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

Abstract

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. 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. Originality/Value. This paper gives a new approach to study a class of natural languages by using lattice-valued logic theory.

1. 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 [3–6], Lee and Zadeh [7], Kumbhojkar and Chaudhari [8, 9], Malik et al. [10–15], 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 lattice-valued 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.

2. Preliminaries

Throughout this paper, a nonempty finite set is called an alphabet. The element in the alphabet is called a symbol or a letter. And, denotes the free monoid, that is, the set of all strings with letters from 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 .

Given an alphabet . 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 .

Let , be two alphabets; the product of and is given by Define power of an alphabet by

The positive closure of is

and the star-closure of 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 an alphabet. We call a map a lattice-value language. , denotes the membership degree of belonging to the lattice-valued language.

Let , , and be the lattice-valued languages on . Then(1) (resp., ) is a lattice-valued language on via Generally, if are the lattice-valued languages on , then (2) is a lattice-valued language on which is defined by (3)Let ; then is a lattice-valued language on which is defined by (4) is a lattice-valued language on 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) (b), ;(c); (d), , and are called power, the positive closure, and the star-closure of , respectively.

3. The Lattice-Valued Turing Machine and the Lattice-Valued Recursively Enumerable Language

A lattice-valued Turing machine is a 7-tuple where  is a finite set of states;  is a finite set of tape symbol and is a special symbol of called the blank;  is a subset of not containing and is called the input symbols set;  is a transition function, that is, a map from to ;  means that 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, means that the read-write head moves left one unit after the same process as above.  is the set of final states.

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 to .

We use to denote an instantaneous description (ID for short) of the lattice-valued Turing machine , where is the current state of and is the string of . When the read-write head of directs symbols right which has nonblank character, is the string which consists of all nonblank symbols of the leftmost position to the rightmost position of the input tape of , otherwise, is the string which consists of all symbols of the leftmost position of the input tape of to the tape location which is directed by the read head of , and is directing the leftmost symbol of .

Now, we define a binary lattice-valued relation on as follows.

Let be a of ; if , then the next of is

The membership degree of replacing is ; that is,

If , then, if , the next of is the membership degree of replacing is ; that is, If , before 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 has not the next .

Obviously, is a binary lattice-valued relation on .

denotes power of ; that is, ; denotes the positive closure of ; that is, ; denotes the star-closure of ; that is, .

Let . By the composition definition of the binary lattice-valued relations, it is not difficult to see the following:

if , which represents that the membership degree turns to after steps is in ; if , which represents that the membership degree changes into after one step at least is in ; if , which represents that the membership degree turns into after several steps is in .

Definition 1. Let be a lattice-valued Turing machine. The acceptable lattice-valued language is defined by The language which is accepted by the lattice-valued Turing machine is called a lattice-valued recursively enumerable language.

4. The Lattice-Valued Type 0 Grammar and Lattice-Valued Phrase Structure Language

The lattice-valued grammar is a 4-tuple , 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 .

is a set of lattice-valued of production; that is, is a binary lattice-valued relationship on . For and a fixed , if and , then it can be denoted by which is called the production on , 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 .

According to Chomsky hierarchy of the grammar and the concept of the fuzzy grammar of Lee and Zadeh, the lattice-valued grammar can be divided into the following four types.

Definition 2. Let be a lattice-valued grammar. Then one has the following.(1) 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 is called the lattice-valued type 0 language or the lattice-valued phrase structure language.(2) is called the lattice-valued type 1 grammar or the lattice-valued context sensitive grammar, if, for any production , one has . The corresponding language is called the lattice-valued type 1 language or the lattice-valued context sensitive language.(3) 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 is called the lattice-valued type 2 language or the lattice-valued context-free language.(4) is called the lattice-valued type 3 grammars or the lattice-valued regular grammar, if, for any production , one has where . The corresponding language is called the lattice-valued type 3 language or the lattice-valued regular language.

Definition 3. Let be a lattice-valued grammar. Define a binary lattice-valued relationship on as follows: for all and , and it can be written as .
If , then we can say that the membership degree that can deduce in the lattice-valued grammar is , or the membership degree that can be reduced into in the lattice-valued grammar is .
denotes the power of ; that is, ; denotes the positive closure of ; that is, ; denotes the star-closure of ; that is, .
Let ; according to the composition definition of the binary lattice-valued relationship, it is not difficult to see the following.
If represents the membership degree that in after n steps can deduce is , or the membership degree that in after n steps can summarize is , it also can be written as .
If represents the membership degree that in after at least 1 step can deduce is , or the membership degree that in after at least 1 step can summarize is , it also can be written as ; if represents the membership degree that in after some steps can deduce is , or the membership degree that in after some steps can summarize is , it also can be written as .

Definition 4. Let be a lattice-valued type 0 grammar, . Define is called the lattice-valued language generated by lattice-valued 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 , are equivalent, if they can generate the same lattice-valued languages; that is, .

5. The Equivalence between the Lattice-Valued Type 0 Grammar and the Lattice-Valued Turing Machine

Theorem 5. If can be recognized by the lattice-valued Turing machine that is, , then there exists the lattice-valued type 0 grammar such that .

Proof. Suppose that can be recognized by lattice-valued Turing machine Formally, we construct a lattice-valued type 0 grammar , where and the production in is as follows: (1), its membership degree is 1, and the following productions which are not labeled with membership degree were regarded as 1;(2);(3);(4);(5);(6), , , , and  ;(7), , , , and ;(8), , , , .
By using rules (1) and (2), we can obtain where every belongs to . Assume that can accept a string , and then there exists such that can use at most units of the string input on the right. Using rules (3), (4) ( times), and (5) in order, we can obtain Then we only use rules (6) and (7), until an accepting state can be generated.
By the induction on movement numbers made by , we can prove that, if then for all , , , and .
Obviously, for the 0 movement, the inductive hypothesis holds, since , .
Assume that the inductive hypothesis holds for the movements. Let By the induction hypothesis, where each and satisfy the following conditions.
If , then the th movement of goes to the right, so and the membership degree . By rule (6), is a production of . Therefore, for , , and the membership degree .
If , then the th movement of goes to the left, we can apply rule (7) to prove the above equation, and we can obtain for the case of
By rule (8), if in , then
Thus

Similarly, we have the following result.

Theorem 6. Assume is a language which generated by the lattice-valued type 0 grammar , and then, can be recognized by the lattice-valued Turing machine ; that is, .

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors thank reviewers for their patient and invaluable advice, all of which have been of inestimable worth to the completion of their paper. They also thank editor for being serious and responsible. The authors acknowledge the support of the National Natural Science Foundation of China (nos. 11161050, 31240020).