﻿ Researches on Six Lattice-Valued Logic

Journal of Computer and Communications
Vol.03 No.10(2015), Article ID:60737,6 pages
10.4236/jcc.2015.310005

Researches on Six Lattice-Valued Logic

Hua Li

Department of Information Engineering, Hangzhou Polytechnic College, Hangzhou, China

Email: zjlihua@126.com   Received 2 September 2015; accepted 26 October 2015; published 29 October 2015

ABSTRACT

Based on the direct product of Boolean algebra and Lukasiewicz algebra, six lattice-valued logic is put forward in this paper. The algebraic structure and properties of the lattice are analyzed profoundly and the tautologies of six-valued logic system L6P(X) are discussed deeply. The researches of this paper can be used in lattice-valued logic systems and can be helpful to automated reasoning systems.

Keywords:

Six Lattice-Valued Logic, Lattice Implication Algebra, Filter, Tautology 1. Introduction

Lattice-valued logic is an important case of multi-valued logic, and it plays more and more important roles in artificial intelligence and automated reasoning. Six lattice-valued is a kind of common lattice, which can express logic in real world, such as language values, and evaluation values. It can deal with not only comparable information but also non-comparable information. Therefore, theoretical researches and logic and reasoning systems based on six lattice-valued logic are of great significance.

2. The Structure of Lattice L6

The set of is a lattice, and the order relation of L is shown in Figure 1. The complement operator “'”and implication operation “®” are defined in Table 1 respectively.

L means an lattice implication algebra.

Then set , . As A is the true set of classical binary logic, the operation rules of the complement operation and the implication operation are the same with the classical two-valued logic systems. B is the true-value set of Lukasiewicz system with three-valued logic, and complement operations and implication operations are defined in Table 2.

Let , the order relations, disjunctive, conjunctive, complement operation and implication operation

Figure 1. Structure of the six-valued lattice.

Table 1. Computing of the six-valued lattice L6.

Table 2. Computing of L3.

on L are defined as follows:

For any , :

(1) , if and only if and .

(2) , if and only if and .

(3) Under other circumstances, (x, y) cannot be compared with (z, r).

(4) ,

(5)

(6)

The L* constitute a six element lattice and its operation diagram is shown in Hasse Figure 2.

Theorem 1. L is isomorphic lattice implication of L*.

Proof:

Obviously, we can construct a upward one-to-one mapping from L to L*:, making

Clearly f is conjunctive homomorphic mapping and disjunctive homomorphism mapping.

Here is the proof that f is complement homomorphic mapping and implication homomorphism mapping.

According to the definition of implication operations and complement operations, it can be easily obtained in Table 3.

Figure 2. Six-valued lattice generated by the direct product.

Table 3. Six-valued lattice generated by the direct product.

It can be seen from the Table 3, f is the implication operations and the complement operations homomorphic.

In summary, we proofed that:

For any, , , where * is one of disjunctive, conjunctive, complement operation.

Thus L and L* is isomorphic lattice implication.

3. The Property and Language of Lattice L6

Due to L6 is a lattice implication algebra, it not only has all the properties of lattice implication algebra but also properties as follows.

Theorem 2. As shown the six-valued lattice L6 in Figure 1, the implication operation satisfies the following properties: For any:

(1) iff

(2)

(3)

(4)

(5)

(6)

Theorem 3. As the true subset of L6, is a sub lattice implication algebra. What’s more, L0 is a Boolean algebra, and the implication arithmetic of it meets that: for any x, ,.

Proof: It is clearly that L0 is a sub lattice of L6. For any, , , therefore when regarding L6, the operation of L0 is closed, that is to say, L0 is a sub lattice implication algebras of L6.

It can be verified easily: for any,. Meeting the of Boolean algebra axiom, L0 is a Boolean algebra.

Any sub-set of power set lattice in a collection is called the set lattice for the collection. The isomorphism from a lattice L to a set lattice B(X) in collection X is named as a isomorphic representation L by B(X), which can be denoted as L for abbreviation. Through establishing the lattice representation, lattice language can be simplified, which is very important for studying the structure and properties of the lattice.

Definition 1  . Let L is a lattice, an element is called as an join-irreducible element, if

(1) x ¹ O (when there is a minimum of O when L);

(2) For any, if, then x = a or x = b.

Assume L is a finite distributive lattice, Á(L) denotes the set of all join-irreducible element in the collection, and all the join-irreducible element in L can form under set lattice (i.e. ideal Lattice) according to the order relation which can be indicated as O(Á(L)). Then we have the following conclusions:

Theorem 4  . Let L is a finite distributive lattice, and mapping can be constructed as follows:

The h is the lattice isomorphism from L to O(Á(L)).

Theorem 5  . Let L is a finite distributive lattice, then the following equivalent hold:

1) L is a distributive lattice;

2);

3) L is isomorphic to a set lattice;

4) For any n ³ 0, L is isomorphic to 2n sub lattice.

According to Theorem 5, theorem representation of six lattice-valued L6 can be got easily.

Theorem 6. As shown the six-valued lattice L6 in Figure 1, conclusions as follows can be got:

(1) The set of join-irreducible element in L6 is, and its order relation are shown in Figure 3.

(2) The under set lattice (i.e. ideal lattice), which is the set of all the join-irreducible element and forms according to its order relation, is.

(3) The Hasse diagram O(Á(L6)) of the ideal lattice of L6, which forms through inclusion relation, is shown in Figure 4. Form the figure, we can see that L6 is isomorphic of lattice implication to its ideal lattice O(Á(L6)). Lattice implication isomorphism h is defined as follows:

4. The Filter of Lattice L6

Since all Lukasiewicz algebras are lattice implication algebra  , it can be proved that Lukasiewicz algebra filters are trivial.

Figure 3. The order of Á(L6).

Figure 4. The ideal lattice of O(Á(L6)).

Theorem 7.

(1) The finite chain of Lukasiewicz only contains trivial filters.

(2) Lukasiewicz algebra [0,1] only contains trivial filters.

Proof: (1) Let’s set. Specific operations are as follows:

For any,

,

,

It is clearly that set {1} and L are trivial filters in L. we can proof that L don’t contain any other trivial filters.

From Theorem 6 we can see that filters in L are ideal dual filters of L, and the set of ideal dual filters of L are upper set of L.

If (where k ³ 1) is a filter of L, then

,

And, so it can be seen that the definition of filters:

This shows that F = L, so it demonstrated that L don’t contain any other trivial filters.

(2) Let L = [0,1], its upper operation is the same as defined C2.

It is clearly that set {1} and L are trivial filters in L. we can proof that L don’t contain any other trivial filters.

We can see that filters in L are ideal dual filters of L, and the set of ideal dual filters of L are upper set of L. So the filter of L must be an interval containing greatest element 1.

Firstly, we can proof that the filter of L must be a closed interval.

Let us set is a filter of L, where, for any x, satisfies, then, and , conclusion can get.

This shows that F is a closed interval.

Secondly, assume is a filter of L, where.

For any x, making and, then

So F is an interval.

This proves that Lukasiewicz interval only have trivial filters.

As a special case of Theorem 7, we have the following corollary.

Corollary 1. and only contain trivial filters.

Theorem 8. The six element lattice only contains the following four filters:

{I}, L6, ,

Proof: According to Theorem 1, L6 can be seen as the direct product of C2 and L3. According to Corollary 1, and only contain trivial filters. As followed:

The filters of are {I} and.

The filters of are {I} and.

It is easy to know, the filters of L6 are the direct products of the filters of C2 and the filters of L3. So the filters of L6 are as followed:

, , and L6 itself.

In other words: The six element lattice L6 only contains the following four filters:

{I}, L6, ,

5. The Tautologies of Lattice-Valued Logic System L6P(X)

Here we take the lattice-valued logic system L6P(X) into consideration, and discuss its tautologies and F-tauto- logies, the true value domain is L6.

It is easy to verify:

where C2 is a Boolean algebra, L3 is a Lukasiewicz algebra.

Theorem 9. (The definition of tautologies in L6P(X)  ) The tautologies in six lattice-valued logic system L6P(X) process the following relationship:

(1)

(2)

(3)

(4)

(5)

Proof: It is noticed that the tautologies in Lukasiewicz three-valued logic system process the following relationship:

Proof of this theorem can be obtained.

From Theorem 7, the six element lattice L6 only contains four filters as followed:

{I}, L6, ,

Therefore, its non-trivial filters are,

We can get the definition of F-tautologies in six lattice-valued logic system L6P(X) as Theorem 8 similarly.

Theorem 10. (The definition of F-tautologies in L6P(X)  ) The F-tautologies in six lattice-valued logic system L6P(X) process the following relationship:

(1)

(2)

Proof:

Since, so T is an injection.

Clearly T is a surjection. For any,

, G has the inverse image.

Thus G is an isomorphic functor of À(L).

As isomorphic relationship means an equivalence relation, so SÀ(L) and Á(L) are isomorphic.

6. Conclusion

In this paper, the six element lattice is built by the direct product of Boolean algebra and Lukasiewicz algebra; the operation of the lattice is defined; the structures, properties and filters are studied; finally the tautologies and F-tautologies of the six lattice-valued logic system are discussed. The results of this paper can be applied to lattice-valued logic systems and automated reasoning applications.

Acknowledgements

The work is supported by the project of Zhejiang province education department of China, Grant No. Y201326675.

Cite this paper

HuaLi, (2015) Researches on Six Lattice-Valued Logic. Journal of Computer and Communications,03,36-42. doi: 10.4236/jcc.2015.310005

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