ABSTRACT

Based on the analysis of the properties of Γ-conclusion by means of deduction theorems, completeness theorems and the theory of truth degree of formulas, the present papers introduces the concept of the membership degree of formulas A is a consequence of Γ (or Γ-conclusion) in Łukasiewicz n-valued propositional logic systems, n-valued propositional logic system and the n-valued propositional logic systems. The condition and related calculations of formulas A being Γ-conclusion were discussed by extent method. At the same time, some properties of membership degree of formulas A is a Γ-conclusion were given. We provide its algorithm of the membership degree of formulas A is a Γ-conclusion by the constructions of theory root.

1. Introduction

Fuzzy logic is the theoretical foundation of fuzzy control. Spurred by the success in its applications, especially in fuzzy control, fuzzy logic has aroused the interest of many famous scholars, a series of important results have been created in documents [1-5]. For the sake of reasoning, we have to choose a subset of well-formed formulas, which can reflect come essential properties, as the axioms of the logical system and we then deduce the so-called -conclusion through some reasonable inference rules [6-9]. So, a natural question then arises: how to judge whether or not a general formula is a conclusion of a given theory, or to what extend the formula is a conclusion of? It is basic problem to judge one thing belong to one kind in artificial intelligence. As is well known, human reasoning is approximate rather than precise in nature. we basic starting point is to establish graded version of basic logical notions. In order to establish a solid foundation for fuzzy reasoning, professor G. J. Wang proposed the concept of root of theory [3], J. C. Zhang proposed the concept of generalized root of theory [10,11], in propositional logic systems. The graded description and properties of formulas being -conclusion were discussed. And provide its algorithm of membership degree of formulas A is a -conclusion, by the constructions of theory root in the above-mentioned logic systems.

2. Preliminaries

It is well known that different implication operators and valuation lattices (i.e., the set of truth degrees for logic) determine different logic systems (see [12]). Here valuation lattices is and three popularly used implication operators and the correspond ing t-norms defined as follows:

These three implication operators and are called Łukasiewicz implication operator, Gödel implication operator, and the -implication operator, respectively. The t-norm, which corresponds to -implication operator, is called also Nilpotent Minimumtnorm [6]. If we fix a t-norm above we then fix a propositional calculus (whose set of truth values is): is taken for the truth function of the strong conjunction &, the residuum of becomes the truth function of the implication operator and R(.,0) is the truth function of the negation. In more details, we have the following definitions.

Definition 1 [7,8]. The propositional calculus given by a t-norm has the set of propositional variables and connectives. The set of well-formed formulas in is defined inductively as follows: each propositional variable is a formula; if, are formulas, then, and are all formulas.

Definition 2 [8,9,13]. The formal deductive systems of given by corresponding to and, are called Łukasiewicz n-valued logic systems, n-valued logic systems, and the nvalued logic systems, respectively.

Define in the above-mentioned logic systems

(1)

and in the corresponding algebras

(2)

where is the t-norm defined on.

Remark 1. It is easy to verify that the following assertions are true:

(1) in, for every .

(2) in, for every and .

(3) in , for every.

Definition 3 [7,8]. (1) A homomorphism of type from into the valuation lattice, i.e. , is called an R-valuation of. The set of all R-valuations will be denoted by.

(2) A formula is called a tautology w.r.t. if holds.

Remark 2 [8,13]. It is not difficult to verify in the above-mentioned three logic systems that , and for every valuation . Moreover, one can check in and that and are logically equivalent.

Definition 4 [8]. Assume that is a formula generated by propositional variables through connectives. Substitute for in and keep the logic connectives in unchanged but explain them as the corresponding operators defined on the valuation lattice. The we get a function and call the truth degree function of.

Definition 5 [7,8]. (1) A subset of is called a theory.

(2) Let be a theory,. A deduction of from, in symbols, , is a finite sequence of formulas such that for each is an axiom of, or, or there are such that follows from and by MP. Equivalently, we say that is a conclusion of (or -conclusion). The set of all conclusions of is denoted by. By a proof of we shall henceforth mean a deduction of from the empty set. We shall also write in place of and call a theorem.

It is easy for the reader to check the following Proposition 1.

Proposition 1. Let be a theory and If then there exist a finite subset of say, such that.

Theorem 1 (Generalized deduction theorems) [7, 8,12]. Suppose that is a theory, , then

(1) in ,

iff s. t..

(2) in,

iff.

(3) in,

iff.

Definition 6 [8,13]. Suppose that is a formula of containing m atomic formulas, and be the truth degree function of. Then

is called the truth degree of, where is the cardinal of set.

Theorem 2. Suppose that and, then in and

iff is a tautology i.e.,.

Proof. Assume that. Since

then. By definite, , thus i.e., , , then is a tautology. Conversely, assume that A is a tautology i.e.,

, then , so

. This completes the proof.

Theorem 3 [8]. Suppose that, then in, iff is a tautology, i.e.,.

Theorem 4. Suppose that. If for every, then.

Proof. Suppose that and are all a formulas of containing atomic formulas, it follows from that

and

hence

.

It is easy to verify that

then.

3. Properties of the Roots of Theories

Definition 7 [3]. Suppose that is a theory,. If for every we have, then is called the root of.

Theorem 5. Suppose that is a finite theory, say, then

(1) in

is a root of;

(2) in,

is a root of;

(3) in, is a root of.

Proof. (1) It following form references [4] that, for every, there exist such that

by Theorem 1. It is easy to check that by Remark 1, it following from that where , thus by Hypothetical, this shows that is a root of.

(2) It following form references [4] that , for every, it following from Theorem 1 that , since and are provably equivalent, and so is. This shows is a root of.

(3) It following from references [4] that for every, we get by Theorem 1, it is easy to verify that and are provably equivalent, hence and are provably equivalent, and so is. This shows that is a root of.

4. Membership Degree of Formulas A Is Γ-Conclusion

In following, let us first take an analysis on the conditions of formulas A is a -conclusion in. Suppose that is a theory and A is a -conclusion , it follows from Proposition 1 and Theorem 1 that there exit a finite string of formulas and such that holds, i.e., the formula is a theorem of, let, hence is a tautology, it follows from Theorem 2 that. Conversely, if there exist a -conclusion such that, then following from Theorem 2 that is a tautology, thus is a theorem of, i.e., holds and, we have that by MP, i.e., is a -conclusion. Moreover, the larger the membership degree of such formulas are, the more closer A is to be -conclusion. Hence it is natural and reasonable for us using the supremum of truth degree of all formulas with the form where to measure A is a -conclusion.

Definition 8. Suppose that is a theory,. Define

then is called the membership degree of formulas A is a -conclusion.

It is easy to verify that and following Proposition 2 by Definition 8.

Proposition 2. In, andIf A is a -conclusion, then.

Theorem 6. In, and, if is a finite theory, say, then A is a -conclusion iff.

Proof. The necessity part by proposition 2, it is only necessary to prove the sufficiency. Let. For every number there exist a formulas such that by Definition 8.

(1) In, it follows from Theorem 5 that is a root of and hold. Hence for every we have it follows from properties of implication operators that, since is arbitrary, we have, thus is a tautology, and is a theorem , together with the result, then by MP, i.e.,

(2) In, notice that is a root of by Theorem 5, hence the proof of (2) is similar to that the proof of (1) and so is omitted.

(3) In, notice that is a root of by Theorem 5, hence the proof of (2) is similar to that the proof of (1). In fact is a theorem by Definition 7, hence we have and , thus , holds, then is a theorem , together with the result we have by MP. The proof is completed.

Theorem 7. Suppose that, then

(1) in,

;

(2) in,

;

(3) in,

.

Proof. (1) Since is a root of by Theorem 5, hence for every we have. Thus for every, , and holds, then by Theorem 4. It follows from that i.e.,.

(2) Notice that in, is a root of by Theorem 5, the proof of (2) is similar to that the proof of (1) and so is omitted.

(3) Notice that in, is a root of by Theorem 5, the proof of (2) is similar to that the proof of (1) and so is omitted.

Theorem 8. Suppose that is a infinite theory. Then

(1) in,

;

(2) ,

;

(3) in,

.

Proof. (1) For every, it following from Proposition 1 that there exist a finite string of formulas such that It follows from Theorem 1 that is a theorem, hence is a tautology by completeness theorem, and for every, , we have

by Theorem 4.

It following form references [14] that , then

.

(2) Notice that in, by Remark 1, the Proof of (2) is similar to that the Proof of (1) and so is omitted.

(3) Notice that in, and is Provably equivalent, the Proof of (3) is similar to that the Proof of (1) and so is omitted.

Theorem 9. Suppose that is a theory, and, then

Proof. (1) If we get, then

(2) If we get and, for any given positive number such that and there exists formulas such that and . It follows from properties of Regular implication operators that and It is easy to verify that and are provably equivalent (i.e., logically equivalent), hence . It follows from the theory of truth degrees of formulas and that .

Bucas and are provably equivalent (i.e., logically equivalent), hence , it is easy to verify that then by the definition of the membership degree of formulas.

Example 1. Suppose that In, and, compute respectively.

Solution. (1) In, assume that Since and

thus

and

We have andhence

then.

(2) In, assume that Since , and

thus

then

then.

(3) In, assume that Since , and

thus

then

Example 2. Suppose that , in, compute

Solution. (1) Assume that Since

, and

thus, then p₂

is a -conclusion.

REFERENCES

NOTES

^*The work was supported by the Science and Technology Item of the Education Department of Fujian Province of China (No. 2010JA10235).