Hierarchical structure and applications of fuzzy logical systems

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Hierarchical structure and applications of fuzzy logical systems ✩ Daowu Pei a,b,∗ , Rui Yang a,b a

Department of Mathematics, Zhejiang Sci-Tech University, Hangzhou 310018, China Key Laboratory of Advanced Textile Materials and Manufacturing Technology, Zhejiang Sci-Tech University, Ministry of Education, Hangzhou, 310018, China b

a r t i c l e

i n f o

Article history: Received 27 January 2012 Received in revised form 21 April 2013 Accepted 7 May 2013 Available online xxxx Keywords: Fuzzy logic Generalized tautology Tautological degree T-norm based fuzzy logic Fuzzy reasoning

a b s t r a c t This paper focuses on hierarchical structures of formulas in fuzzy logical systems. Basic concepts and hierarchical structures of generalized tautologies based on a class of fuzzy logical systems are discussed. The class of fuzzy logical systems contains the monoidal t-norm based system and its several important schematic extensions: the Łukasiewicz logical system, the Gödel logical system, the product logical system and the nilpotent minimum logical system. Furthermore, hierarchical structures of generalized tautologies are applied to discuss the transformation situation of tautological degrees during the procedure of fuzzy reasoning. © 2013 Elsevier B.V. All rights reserved.

1. Introduction In the classical logic if a proposition is not true then it must be false, and formal formulas are divided into three families: tautologies, contradictions and satisfiable formulas, which are not tautologies according to their values under all valuations. In the third family, there are not suitable methods to distinguish these formulas in the classical logic. For example, the formulas A ∨ B and A → B are obviously more true than formula A ∧ B, but all of them are satisfiable. Furthermore, in the classical logic one only considers inference issues based on tautologies, i.e., true conclusions are derived from true premises. In multi-valued logics and fuzzy logics, however, the situations are completely different. One expects to infer from some partly true premises to obtain other partly true conclusions. Thus various generalized tautologies are intensively studied in the literature. In this direction, we must mention the excellent work completed by Pavelka [12] (also see Novák et al. [11], Pei and Wang [20]). Based on the Tarski’s idea and Łukasiewicz logic, Pavelka [12] proposed syntactic and semantic conclusion operators, and then he graded formulas based on the conclusion operators. Recently Carotenuto and Gerla [2] propose a framework for the deduction apparatus of multi-valued logics based on the Pavelka’s theory. In the past several years, some authors considered generalized tautologies in several important fuzzy logical systems. Some new concepts and methods are proposed, and structures of generalized tautologies in these fuzzy logical systems are investigated. In 1998, Wang [22] proposed a framework of generalized tautology theory for the so-called revised Kleene logical system (Pei [14] proved that the formal system L∗ of this logic is equivalent with the formal system NM of the nilpotent minimum logic proposed by Esteva and Godo [3]). According to Wang’s results, in the revised Kleene logical system, there are only three classes of generalized tautologies: 12 -tautologies, ( 12 )+ -tautologies and tautologies. Wu [34] investigated ✩

*

This work is supported by National Science Foundation of China (Grant Nos. 10871229 and 11171308). Corresponding author at: School of Sciences, Zhejiang Sci-Tech University, Hangzhou 310018, China. E-mail address: [email protected] (D. Pei).

0888-613X/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.ijar.2013.05.003

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generalized tautologies of the parameterized Kleene systems, and gave similar conclusions. And following Wang’s work, Yang and Zhang [35] in 1998, Wu and Wang [33] in 2000 discussed the generalized tautology theory of the Łukasiewicz logical system, proved that in this system all classes of accessible tautologies are not empty. This conclusion shows that the Łukasiewicz logical system possesses rich hierarchical structures, and it is suitable to deal with fuzzy reasoning in some sense. Wu [31] discussed the generalized tautology theory of the Gödel logical system in 2000. Pei and Li [18] discussed the generalized tautology theory of the product logical system in 2002. These results show that both of the Gödel logical system and product logical system have poor hierarchical structures. There are still some open problems with respect to hierarchical structures of generalized tautologies in these fuzzy logical systems. Up to now, there is not a unified framework for the generalized tautology theory. Also, the possible applications of generalized tautologies in fuzzy reasoning have not been discussed yet. In fact, we should compare different methods of fuzzy reasoning by analyzing the transformation situation of tautological degrees from premises to conclusions in the procedure of fuzzy reasoning. Therefore, systematic discussions to these problems are both necessary and interesting. In this paper, we have two main goals: to build a unified framework for the generalized tautology theory of general t-norm based fuzzy logical systems, and to consider fuzzy reasoning based on generalized tautologies. The remainder parts of the paper are organized as follows. Section 2 introduces some basic concepts and conclusions of t-norm based fuzzy logical systems and generalized tautologies. Section 3 gives systematic discussions to generalized tautology theories of four important t-norm based fuzzy logical systems. Section 4 considers fuzzy reasoning based on generalized tautologies. Finally, Section 5 concludes the paper. 2. Basic framework of hierarchical structure of fuzzy logical systems In the theory of fuzzy logic, for a given t-norm based fuzzy logic, its propositional calculus contains a countable set of propositional variables S = { p 1 , p 2 , . . .} and four main connectives: strong conjunction &, implication →, conjunction ∧ and disjunction ∨. Then the formula set F ( S ) can be freely generated by S and the above mentioned four connectives by using the usual manner (see Hájek [6], or Gottwald [5]). In a t-norm based fuzzy logic, one introduces negation connective ¬ and equivalence connective ↔ as derived symbols as follows:

¬ A = A → 0,

A ↔ B = ( A → B ) ∧ (B → A)

where 0 is a propositional constant interpreted as 0. In fuzzy logics, one takes some subset E containing 0 and 1 of the real unit interval [0, 1] as the truth value set. Under a given valuation, four connectives ∧, ∨, & and → are interpreted by the minimum operator, maximum operator, a left continuous t-norm T (see Klement et al. [7], or Wang et al. [30], Ying [36]) and the residuated implication → T respectively, where → T is induced by T according to the following manner:







x → T y = sup z ∈ [0, 1]  T (x, z)  y ,

x, y ∈ [0, 1].

It is interesting to notice that the concept of residuated implication given here is related the corresponding concept of T -residuation given by Blyth [1]. Suppose that E is a subset of [0, 1] which contains 0 and 1, and is closed under operations T and → T . Usually, one calls a propositional calculus with the algebraic structure of truth values

FL E ( T ) = ( E , T , → T , min, max) the t-norm based fuzzy logic [5]. In this paper, we call FL E ( T ) the t-norm based fuzzy algebraic system (shortly, t-norm algebra) on the set E determined by the left continuous t-norm T . If E = [0, 1], we write FL( T ) instead of FL[0,1] ( T ). Naturally, FL E ( T ) is a (2, 2, 2, 2)-type subalgebra of FL( T ). Four important t-norm algebras are defined as follows: (i) Łukasiewicz t-norm algebra:



L = [0, 1], T L , → L , min, max



where

T L (x, y ) = max(0, x + y − 1),

x → L y = min(1, 1 − x + y ).

(ii) Gödel t-norm algebra:



G = [0, 1], T G , →G , min, max



where

 T G (x, y ) = min(x, y ),

x →G y =

1, y,

if x  y , otherwise.

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(iii) Product t-norm algebra:



P = [0, 1], T P , → P , min, max where



 T P (x, y ) = xy ,

x →P y =

if x  y , otherwise.

1,

y , x

(iv) Revised Kleene t-norm algebra, or nilpotent minimum t-norm algebra:



W = [0, 1], T 0 , →0 , min, max where

 T 0 (x, y ) =



if x + y  1, otherwise.

0, min(x, y ),

 x →0 y =

1, max(1 − x, y ),

if x  y , otherwise.

We observe that in the systems L and W, the negation connective ¬ are interpreted as the standard negation operator  with

x = x → L 0 = 1 − x,

x ∈ [0, 1],

and in the systems G and P, ¬ are interpreted as the Gödel negation operator N G with



N G (x) = x →G 0 =

1, if x = 0, 0, otherwise,

x ∈ [0, 1].

Based on the above definitions, the formula set F ( S ) and a t-norm algebra FL E ( T ) are the same (2, 2, 2, 2)-type algebras. The corresponding four binary operations of F ( S ) are &, →, ∧ and ∨. An algebraic homomorphism v : F ( S ) → FL E ( T ) is called an E-valuation of F ( S ). We use Ω(FL E ( T )), or just, Ω E ( T ) to denote the set of all E-valuations of F ( S ). Now we can introduce the basic concepts of various generalized tautologies based on a t-norm algebra. For convenience to compare, we first give the following concepts of tautology, contradiction and satisfiable formula in a t-norm algebra. Definition 2.1. (See [6] or [5].) Let FL( T ) be a t-norm algebra and A ∈ F ( S ), E be a subalgebra of FL( T ). (i) If for all v ∈ Ω E ( T ) we have v ( A ) = 1 then we call A a tautology on FL E ( T ). (ii) If for all v ∈ Ω E ( T ) we have v ( A ) = 0 then we call A a contradiction on FL E ( T ). (iii) If there is v ∈ Ω E ( T ) such that v ( A ) = 1 then we call A a satisfiable formula on FL E ( T ). We use





Tau FL E ( T )

or

Tau E ( T )

to denote the set of all tautologies on FL E ( T ),





Contr FL E ( T )

or

Contr E ( T )

to denote the set of all contradictions on FL E ( T ). According to Definition 2.1, all formulas are divided into three disjoint classes: First class consists of all tautologies:

Tau E ( T ); Second class consists of all contradictions:







Contr E ( T ) = A ∈ F ( S )  ¬ A ∈ Tau E ( T ) ; Third class consists of all satisfiable formulas which are not tautologies:

Sat0E ( T ) = F ( S ) − Tau E ( T ) − Contr E ( T ). However, the discrepancy between different satisfiable formulas of the third class are possibly very large. For example, in the Łukasiewicz t-norm algebra L, all three formulas

p,

p ∨ ¬ p,

¬( p ∨ ¬ p )

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are satisfiable where p is a proposition variable, but they are completely different in nature. Hence it is necessary to introduce some new standards to distinguish them. The following concepts are important for discussion of hierarchical structure of fuzzy logic systems. Definition 2.2. Let FL E ( T ) be a t-norm algebra,

α ∈ [0, 1] and A ∈ F ( S ).

(i) If v ( A )  α for all v ∈ Ω E ( T ) then we call A an α -tautology on FL E ( T ). We use Tauα (FL E ( T )) to denote the set of all α -tautologies on FL E ( T ). (ii) If v ( A ) > α for all v ∈ Ω( T ) then we call A an α + -tautology on FL E ( T ). In particular, a 0+ -tautology is also called a quasi-tautology. + We use Tau+ α (FL E ( T )), QL(FL E ( T )) to denote the sets of all α -tautologies and all quasi-tautologies on FL E ( T ), respectively. (iii) If A ∈ Tauα (FL E ( T )), and there is v ∈ Ω E ( T ) such that v ( A ) = α then we call A an accessible α -tautology on FL E ( T ). We use Tauα (FL E ( T )) to denote the set of all accessible α -tautologies on FL( T ). (iv) If A ∈ Tau+ α (FL E ( T )), and for any small positive real number  there is v ∈ Ω E ( T ) such that v ( A )  α +  then we call A an asymptotic α -tautology on FL E ( T ). ↓

We use Tauα (FL E ( T )) to denote the set of all asymptotic α -tautologies on FL E ( T ). For any α ∈ (0, 1], we use the unified name generalized tautologies to call the above defined four classes of formulas, and the name tautological degree for the real number α . ↓ For our convenience, we also use E ( T ) to replace the notation (FL E ( T )) in notations such as Tauα (FL E ( T )) in Defini↓ ↓ tion 2.2. Thus, Tauα (FL E ( T )) and Tauα , E ( T ) are the same formula set. If E = [0, 1], then we can omit the subscript “E”. Obviously, for several classes of generalized tautologies, we have the following relations: (i) Tau E ( T ) ⊆ Tau+ α , E ( T ) ⊆ Tauα , E ( T ),

(ii) Tauα , E ( T ) ⊆ Tauα , E ( T ), ↓

α ∈ [0, 1);

α ∈ [0, 1);

(iii) Tauα , E ( T ) ⊆ Tau+ α , E ( T ) ⊆ Tauα , E ( T ),

α ∈ [0, 1).

Here for the sake of simplicity, we do not use the same nomenclature as in Refs. [22,33]. Remarks. (i) The concept of α -tautology has already been introduced naturally by Pavelka [12] and Novák [11] in the fuzzy logic with evaluated syntax. The concept of the α -tautology stated in Definition 2.2 is similar to the corresponding concept of the α -conclusion of ∅ under the syntactic conclusion operator. However, both logical frameworks are not the same (also see [2]). (ii) In order to state the idea of “grading formulas”, Wang [22] proposed concepts of α -tautology and α + -tautology for the revised Kleene t-norm algebra W in 1998. However, there is a new problem: if α < β then β -tautologies and β + -tautologies must be α -tautologies and α + -tautologies, respectively. In order to solve this problem, Wang [22] further proposed the concept of accessible α -tautology, and Wu and Wang [33] proposed the concept of accessible α + -tautology (here we call the later asymptotic α -tautology). C Hájek [6, p. 154, Definition 6.2.1] defined the notations TAUT pos and TAUT 1C as follows:

 



TAUT 1C = A  for each [0, 1]-evaluation v , v ( A ) = 1 ,

 



C TAUT pos = A  for each [0, 1]-evaluation v , v ( A ) > 0 ,

where C stands for anyone of {L, G, P}. Thus we have C QT (C ) = TAUT pos ,

Tau(C ) = TAUT 1C .

The following proposition and corollary are obvious. Proposition 2.1. Let ( E , T , → T , min, max) be a t-norm algebra of truth values, and E 1 be a subalgebra of E. Then (i) v | E 1 ∈ Ω E 1 ( T ) whenever v ∈ Ω E ( T ); (ii) Tauα , E ( T ) ⊆ Tauα , E 1 ( T ) for any α ∈ [0, 1]. In particular, Tau E ( T ) ⊆ Tau E 1 ( T ). Corollary 2.1. If E and E  are isomorphic subalgebras of FL( T ), then Tauα , E ( T ) = Tauα , E  ( T ) for any α ∈ [0, 1].

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A natural problem arises: for a given t-norm algebra FL( T ) of truth values and a parameter α ∈ [0, 1], or α ∈ [0, 1] ∩ Q, are all classes Tauα ( T ) of accessible α -tautologies nonempty? The coming section will give answers for several important t-norm algebras. 3. Structures of generalized tautologies in several t-norm algebras Up to now, about structures of generalized tautologies in a general t-norm algebra has not investigated in depth in the literature. But we do clearly know structures of generalized tautologies on four important t-norm algebras L, G, P and W introduced in the previous section. These results will be reviewed and unified in the following four subsections. 3.1. Nilpotent minimum t-norm algebra The main results of this subsection come from Wang [22] and Wu [32]. In order to discuss structure of generalized tautologies on the algebra W, Wang [22] constructed the following algebraic homomorphism and isomorphism corresponding to different values of tautological degree α in 1998. For α ∈ (0, 12 ], define ϕ1 : [0, 1] → {0, 12 , 1} as follows:

⎧ ⎪ 0, if x < 12 , ⎪ ⎨ ϕ1 (x) = 12 , if x = 12 , ⎪ ⎪ ⎩ 1, if x > 1 . 2

For

α ∈ ( 12 , 1], define ϕ2 : [0, 1] → {0, 1} ∪ (1 − α , α ) as follows: 0, if x = 0, ϕ2 (x) = (2α − 1)x + (1 − α ), if x ∈ (0, 1), 1, if x = 1.

Based on the above defined two mappings, the following main results of the system W are reported. Proposition 3.1. (See [22].) For the system W, (i) if α ∈ (0, 12 ), then Tauα (W) = Tau+ α (W) = Tau1/2 (W);

(ii) if α ∈ ( 12 , 1), then Tauα (W) = Tau+ α (W) = Tau(W).

In addition, the system W has the following property (see [30]): (iii) Tau+ 0 (W) = Tau1/2 (W); (iv) For α ∈ (0, 12 ) ∪ ( 12 , 1), ↓

Tauα (W) = ∅. Thus in the revised Kleene t-norm algebra W, all possible disjoint classes of generalized tautologies are:





↓

Tau1/2 (W), Tau1/2 (W), Tau(W) .

From the following facts given by Wang [22] we see that these classes are not empty:

A = p 1 → p 1 ∈ Tau(W), B = p 1 ∨ ¬ p 1 ∈ Tau1/2 (W), ↓

C = ( p 2 → B ) ∨ p 2 ∈ Tau1/2 (W), where p 1 and p 2 are propositional variables or atomic formulas. 3.2. Łukasiewicz t-norm algebra The main results of this subsection come from Yang and Zhang [35], Wu and Wang [33]. For a formula A, Yang and Zhang [35] construct a sequence of formulas as follows:

A1 = A,

A k +1 = ¬ A k → A ,

k = 1, 2, . . . .

For v ∈ Ω( T L ), the sequence has two interesting properties:

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(i) v ( A k ) = min(1, kv ( A )) for any positive integer k; m (ii) v (¬ A n ∨ A m )  n+ , and the equality holds if and only if v ( A ) = n+1m . m Based on the above properties of the sequence { A n }, Yang and Zhang [35] reported the following results. Proposition 3.2. (See [35].) In the Łukasiewicz t-norm algebra L, the following inequalities hold for any rational number r ∈ [0, 1] and

α , β ∈ [0, 1]:

(i) Taur (L) = ∅; (ii) Tauα (L) = ∅; (iii) Tauβ (L) = Tauβ (L) whenever α = β . According to this proposition, in the Łukasiewicz t-norm algebra, the structure of generalized tautologies is rich and suitable for fuzzy reasoning. Following Wang [22], the property (ii) of Proposition 3.2 is called every class is nonempty, and the property (iii) is called every pair of classes are not equal. We point out an obvious fact: there must be a real r ∈ [0, 1] such that (i) of the above proposition does not hold since the formula set F ( S ) is a countable set. However, a natural problem arises from the property (i) of the above proposition: is there an irrational number r in [0, 1] such that the condition (i) holds? The answer given by Wu and Wang [33] is negative. Proposition 3.3. (See [33].) In the Łukasiewicz t-norm algebra L, Taur (L) = ∅ for any irrational number r ∈ [0, 1]. Thus we obtain all disjoint classes of generalized tautologies of Łukasiewicz t-norm algebra L as follows:





Tauα (L)  α ∈ (0, 1] ∩ Q



where Q is the set of all rational numbers. Remark. The conclusion given by Proposition 3.3 is trivial from the point of view of the known MacNaughton theorem for Łukasiewicz logic since α -tautologies are just minimal points in the piecewise linear representation of formulas (see [11]). 3.3. Gödel t-norm algebra From the viewpoint of algebra, we can easily see that as a (min, →, max)-type algebra, a subset E of the Gödel t-norm algebra [0, 1] is a subalgebra of [0, 1] if and only if {0, 1} ⊆ E. In particular, [0, α ) ∪ {1} is a subalgebra of [0, 1] for any α ∈ (0, 1]. In order to discuss the structure of the Gödel t-norm algebra G, we introduce the following isomorphism ϕ from [0, 1] onto its subalgebra [0, α ) ∪ {1}.



ϕ (x) =

α x, if x ∈ [0, α ), 1,

(1)

if x = 1.

About the structure of generalized tautologies of G, we have the following conclusion. Proposition 3.4. In the Gödel t-norm algebra G, for any α ∈ (0, 1), we have

Tauα (G) = Tau+ α (G) = Tau(G). Proof. Suppose that

α ∈ (0, 1), then obviously we have

Tau(G) ⊆ Tau+ α (G) ⊆ Tauα (G). Now we suppose that A ∈ Tauα (G), then for any valuation v ∈ Ω(G), we have v ( A )  α . In particular, for the valuation

ϕ ◦ v, we have   ϕ v ( A) = ϕ ◦ v ( A)  α.

Furthermore, we have ϕ ( v ( A )) = 1 by the definition of ϕ . Thus v ( A ) = 1 since This proves that A ∈ Tau(G), and thus we complete the proof. 2

ϕ is an isomorphism.

From the result we see that comparing with the Łukasiewicz t-norm algebra L the structure of generalized tautologies of the Gödel t-norm algebra G is not too rich for fuzzy reasoning. Thus in the Gödel t-norm algebra G, all possible disjoint classes of generalized tautologies are:

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↓

7



Tau0 (G), Tau(G) .

From the following facts we see that these classes are not empty:

A = p 1 → p 1 ∈ Tau(G); ↓

B = p 1 ∨ ¬ p 1 ∈ Tau0 (G); ↓

C = ( p 2 → ¬ A ) ∨ p 2 ∈ Tau0 (G); where p 1 and p 2 are propositional variables. Wu [33] considered a fuzzy algebra G which is similar to G. In the system G, there are four operations T G , →G , max and the standard negation  . In order to discuss structure of generalized tautologies of the algebra G, Wu [31] constructed the following two algebraic isomorphisms corresponding to different values of tautological degree α . For α ∈ (0, 12 ], define ϕ1 : [0, 1] → [0, α ) ∪ { 12 } ∪ (1 − α , 1] as follows:

⎧ ⎪ 2α x, if x ∈ [0, 12 ), ⎪ ⎨ ϕ1 (x) = 12 , if x = 12 , ⎪ ⎪ ⎩ 2α (x − 1 ) + (1 − α ), if x ∈ ( 1 , 1]. 2 2

For

α ∈ ( 12 , 1], define ϕ2 : [0, 1] → {0, 1} ∪ (1 − α , α ) as follows: ⎧ if x = 0, ⎨ 0, ϕ2 (x) = (2α − 1)x + (1 − α ), if x ∈ (0, 1), ⎩ 1, if x = 1.

Based on the above defined two isomorphisms, the following main results of the system G are proved by [31]: For the system G,

α ∈ (0, 12 ), then Tauα (G) = Tau+ α (G) = Tau1/2 (G); (ii) if α ∈ ( 12 , 1), then Tauα (G) = Tau+ α (G) = Tau(G); ↓ (iii) for α ∈ (0, 12 ) ∪ ( 12 , 1), Tauα (G) = ∅. (i) if

We point out that the t-norm algebra G is different from the algebra G. The unique difference is that in [31] the negation operator is the standard negation  and here we use the Gödel negation N G as negation operator. We have showed that in the system G, the structure of generalized tautologies is different from that of the system G. 3.4. Product t-norm algebra The structure of the product t-norm algebra P is the same as the Gödel t-norm algebra G showed in the above subsection. These results partly come from Pei and Li [18]. We know that in the product t-norm algebra P, the negation operator induced by the product t-norm is also the Gödel negation. The following results have been reported by Pei and Li [18]. Proposition 3.5. (See [18].) (i) Let a ∈ [0, 1]. Then P a = {an | n = 1, 2, . . .} ∪ {0, 1} is a subalgebra of P. (ii) If L is a subalgebra of P and L contains a subinterval [a, b] ⊆ [0, 1] with a < b, then L = P. According to the above properties of P, for α ∈ (0, 1), [0, 1]α = [0, α ) ∪ {1} is not a subalgebra of P. Therefore, the mapping ϕ defined by (1) is not an isomorphism. In order to solve this problem, we recall the concept of product t-norm algebras proposed by Hájek [6]. Definition 3.1. (See [6].) The algebraic system ( L , ⊗, →, ∧, ∨, ¬, 0, 1) is called a product algebra, if the following conditions hold for any x, y , z ∈ L: (i) ( L , ⊗, →, 0, 1) is a residuated lattice; (ii) x ∧ y = x ⊗ (x → y );

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(iii) (iv) (v) (vi)

(x → y ) ∨ ( y → x) = 1; ¬x = x → 0; ¬¬ z  (x ⊗ z → y ⊗ z) → (x → y ); x ∧ ¬x = 0.

Now we construct a product t-norm algebraic structure on [0, 1]α as follows:

 x ⊗ y = xy ,

x →α y =

1,

if x  y ,

yα , x

if x > y .

(2)

Furthermore, we can define an isomorphism ϕ as (1) from [0, 1] to [0, 1]α . Now we can prove the similar conclusion for product t-norm algebra P. Proposition 3.6. In the product t-norm algebra P , the following equalities hold for any α ∈ (0, 1):

Tauα (P) = Tau+ α (P) = Tau(P). Proof. Suppose that

α ∈ (0, 1), then obviously we have

Tau(P) ⊆ Tau+ α (P) ⊆ Tauα (P). Now we suppose that A ∈ Tauα (P), then for any valuation v ∈ Ω(P), we have v ( A )  α . In particular, for the valuation

ϕ ◦ v, we have   ϕ v ( A) = ϕ ◦ v ( A)  α.

Furthermore, we have ϕ ( v ( A )) = 1 by the definition of ϕ . Thus v ( A ) = 1 since This proves that A ∈ Tau(P), and thus we complete the proof. 2

ϕ is an isomorphism.

From the result we see that comparing with Łukasiewicz t-norm algebra L the structure of generalized tautologies of the product t-norm algebra P is also not too rich for fuzzy reasoning. Thus in the product t-norm algebra P, all possible disjoint classes of generalized tautologies are:





↓

Tau0 (P), Tau(P) .

Similarly, from the following facts we see that these classes are not empty:

A = p → p ∈ Tau(P),

↓

B = p ∨ ¬ p ∈ Tau0 (P)

where p is a propositional variable. For the sake of convenience for applications, we list all generalized tautologies of four important t-norm algebras by the following Table 1 where α ∈ (0, 1) ∩ Q and α = 12 . Table 1 Generalized tautologies of fuzzy logics. ↓

FL

Tau0 (FL)

L G P W

∅ √ √ ∅

Tau1/2 (FL)

Tauα (FL)

√

√

∅ ∅ √

∅ ∅ ∅

↓

Tau1/2 (FL)

∅ ∅ ∅ √

Tau(FL)

√ √ √ √

According to the above obtained results, we can say that about the structures of generalized tautologies, the Łukasiewicz t-norm algebra L is the richest, and the Gödel t-norm algebra G and the product t-norm algebra is the simplest in four important t-norm algebras. So we should notice that none of the algebras G and P is suitable to cope with fuzzy reasoning. 4. Applications of generalized tautologies in fuzzy reasoning In fuzzy reasoning, two basic inference schemas are generalized modus ponens and generalized modus tollens, shortly, GMP and GMT, respectively. In this section, we discuss possible applications of generalized tautologies in GMP and GMT, respectively in two separate subsections.

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4.1. GMP based on generalized tautologies In this subsection, we try to formally solve GMP-type problems of fuzzy reasoning based on the formal system MTL (monoidal t-norm based logic) proposed by Esteva and Godo [3] and the fully implicative method proposed by Wang [23–25]. The generalized modus ponens is of the following form: GMP

From A → B and A ∗ infer B ∗ ,

where A, B, A ∗ and B ∗ are formulas of the system MTL. Naturally, in the rule GMP, the conclusion B ∗ should be determined by both formulas A → B and A ∗ by using a suitable method. More general GMP is of the following form: GGMP

From A i1 ∧ A i2 ∧ · · · ∧ A in → B i for i = 1, 2, . . . , m and A ∗1 ∧ A ∗2 ∧ · · · ∧ A n∗ infer B ∗ .

According to the full implication inference approach (see [23], or [8,15–17,26]) of fuzzy reasoning and the concept of root of a formula set (see [28]), we propose the concept of solutions of the problem GMP. Definition 4.1. Based on the system MTL, a formula B ∗ is called a full implication inference solution of the problem GMP if the following conditions hold: (i) (( A → B ) → ( A ∗ → B ∗ )); (ii) if a formula C satisfies

    ( A → B ) → A∗ → C then we have  ( B ∗ → C ). In the above definition, the condition (i) guarantees that the obtained solutions are ideal in some sense, or that the rule A → B sustains A ∗ → B ∗ as large as possible; the condition (ii) guarantees that the obtained solutions are optimal in some sense, or that the solution is minimal with respect to the following partial ordering ≺ of the formula set F ( S ) defined as follows:

A≺B

⇐⇒

( A → B ).

(3)

We say that two formulas A and B are provably equivalent, denoted A ∼ B, if A ≺ B and B ≺ A. Definition 4.2. An algorithm to solve GMP is said to be consistent if B ∗ and B are provably equivalent whenever A ∗ = A. The consistency of an algorithm to solve GGMP can be defined similarly. Based on the known conclusions of the system MTL (see [3,5]), we give the following results and omit their proofs. Proposition 4.1. A full implication inference solution of GMP in MTL is of the following form for the given formulas A → B and A ∗ :

B ∗ = ( A → B )& A ∗ , and all full implication inference solutions are provably equivalent. Moreover, the full implication inference algorithm to solve GMP is consistent. For the model GGMP, we can similarly obtain the following conclusion: Proposition 4.2. A full implication inference solution of GGMP in MTL is of the following form for the given formulas A i j , B i , A ∗j , i = 1, . . . , m, j = 1, . . . , n:

B∗ =

m

  ( A i1 ∧ · · · ∧ A in → B i )& A ∗1 ∧ · · · ∧ An∗ ,

i =1

and all full implication inference solutions are provably equivalent. Moreover, the full implication inference algorithm to solve GGMP is consistent. The following propositions show transformation situation of tautological degrees during the procedure of fuzzy reasoning.

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Proposition 4.3. Let FL be a t-norm algebra determined by a left continuous t-norm ⊗. If

A → B ∈ Tauα (FL),

A ∗ ∈ Tauβ (FL),

then the conclusion B ∗ given by Proposition 4.1 is an α ⊗ β -tautology of FL, i.e.,

B ∗ ∈ Tauα ⊗β (FL). For four important t-norm algebras L, G, P and W, we have more concrete results given by Table 2, where B ∗L , B ∗G , B ∗P and B ∗0 are inference conclusions given by Proposition 4.1 in the Łukasiewicz, Gödel, product and revised Kleene t-norm algebras, respectively. Table 2 Tautological degrees of inference conclusions. A→B

α

1

A∗

β

1

↓

Tau0 (FL) ↓

Tau0 (FL)

B ∗L

min(1, 1 − α + β)

1

B ∗G

min(α , β)

1

Tau0 (G ) ∪ Tau(G )

B ∗P

αβ

1

Tau0 ( P ) ∪ Tau( P )

B∗

min(α , β) if

0

α+β >

1 2

↓

Tau1/2 (FL) ↓

Tau1/2 (FL)

↓ ↓

1

↓

Tau1/2 ( W ) ∪ Tau( W )

We know that as a continuous t-norm, T L is nilpotent, i.e., for any a ∈ [0, 1) there is a positive integer n such that an = 0. Therefore, generally speaking, we have min(1, 1 − α + β) < min(α , β). The conclusion B ∗ of inference possibly has smaller tautological degree even if the both premisses A → B and A ∗ have bigger tautological degrees. For example, in the system L, we take

A = A∗ = p → p,

B = ¬ p ∨ p,

then we can obtain

B ∗ = B ∈ Tau1/2 (L). In this example, α = 1/2, β = 1. Furthermore, the final conclusion obtained from a long inference chain possibly has very small tautological degree. This fact shows that the transitive property of the system L is not very satisfactory. Based on this observation, one should control the tautological degrees of conclusions in long inference chains provided the system L is used. In addition, for the system W, based on the properties of nilpotent minimum t-norm, it is possible that the tautological degree of the conclusion B ∗ less than 1/2 even if both A → B and A ∗ are accessible 1/2-tautologies. At the end of this subsection, we propose the following inference model of the Generalized Hypothetical Syllogism (GHS for short). GHS

If A → B, B → C and A ∗ then C ∗ ,

where A, B, C , A ∗ and C ∗ are formulas of the system MTL. In order to solve this model, we can divide the goal into two steps: First step. To solve the following model by the full implication inference method: GMP1

If A → B and A ∗ then B ∗ ;

Second step. To solve the following model by the full implication inference method again: GMP2

If B → C and B ∗ then C ∗ .

Similarly, we can consider the tautological degree of C ∗ based on tautological degrees of A → B, B → C and A ∗ . 4.2. GMT based on generalized tautologies In this subsection, we try to formally solve the inference problem of the generalized modus tonens based on the formal system IMTL (a schematic extension of MTL by double negation law) proposed by Esteva and Godo [3]. The generalized modus tonens problem is of the following form:

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If A → B and B ∗ then A ∗ ,

GMT

where A, B, B ∗ and A ∗ are formulas of the system IMTL. In the system IMTL, we transform the inference model GMT into the following model: If ¬ B → ¬ A and B ∗ then A ∗ ,

GMT’

where A, B, B ∗ and A ∗ are formulas of the system IMTL. According to the full implication inference approach of fuzzy reasoning and the concept of root of a formula set again, we propose the concept of solutions of the generalized modus tonens problem. Definition 4.3. Based on the system IMTL, a formula A ∗ is called a full implication inference solution of the problem GMT’ if the following conditions hold: (i) ((¬ B → ¬ A ) → ( B ∗ → A ∗ )); (ii) if a formula D satisfies

    (¬ B → ¬ A ) → B ∗ → D then we have ( A ∗ → D ). The meaning of conditions (i) and (ii) of Definition 4.3 is similar to that of the corresponding conditions of Definition 4.1. In particular, any full implication inference solution of GMP is minimal with respect to the partial ordering ≺ of the formula set F ( S ) defined by (3). Proposition 4.4. A full implication inference solution of GMT’ in IMTL is of the following form:

A ∗ = (¬ B → ¬ A )&B ∗ , and all full implication inference solutions are provably equivalent. For transformation situation of tautological degrees during the procedure of fuzzy reasoning for the model GMT’, we have the following conclusion. Proposition 4.5. Given a t-norm algebra FL determined by a left continuous t-norm ⊗ satisfying the so-called contrapositive symmetry, i.e., for all x, y ∈ [0, 1],

x → y  = y → x. If A →

B ∈ Tauα (FL) and A ∗

(4)

∈ Tauβ (FL), then the inference conclusion

B ∗ ∈ Tauα ⊗β (FL). We know that residuated implications induced by T L and T 0 satisfy the contrapositive symmetry (see [4,13,19]). For two important t-norm algebras L and W, we can obtain more concrete results. (i) In Łukasiewicz t-norm algebra L, if

A → B ∈ Tauα (L),

B ∗ ∈ Tauβ (L),

then A ∗ ∈ Taumin(1,1−α +β) (L). (ii) In the revised Kleene t-norm algebra W, if

A → B ∈ Tauα (W), where

B ∗ ∈ Tauβ (W),

α + β > 12 , then A ∗ ∈ Taumin(α ,β) (W).

5. Conclusions One of main objectives of fuzzy reasoning is to design some algorithms for solving two basic fuzzy inference models, FMP and FMT. In fuzzy reasoning, one needs to produce some partly true conclusion from some partly true premises. Therefore, it is meaningful to discuss fuzzy reasoning problems based on various generalized tautologies.

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In this paper, we reviewed some related concepts and main results of generalized tautologies based on the logic MTL of left continuous t-norms and their residua. Some new results are presented. In particular, we listed and investigated structure of generalized tautologies of four important t-norm algebras including the Łukasiewicz t-norm algebra L, the Gödel t-norm algebra G, the product t-norm algebra P and the revised Kleene t-norm algebra W. In the four algebras, L has the richest hierarchical structure of generalized tautologies: for every rational number r ∈ [0, 1], the class of accessible r-tautologies is not empty; both algebras G and P have poorer structures of generalized tautologies than other t-norm algebras: in the systems G and P, only two classes of generalized tautologies are not empty: tautologies and asymptotic 0-tautologies. Based on the theory of generalized tautologies, we investigated the transformation situation of tautological degrees in fuzzy reasoning. In the framework of the formal system MTL, an algorithm for solving the inference model GMP, i.e., generalized modus ponens, is proposed. And similarly, in the framework of the formal system IMTL, an algorithm for solving the inference model GMT, i.e., generalized modus tonens, is designed. Making use of the proposed algorithm, we gave some new results to determine tautological degrees of conclusions from given tautological degrees of premisses. The main results of the present paper can be transformed from fuzzy logics to multiple valued logics. Also, we can consider the tautological degree transformation properties of CRI based fuzzy reasoning (see Zadeh [37,38]) and the similarity based inference method (see Turksen and Zhong [21]). In addition, the relationships between tautological degrees and integral truth degree (see Wang et al. [27,29]), state theory (see Mundici [10], or Liu et al. [9]) are interesting topics for future discussion. 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