Researches on Six Lattice-Valued Logic ()

Hua Li^{}

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

**DOI: **10.4236/jcc.2015.310005
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Department of Information Engineering, Hangzhou Polytechnic College, Hangzhou, China.

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.

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Li, H. (2015) Researches on Six Lattice-Valued Logic. *Journal of Computer and Communications*, **3**, 36-42. doi: 10.4236/jcc.2015.310005.

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 L_{6}

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 L_{6}.

Table 2. Computing of L_{3}.

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 L_{6}

Due to L_{6} 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 L_{6} 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 L_{6}, is a sub lattice implication algebra. What’s more, L_{0} is a Boolean algebra, and the implication arithmetic of it meets that: for any x, ,.

Proof: It is clearly that L_{0} is a sub lattice of L_{6}. For any, , , therefore when regarding L_{6}, the operation of L_{0} is closed, that is to say, L_{0} is a sub lattice implication algebras of L_{6}.

It can be verified easily: for any,. Meeting the of Boolean algebra axiom, L_{0} 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 [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 [2] . 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 [2] . 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 2^{n} sub lattice.

According to Theorem 5, theorem representation of six lattice-valued L_{6} can be got easily.

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

(1) The set of join-irreducible element in L_{6} 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(L_{6})) of the ideal lattice of L_{6}, which forms through inclusion relation, is shown in Figure 4. Form the figure, we can see that L_{6} is isomorphic of lattice implication to its ideal lattice O(Á(L_{6})). Lattice implication isomorphism h is defined as follows:

4. The Filter of Lattice L_{6}

Since all Lukasiewicz algebras are lattice implication algebra [1] , it can be proved that Lukasiewicz algebra filters are trivial.

Figure 3. The order of Á(L_{6}).

Figure 4. The ideal lattice of O(Á(L_{6})).

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 C_{2}.

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

thereby, that is contradictory, because.

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}, L_{6}, ,

Proof: According to Theorem 1, L_{6} can be seen as the direct product of C_{2} and L_{3}. 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 L_{6} are the direct products of the filters of C_{2} and the filters of L_{3}. So the filters of L_{6} are as followed:

, , and L_{6} itself.

In other words: The six element lattice L_{6} only contains the following four filters:

{I}, L_{6}, ,

5. The Tautologies of Lattice-Valued Logic System L_{6}P(X)

Here we take the lattice-valued logic system L_{6}P(X) into consideration, and discuss its tautologies and F-tauto- logies, the true value domain is L_{6}.

It is easy to verify:

where C_{2} is a Boolean algebra, L_{3} is a Lukasiewicz algebra.

Theorem 9. (The definition of tautologies in L_{6}P(X) [3] ) The tautologies in six lattice-valued logic system L_{6}P(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 L_{6} only contains four filters as followed:

{I}, L_{6}, ,

Therefore, its non-trivial filters are,

We can get the definition of F-tautologies in six lattice-valued logic system L_{6}P(X) as Theorem 8 similarly.

Theorem 10. (The definition of F-tautologies in L_{6}P(X) [4] ) The F-tautologies in six lattice-valued logic system L_{6}P(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.

Conflicts of Interest

The authors declare no conflicts of interest.

[1] | Yang, X. and Yun, Q.K. (1995) Fuzzy Lattice Implication Algebra. Southwest Jiaotong University, 2, 121-127. |

[2] |
Xu, Y., Ruan, D. and Liu, J. (2004) Progress and Prospect in Lattice-Valued Logic Systems Based on Lattice Implication Algebras. Proceedings of the 6th International FLINS Conference Applied Computational Intelligence, 29-34. http://dx.doi.org/10.1142/9789812702661_0009 |

[3] | Min, H.T. (1996) Georgia, Sequencing Primer Theory and Its Application. Southwest Jiaotong University Press. |

[4] | Fang, S. and Mei, Z.F. (2010) Six Yuan-Based Language of Logic Attributed True Value Method. Guangxi Normal University, 3, 118-122. |

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