplotent Archimedeam t-norm, then it satisfies FGC, i.e., Rðf Þ is a finite subset of SðDÞ by Lemma 3.3, where D ¼ RðhÞ [ f1g is a finite subset of [0,1]. By Theorem 3.2, fa is a CFL for any a 2 Rðf Þ: Since the family of CFL is closed under S finite union, thus suppðf Þ ¼ a2Rðf Þ fa is a CFL. h It is worth noting that Corollary 3.1 remains valid if the Archimedeam t-norm is replaced by general t-norm operation satisfying WFGC.

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Theorem 3.3. Let H be a t-norm satisfying WFGC. If the fuzzy language f is generated by an FCFGH G, then for any a 2 ð0; 1; the set

Rðfa Þ ¼ ff ðxÞ : x 2 T  ; f ðxÞ P ag is always finite, and there exists an FCFG^ G0 such that fG0 ¼ fG _ a. Proof. Suppose that G ¼ ðT; N; P; A0 ; hÞ is an FCFGH such that fG ¼ f . Let D ¼ RðhÞ [ f1g. Then it is obvious that Rðf Þ # SðDÞ. Consequently, Ra ðf Þ # Sa ðDÞ for any a 2 ð0; 1. Note that Sa ðDÞ is finite, because H satisfies WFGC, so Ra ðf Þ is finite for any a 2 ð0; 1. For fuzzy language g ¼ f _ a, and any b 2 D, if b 6 a, then g b ¼ T  and thus is a CFL. If b > a, then g b ¼ fb is a CFL because of Theorem 3.2. From Theorem 3.1, g can be generated by an FCFG^ . That is to say, there exists an FCFG^ G0 such that fG0 ¼ g ¼ f _ a. h Obviously, the FCFG^ constructed in Theorem 3.3 satisfies the condition jfG0 ðxÞ  fG ðxÞj 6 a for any x 2 T  . Therefore, let us assume that H is any t-norm satisfying WFGC,  is a given approximation error. Take a positive integer n such that 1n 6 , ; 1g; then FCFGH can be approximated by some FCFG^ with any given accuracy. We and let fhðpÞ : p 2 Pg # f0; 1n ; 2n ; . . . ; n1 n present it in the following theorem. Theorem 3.4. Let H be a t-norm satisfying WFGC. For any FCFGH G, and any small positive number , there always exists an FCFG^ G0 such that the following inequality holds for any x 2 T  ,

jfG0 ðxÞ  fG ðxÞj 6 : Remark 3.1. By Theorem 3.4, FCFGH can be approximated by FCFG^ with given accuracy . Correspondingly, we can construct a class of FCFG^ s in which all the fuzzy productions take values in a fixed finite subset, written as D of the unit interval [0, 1], for each FCFGH can be approximated by an FCFG^ in this specified class with given accuracy . That is to say, an FCFGH G can be approximated by an FCFG^ G0 with accuracy , i.e., the inequality jfG ðxÞ  fG0 ðxÞj 6  holds for any generated string x. We will illustrate the construction of the class of FCFG^ s. Suppose that H is any t-norm satisfying WFGC,  is a given approximation error, and T is a given generated terminal alphabet. Take a positive integer n such that 1n 6 , and let

D¼

  1 2 n1 0; ; ; . . . ; ;1 : n n n

We construct a class of FCFG^ s with generated terminal alphabet T, denoted as n-FCFG^ , such that n-FCFG^ satisfies the following property: For any FCFGH G with generated terminal alphabet T, there is an FCFG^ G0 in the class n-FCFG^ , such that G can be approximated by G0 with accuracy , i.e., jfG ðxÞ  fG0 ðxÞj 6  for any generated string x 2 T  . The construction of n-FCFG^ is based on Theorems 3.3 and 3.4. The element B in n-FCFG^ satisfies the following condition. B is an FCFG^ , written as ðT; N; P; A0 ; hÞ, then the fuzzy set h only takes values in the given finite subset D of the unit interval [0, 1]. Next, we show that the class n-FCFG^ satisfies the mentioned approximation property. Take any FCFGH A with generated terminal alphabet T. By Theorem 3.3, there is an FCFG^ C such that fC ¼ fA _ 1n. Write C ¼ ðT; N; P; A0 ; hC Þ. Construct a new FCFG^ B from C as follows, where B ¼ ðT; N; P; A0 ; hB Þ, the only difference between B and C lies in the definitions of fuzzy sets hC and hB , where hB is defined in the following manner: for any production A ! x in P; hB ðA ! xÞ ¼ ni for some nonnegative integer i 6 n if ni 6 hC ðA ! xÞ < iþ1 n . It is clear that hB is well-defined, and jhB ðqÞ  hC ðqÞj 6 1n 6  holds for any production q 2 P. Evidently, B is in the class n-FCFG^ and B is an FCFG^ . For any generated string x 2 T  , it can be readily verified that jfA ðxÞ  fB ðxÞj 6 . Therefore, B is an FCFG^ in the class n-FCFG^ which can approximate A with accuracy . In the next example, we illustrate the constructions given in Remark 3.1. Example 3.2. Let T ¼ fx; yg; N ¼ fA0 ; S; A; Bg and H be the usual product operation of numbers. Consider the FCFGH A ¼ ðT; N; PA ; A0 ; hA Þ; where P A with the grade of membership is given by

A0 ! S=1; S ! AB=0:8jBA=0:6; A ! AS=0:6jSA=0:7jx=0:8; B ! BS=0:7jSB=0:6jy=0:8: It is easy to see that fA ðxÞ 6 0:512 for any generated string x 2 T  : By the construction given in Remark 3.1, let n ¼ 5 and  ¼ 0:3. Then there exists an FCFG^ C ¼ ðT; N; P; A0 ; hC Þ such that fC ¼ fA _ 1n ; where P with the grade of membership is defined by

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A0 ! S=1; S ! xSB=0:2jxBS=0:2jySA=0:2jyAS=0:2jxB=0:512jyA=0:384; A ! xS=0:2jx=0:8;

B ! yS=0:2jy=0:8:

It is straightforward to prove that both A and C generate the same crisp language. Now we construct a new FCFG^ B ¼ ðT; N; P; A0 ; hB Þ from C, the only difference between B and C lies in the definitions of fuzzy sets hC and hB , where hB is defined as follows: hB ðA0 ! SÞ ¼ 1; hB ðS ! xSBÞ ¼ hB ðS ! xBSÞ ¼ hB ðS ! ySAÞ ¼ hB ðS ! yASÞ ¼ hB ðA ! xSÞ ¼ hB ðB ! ySÞ ¼ 0:2; hB ðS ! xBÞ ¼ 0:4; hB ðS ! yAÞ ¼ 0:2; hB ðA ! xÞ ¼ hB ðB ! yÞ ¼ 0:8: Obviously, for any generated string x 2 T  , if fA ðxÞ < 15 ; then fC ðxÞ ¼ fA ðxÞ _ 15 ¼ 15 : By the definition of hB ; we have fB ðxÞ ¼ 15 and jfA ðxÞ  fB ðxÞj ¼ 15  fA ðxÞ < 15 < . If fA P 15 ; then fC ðxÞ ¼ fA ðxÞ _ 15 ¼ fA ðxÞ: Recall that by definitions of hB and hC ; it follows that jfA ðxÞ  fB ðxÞj ¼ jfC ðxÞ  fB ðxÞj 6 15 < . Therefore, B is an FCFG^ in the class n-FCFG^ which can approximate A with accuracy . Remark 3.2. In practice, as pointed out in Theorems 3.2 and 3.4 and the constructions given in Remark 3.1, if H satisfies WFGC, we know that FCFGH s under max-H compositional inference and FCFG^ s under max–min compositional inference are equivalent in the approximation sense. Therefore, for different kinds of t-norms satisfying WFGC, FCFGH s under maxH compositional inference are equivalent in the approximation sense. This gives an affirmative answer to the mentioned problem in the introduction section, and FCFGH s under different compositional inference mode have the ability to describe fuzzy systems with fuzzy uncertainty. That is to say, just as fuzzy systems under different compositional reasoning methods are universal approximators to the general fuzzy control systems [16,34], FCFGH s under different compositional inference are also universal to fuzzy systems. This shows the significance of Theorem 3.4. Furthermore, since a continuous t-norm can be uniformly approximated by a sequence of Archimedean t-norms [12], the results presented in this paper can also be extended to any continuous t-norm if the t-norm satisfies WFGC. This fact could be observed, we shall not expose this issue further in this paper. For an FCFGH G with max-H composition, the codomain of fG can be infinite if the t-norm H satisfying WFGC, i.e., RðfG Þ can be infinite since the t-norm H can produce new elements. This fact could be observed in [1,2]. But the codomain of FCFG^ is always finite as shown in Theorem 3.1, so we have the following distinguished result. Theorem 3.5. Let H be a t-norm satisfying FGC, then for any FCFGH G, there is an FCFG^ G0 such that fG ¼ fG0 , that is, G and G0 are equivalent. If H does not satisfy FGC, then there is an FCFGH G such that RðfG Þ is infinite and there is no FCFG^ equivalent to G. Proof. If the t-norm H satisfies FGC, and a fuzzy language fG can be generated by an FCFGH . Then RðfG Þ is finite and by Theorem 3.2 ðfG Þa is a CFL for any a 2 Rðf Þ. By Theorem 3.1, fG can be generated by an FCFG^ . If H does not satisfy FGC, by Lemma 3.2, there exists a 2 ð0; 1Þ such that ai –aj whenever i–j. Consider the 1 a a FCFGH G ¼ ðfx1 ; x2 ; x3 g; fA0 ; A1 ; A2 g; P; A0 ; hÞ, where P consists of the fuzzy productions A0 ! A1 A2 , A1 ! x1 A1 x2 , A1 ! x1 x2 , a a  n1 n1 m1 n n m A2 ! x3 A2 ; and A2 ! x3 . Then for any x 2 fx1 ; x2 ; x3 g , if x R fx1 x2 x3 jn; m P 1g; then fG ðxÞ ¼ 0; and if x ¼ x1 x2 x3 for some nonnegative integers n1 ; m1 ; we have fG ðxÞ ¼ an1 þm1 . Obviously, RðfG Þ is infinite, so fG can not be generated by any FCFG^ . h Theorem 3.6. Let H be a t-norm satisfying FGC. For a fuzzy language f : T  ! ½0; 1; the following statements are equivalent. (1) f is generated by an FCFG^ . (2) Rðf Þ is finite and fa is a CFL for any a 2 Rðf Þ. (3) f is generated by an FCFGH . Proof. By Theorems 3.1 and 3.5, the proof is straightforward.

h

Theorem 3.7. Let H be an Archimedean t-norm. For any FCFGH G, let D ¼ RðhÞ [ f1g. If RðfG Þ is infinite, then RðfG Þ is a sequence of SðDÞ converging to 0. Theorem 3.8. Let H be an Archimedean t-norm. If f : T  ! ½0; 1 satisfies the following conditions: (i) for any a 2 ð0; 1, Ra ðf Þ ¼ ff ðxÞ : x 2 T  ; f ðxÞ P ag is finite, and (ii) for any a 2 ð0; 1; f a is a CFL, then for any  2 ð0; 1, there exists an FCFG^ G0 such that the following inequality holds for any generated x 2 T  ,

jf ðxÞ  fG0 ðxÞj 6 : Proof. The proof follows from Theorem 3.2, Lemma 3.3 and Theorem 3.3. h

4. Sensitivity of fuzzy context-free max- grammars Motivated by notions of robustness and sensitivity in fuzzy computations and in parsing context-free languages [2], some general and systematic results have been obtained. Now our focus will be on the sensitivity of fuzzy computations for

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FCFGH s. In other words, how does imprecision of fuzzy derivation of an FCFGH influence the overall imprecision of the fuzzy computations? How do we precisely give the membership degree of fuzzy derivation if the overall precision of computations is not to be compromised? To address this problem, we present the concept of ‘‘closeness” of FCFGH s, which is similar to that given in fuzzy finite automata and quantum automata in [3,17,18]. 0

Definition 4.1. Two FCFGH s, G ¼ ðT; N; P; g; hÞ and G0 ¼ ðT; N; P0 ; g0 ; h Þ are -close,  > 0, if they have the same sets of terminals and nonterminals and the difference between pairs of the corresponding membership degrees is at most . That is to say, jgðAÞ  g0 ðAÞj <  and jlða ) bÞ  l0 ða ) bÞj <  for any A 2 N; a; b 2 ðT [ NÞ ; where a ¼ c1 Ai c2 ; b ¼ c1 xi c2 ; i.e., a qi directly derives b by applying the same production Ai ! xi ; i 2 N; but with different membership degree. We need the following two lemmas, which are well-known results encountered mathematical analysis, we may refer to [16,19]. Lemma 4.1. Assume that f ; g : U ! ½0; 1 are two functions, then

   _ _ _   gðxÞ 6 jf ðxÞ  gðxÞj;  f ðxÞ   x2U x2U x2U    _ ^ ^   gðxÞ 6 jf ðxÞ  gðxÞj:  f ðxÞ   x2U x2U x2U Lemma 4.2. Assume that ai ; bi 2 ½0; 1 such that jai  bi j <  for i ¼ 1; 2; . . . ; k, then

ja1 Ha2 H    Hak  b1 Hb2 H    Hbk j < 1  ð1  Þk : Theorem 4.1. If two FCFGH s, G and G0 are -close, with the set of terminals T and nonterminals N are -close, then

jfG ðxÞ  fG0 ðxÞj < 1  ð1  Þnþ1 holds for any generated string x 2 T  , where x is obtained by n-steps derivation from the starft nonterminal A of G or G0 . W W W   Proof. From jfG ðxÞ  fG0 ðxÞj = j fgðAÞHlðA ) xÞ : A 2 N; x 2 T  g  fg0 ðAÞHl0 ðA ) xÞ : A 2 N; x 2 T  gj = j fgðAÞHlðA ) W 0  0 0 a1 ÞHlða1 ) a2 ÞH    Hlðan1 ) xÞ : A 2 N; ai 2 ðT [ NÞ ; i ¼ 1; 2; . . . ; n  1g  fg ðAÞHl ðA ) a1 ÞHl ða1 ) a2 ÞH    Hl0 ðan1 ) xÞ : A 2 N; ai 2 ðT [ NÞ ; i ¼ 1; 2; . . . ; n  1gj. Since jgðAÞ  g0 ðAÞj <  and jlðai1 ) ai Þ  l0 ðai1 ) ai Þj <  for any i ¼ 1; 2; . . . ; n and let A ¼ a0 ; x ¼ an , using Lemmas 4.1 and 4.2, we have jfG ðxÞ  fG0 ðxÞj < 1  ð1  Þnþ1 for any x 2 T  , where n means n-steps derivation. h Corollary 4.1. Assume that H is an Archimedean t-norm. Let G be an FCFGH , and let G0 be another FCFGH which is a-closed to G, 1 where a ¼ 1  ð1  Þ1þn ;  > 0; n 2 N. Then G0 simulates G for n-steps derivation with accuracy . Proof. In the n-steps derivation, for any x 2 T  ; x can be generated by FCFGH with less than n-steps derivation. By Theorem 4.1, jfG ðxÞ  fG0 ðxÞj < 1  ð1  aÞnþ1 6  for any x 2 T  satisfying jxj 6 n. Thus G0 simulates G for n-steps with accuracy . h Corollary 4.2. Assume that H is the minimum operation. Let G be an FCFGH , and let G0 be another FCFGH which is -closed to G;  > 0. Then G0 simulates G with accuracy . Proof. If H ¼ ^, then jfG ðxÞ  fG0 ðxÞj <  for any x 2 T  as shown in Theorem 4.1. That is to say, G0 simulates G with accuracy . h 5. Conclusions In this paper, we have dealt with FCFG^ under max–min compositional inference and FCFGH under max-H compositional inference for some t-norm H satisfying weakly finite generated condition. We have shown that an FCFG^ is in general not equivalent to an FCFGH in recognizing fuzzy languages. In particular, we presented the fact that an FCFGH can be approximated by some FCFG^ with any given accuracy. In the point of this view, the related construction is also presented in the Remark 3.1. Furthermore, they are equivalent if the t-norm H satisfies a certain condition, which is essential in classical situation. Therefore, as the models of discrete systems with fuzzy uncertainty, FCFGH s can be universally used in such as learning systems, neural networks and intelligent interface design. The sensitivity of FCFGH s against the imprecisions of fuzzy derivation is also considered. To a certain extent, these results are some generalizations of the previous conclusions in literature [2].

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Undoubtedly, much work remains to be completed along this line. For example, approximation properties and sensitivity may be readily extend to fuzzy type 0 grammars [22] as we have done in this paper. In fact, some related work has been done in [2]. Further work should be done on the fuzzy computational complexity theory using fuzzy grammars theory. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36]

P.R.J. Asveld, Fuzzy context-free languages Part 1: generalized fuzzy context-free grammars, Theoretical Computer Science 347 (1–2) (2005) 167–190. P.R.J. Asveld, Fuzzy context-free languages Part 2: recognition and parsing algorithms, Theoretical Computer Science 347 (1–2) (2005) 191–213. E. Bernstein, U. Vazirani, Quantum complexity theory, SIAM Journal of Computing 26 (5) (1997) 1411–1473. J. Berstel, Transductions and Context-Free Languages, Teubner, Stuttgart, 1979. G.F. DePalma, S.S. Yau, Fractionally fuzzy grammars with application to pattern recognition, in: L.A. Zadeh, K.S. Fu, K. Tanaka, M. Shimura (Eds.), Fuzzy Sets and Their Applications to Cognitive and Decision Processes, Academic Press, 1975, pp. 329–351. R. Beˇlohlávek, Determinism and fuzzy automata, Information Sciences 143 (2002) 205–209. F. Lin, H. Ying, Modeling and control of fuzzy discrete event systems, IEEE Transactions on Systems, Man and Cybernetics B 32 (4) (2002) 408–415. C.L. Gile, C.W. Omlin, K.K. Thornber, Equivalence in knowledge representation: automata, recurrent neural networks, and dynamical fuzzy systems, Proceedings of IEEE 87 (1999) 1623–1640. P. Hájak, Metamathematics of Fuzzy Logic, Kluwer Academic Publisher, Dordrecht, 1998. S. Hikmet, Fuzzy command grammars for intelligent interface design, IEEE Transactions on Systems, Man and Cybernetics 22 (1992) 1124–1131. J.E. Hopcroft, J.D. Ullman, Introduction to Automata Theory, Languages and Computation, Addison-Wesley, New York, 1979. E.P. Klement, E. Pap, Triangular Norms, Kluwer Academic Publisher, Dordrecht, 2000. E.T. Lee, L.A. Zadeh, Note on fuzzy languages, Information Sciences 1 (1969) 421–434. J. Ignjatovic´, M. C´iric´, S. Bogdanovic´, Determinization of fuzzy automata with membership values in complete residuated lattices, Information Sciences 178 (2008) 164–180. 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. Y.M. Li, Analysis of Fuzzy Systems, Science Press, Beijing, 2005 (in Chinese). Y.M. Li, Approximation and robustness of fuzzy finite automata, International Journal of Approximate Reasoning 47 (2) (2008) 247–257. Y.M. Li, Approximation and universality of fuzzy Turing machines, Science in China Series F: Information Sciences 51 (10) (2008) 1445–1465. Y.M. Li, D. Li, W. Pedrycz, J. Wu, An approach to measure the robustness of fuzzy reasoning, International Journal of Intelligent Systems 20 (4) (2005) 393–413. Z.H. Li, P. Li, Y.M. Li, The relationships among several types of fuzzy automata, Information Sciences 176 (2006) 2208–2226. P. Li, Y.M. Li, Algebraic properties of LA-languages, Information Sciences 176 (2006) 3232–3255. J.N. Mordeson, D.S. Malik, Fuzzy Automata and Languages: Theory and Applications, Chapman & Hall/CRC, Boca Raton, London, 2002. S. Bozapalidis, O. Louscou-Bozapalidou, On the recognizability of fuzzy languages I, Fuzzy Sets and Systems 157 (2006) 2394–2402. E.S. Santos, Fuzzy automata and language, Information Sciences 10 (1976) 193–197. M.G. Thomson, P.N. Marinos, Deterministic acceptors of regular fuzzy languages, IEEE Transactions on Systems, Man and Cybernetics 4 (1974) 228– 230. L. Sheng, Y.M. Li, Regular grammars with truth values in lattice-ordered monoid and their languages, Soft Computing 10 (2006) 79–86. X. Guo, Grammar theory based on lattice-ordered monoid, Fuzzy Sets and Systems (2008), doi:10.1016/j.fss.2008.07.009. W. Kuich, G. Rahonis, Fuzzy regular languages over finite and infinite words, Fuzzy Sets and Systems 157 (2006) 1532–1549. A. Paz, Introduction to Probabilistic Automata, Academic, New York, 1971. E. Trillas, On the use of words and fuzzy sets, Information Sciences 176 (11) (2006) 1463–1487. D.W. Qiu, H. Wang, A probabilistic model of computing with words, Jounal of Computer and System Sciences 70 (2005) 176–200. D.W. Qiu, Automata theory based on quantum logic: reversibilities and pushdown automata, Theoretical Computer Science 386 (2007) 38–56. F. Steimann, K.P. Adlassning, Clinical monitoring with fuzzy automata, Fuzzy Sets and Systems 61 (1994) 37–42. L.X. Wang, A Course in Fuzzy Systems and Control, Prentice-Hall PTR, Englewood Cliffs, NJ, 1997. L.A. Zadeh, A note on web intelligence, world knowledge and fuzzy logic, Data and Knowledge Engineering 50 (2004) 291–304. L.A. Zadeh, Is there a need for fuzzy logic?, Information Sciences 178 (2008) 2751–2779.