The Lattice-Valued Turing Machines and the Lattice-Valued Type 0 Grammars

Purpose. The purpose of this paper is to study a class of the natural languages called the lattice-valued phrase structure languages, which can be generated by the lattice-valued type 0 grammars and recognized by the lattice-valued Turing machines. Design/Methodology/Approach. From the characteristic of natural language, this paper puts forward a new concept of the l-valued Turing machine. It can be used to characterize recognition, natural language processing, and dynamic characteristics. Findings. Themechanisms of both the generation of grammars for the lattice-valued type 0 grammar and the dynamic transformation of the lattice-valued Turing machines were given.Originality/Value.This paper gives a new approach to study a class of natural languages by using lattice-valued logic theory.


Introduction
According to Chomsky's rationalist theory, the language can be divided into types by the grammar, and then, types can be recognized and processed, respectively.The fuzziness is the typical trait of the natural languages, and then, how to recognize and deal with the fuzzy natural language is becoming the more and more important subject.From the point of the reorganization for the natural language, in 1967, Wee [1] firstly introduced the fuzzy finite automata, which generated many interesting explorations.Its important applications in learning systems, automatic control, pattern recognition and database, and so on were also studied by many researchers; the more details and the fuzzy finite automata were studied by Wee and Fu [2], Santos [3][4][5][6], Lee and Zadeh [7], Kumbhojkar and Chaudhari [8,9], Malik et al. [10][11][12][13][14][15], and so on; the authors can also refer to [16,17].
In this paper, our main purpose is to consider a class of the natural languages which is called the lattice-valued phrase structure languages.In fact, these natural languages can be generated by the lattice-valued type 0 grammars and recognized by the lattice-valued Turing machines.Moreover, the natural languages can be described by the formation mechanism and the transferred mechanism of the grammar.
This paper is organized as follows.Some basic concepts such as the complete Heyting algebra, the lattice-valued language, and the corresponding membership were recalled in Section 2. In Section 3, we mainly studied the latticevalued Turing machine and the lattice-valued recursively enumerable language.Especially, the mechanisms of both the generation of grammars for the lattice-valued type 0 grammar and the dynamic transformation of the lattice-valued Turing machines were given; we used the lattice-valued logic theory to study the class of the natural languages; it seems to be that the approach which we used is new as far as we know.Finally, we study the equivalence between the lattice-valued type 0 grammar and the lattice-valued Turing machine in Section 4.

Preliminaries
Throughout this paper, a nonempty finite set Σ is called an alphabet.The element in the alphabet is called a symbol or a letter.And, Σ * denotes the free monoid, that is, the set of all 2 Mathematical Problems in Engineering strings with letters from Σ and the empty string, where the empty string is denoted by ; moreover, the length of a string  is denoted by ||, which is the number of symbols in the string; for the empty string , we set || = 0.
Given an alphabet Σ.The concatenation of two strings  and  is the string which was denoted by  and was obtained by appending the symbols of  to the right end of .Assume  is a string, and then   stands for the string obtained by repeating the string  by  times.As a special case, we can define  0 = .
Let Σ 1 , Σ 2 be two alphabets; the product of Σ 1 and Σ 2 is given by Define  power of an alphabet Σ by ( The positive closure of Σ is and the star-closure of Σ is Suppose  is a complete lattice; the least element and the greatest element are 0 and 1, respectively, and also satisfy the infinite distributivity law.That is, ∀ ∈ ,   ∈ ,  ∈ , we have Then  is called a complete Heyting algebra. Let  be a complete Heyting algebra and Σ an alphabet.We call a map  : Σ * →  a lattice-value language.∀ ∈ Σ * , () denotes the membership degree of  belonging to the lattice-valued language.
Let ,  1 , and  2 be the lattice-valued languages on Σ.Then (1)  1 ∨  2 (resp.,  1 ∧  2 ) is a lattice-valued language on Σ via Generally, if  1 ,  2 , . . .,   are the lattice-valued languages on Σ, then (2)  1 ∘  2 is a lattice-valued language on Σ which is defined by (3) Let  ∈ ; then  is a lattice-valued language on Σ which is defined by (4)  * is a lattice-valued language on Σ which is defined by Let  be a nonempty set and  a complete Heyting algebra.The map is called the binary lattice-valued relation on .If (, ) = , we denote it by   , which can be understood as the membership degree  of  and  satisfying the relation .
Let ,  be binary lattice-valued relations on ; the composition operation of  and  is defined by Consider the following: (a)

The Lattice-Valued Turing Machine and the Lattice-Valued Recursively Enumerable Language
A lattice-valued Turing machine M is a 7-tuple where The above description is the definition of a single tape lattice-valued Turing machine.For the  tapes lattice-valued Turing machine, the transition function is defined as the map from  × Γ  to  × Γ  × {−1, +1}  × .
We use  1  2 to denote an instantaneous description (ID for short) of the lattice-valued Turing machine M, where  0 ∈  is the current state of M and  1  2 is the string of Γ * .When the read-write head of M directs symbols right which has nonblank character,  1  2 is the string which consists of all nonblank symbols of the leftmost position to the rightmost position of the input tape of M, otherwise,  1  2 is the string which consists of all symbols of the leftmost position of the input tape of M to the tape location which is directed by the read head of M, and M is directing the leftmost symbol of  2 .Now, we define a binary lattice-valued relation on Γ * Γ * as follows. Let be a ID of M; if (,   ) = (, , +1, ), then the next ID of M is The membership degree of  replacing   is ; that is, If (,   ) = (, , +1, ), then, if the membership degree of  replacing   is ; that is, If  = 1, before M moves on, the read-write head has been at the far left of the input tape, then the read-write head moves left, which would make the read-write head away from the input tape, which is not allowed.In order to avoid this phenomenon, in this case, we have defined that M has not the next ID.
Obviously, ⊢ M is a binary lattice-valued relation on By the composition definition of the binary lattice-valued relations, it is not difficult to see the following: if ⊢  M (ID 1 , ID 2 ) =  ̸ = 0, which represents that the membership degree ID 1 turns to ID 2 after  steps is  in M; if ⊢ + M (ID 1 , ID 2 ) =  ̸ = 0, which represents that the membership degree ID 1 changes into ID 2 after one step at least is  in M; if ⊢ * M (ID 1 ,ID 2 ) =  ̸ = 0, which represents that the membership degree ID 1 turns into ID 2 after several steps is  in M. Definition 1.Let M = (, Γ, Σ, ,  0 , , ) be a lattice-valued Turing machine.The acceptable lattice-valued language M is defined by The language which is accepted by the lattice-valued Turing machine is called a lattice-valued recursively enumerable language.

The Lattice-Valued Type 0 Grammar and Lattice-Valued Phrase Structure Language
The lattice-valued grammar G is a 4-tuple G = (, , , ), where we have the following.
is a nonempty finite set of variables and ∀ ∈ ,  is called a syntactic variable (variable for short) or a nonterminal symbol.It represents a syntactic category.
is a nonempty finite set of the terminal symbols and ∀ ∈ ,  is called a terminal symbol.Since the variables in  represent the syntactic category and the characters in  are the characters that appear in the sentence of language, so we have  ∩  = 0.
is a set of lattice-valued of production; that is,  is a binary lattice-valued relationship on ( ∪ ) * .For ,  ∈ ( ∪ ) * and a fixed  ∈ , if (, ) =  and  ̸ = 0, then it can be denoted by  →   which is called the production on G, and the production means that the membership degree that  is defined as  is . is called the left part of  →  ,  is the right part of  →  , and  is the membership degree that  is defined as .The productions are termed the definitions or the grammar rules.
∈  is the start symbol of the grammar G.
According to Chomsky hierarchy of the grammar and the concept of the fuzzy grammar of Lee and Zadeh, the latticevalued grammar can be divided into the following four types.Definition 2. Let G = (, , , ) be a lattice-valued grammar.Then one has the following.
(1) G is called the lattice-valued type 0 grammar or the lattice-valued phrase structure grammar, if the production in  is without any constraint conditions.
The corresponding language  G is called the latticevalued type 0 language or the lattice-valued phrase structure language.
(2) G is called the lattice-valued type 1 grammar or the lattice-valued context sensitive grammar, if, for any production  →   ∈ , one has || ≤ ||.The corresponding language  G is called the lattice-valued type 1 language or the lattice-valued context sensitive language.
(3) G is called the lattice-valued type 2 grammar or the lattice-valued context-free grammar, if, for any production  →   ∈ , one has || ≤ || and  ∈ .
The corresponding language  G is called the latticevalued type 2 language or the lattice-valued contextfree language.
(4) G is called the lattice-valued type 3 grammars or the lattice-valued regular grammar, if, for any production  →   ∈ , one has where ,  ∈ ,  ∈  * .The corresponding language  G is called the lattice-valued type 3 language or the lattice-valued regular language.If ⇒ G (, ) =  ̸ = 0, then we can say that the membership degree that  can deduce  in the latticevalued grammar G is , or the membership degree that  can be reduced into  in the lattice-valued grammar G is .
Let ,  ∈ ( ∪ ) * ; according to the composition definition of the binary lattice-valued relationship, it is not difficult to see the following.( G is called the lattice-valued language generated by latticevalued type 0 grammar which is termed the lattice-valued type 0 language or lattice-valued phrase structure language.
One calls that two lattice-valued grammars G 1 , G 2 are equivalent, if they can generate the same lattice-valued languages; that is,  G 1 =  G 2 .

(Definition 4 .
) =  ̸ = 0 represents the membership degree that  in G after n steps can deduce  is , or the membership degree that  in G after n steps can summarize  is , it also can be written as   , ) =  ̸ = 0 represents the membership degree that  in G after at least 1 step can deduce  is , or the membership degree that  in G after at least 1 step can summarize  is , it also can be written as  + the membership degree that  in G after some steps can deduce  is , or the membership degree that  in G after some steps can summarize  is , it also can be written as  * Let G = (, , , ) be a lattice-valued type 0 grammar,  ∈  * .Define  G () = ( *  ⇒ G ) (, ) .
is a transition function, that is, a map from  × Γ to  × Γ × {−1, +1} × ; (, ) = (, , +1, ) means that M reads a symbol  in the state , the next state is , and the read-write head moves right one unit after the symbol  in place of  on the tape. represents the membership degree of  in place of .Similarly, (, ) = (, , −1, ) means that the read-write head moves left one unit after the same process as above. ⊆  is the set of final states.
is a finite set of states; Γ is a finite set of tape symbol and  is a special symbol of Γ called the blank; Σ is a subset of Γ not containing  and is called the input symbols set;