Some properties of fuzzy reasoning in propositional fuzzy logic systemsq
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Some properties of fuzzy reasoning in propositional fuzzy logic systems q Jiancheng Zhang *, Xiyang Yang Department of Mathematics, Quanzhou Normal University, Fujian 362000, China
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 8 January 2010 Received in revised form 15 July 2010 Accepted 29 July 2010
In order to analyze the logical foundation of fuzzy reasoning, this paper first introduces the concept of generalized roots of theories in Łukasiewicz propositional fuzzy logic Łuk, Gödel propositional fuzzy logic Göd, Product propositional fuzzy logic P, and nilpotent minimum logic NM (the R0-propositional fuzzy logic L ). Next, it is proved that all consequences of a theory C, named D(C), are completely determined by its generalized root whenever C has a generalized root. Moreover, it is proved that every finite theory C has a generalized root, which can be expressed by a specific formula. Finally, we demonstrate the existence of a non-fuzzy version of Fuzzy Modus Ponens (FMP) in Łuk, Göd, P and NM ðL Þ, and we provide its numerical version as a new algorithm for solving FMP. 2010 Elsevier Inc. All rights reserved.
Keywords: Fuzzy reasoning Algorithm Propositional fuzzy logic Generalized root Deduction theorem
1. Introduction Fuzzy reasoning is the theoretical foundation of fuzzy control. As fuzzy control became more widely applied and different kinds of fuzzy reasoning methods were introduced, researchers became more interested in the logical foundation of fuzzy reasoning. One of the main deduction rules in propositional logic is Modus Ponens (MP), which can be expressed as follows:
from A ! B and A infer B:
ð1Þ
Where A, B are two propositions. A and A ? B are called premises and B is called a conclusion. In cases where A ? B and a proposition A* that is close to A are given, a conclusion B* close to B must be drawn. This deduction can be written as:
from A ! B and A
infer B :
ð2Þ
For example, if we know that young people (A) are usually physically fit (B), and that some person is almost young (A*), then we can reasonably conclude that this person (A*) is probably nearly physically fit (B*). Examples like this can be found everywhere in common life. In what follows, we call the inference model (2) Generalized Modus Ponens (GMP). GMP is much more general and useful than deduction form (1), because A* is allowed to be different from A, thus many different propositions A* adapt to (2) (see [8]). More generally, consider the following inference model:
from Ai ! B and Ai
ði ¼ 1; 2; . . . ; mÞ infer B ;
ð3Þ
where the major premises Ai ? B(i = 1, . . . , m) and the minor premise Ai ði ¼ 1; . . . ; mÞ are given and a conclusion B* is to be collectively determined by the pairs fAi ! B; Ai gði ¼ 1; . . . ; mÞ. We call this inference model (3) Collective Generalized Modus Ponens (CGMP). q
Projected by the Natural Science Foundation of Fujian Province of China (No. 2006J0221).
* Corresponding author. Tel.: +86 059522983326; fax: +86 059522917084. E-mail address: [email protected] (J. Zhang). 0020-0255/$ - see front matter 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.ins.2010.07.035
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GMP and CGMP have a lot of applications, therefore it is necessary to put forward a program to find a reasonable B* in (2) and (3). It is Zadeh who first investigated the GMP-like problem where the propositions A, A* and B, B* appeared in (2) were regarded as fuzzy subsets on universes X and Y, respectively. In this framework, (2) turns out to be
from AðxÞ ! BðyÞ and A ðxÞ; x 2 X; y 2 Y;
calculate B ðyÞ:
ð4Þ
We call (4) Fuzzy Modus Ponens (FMP) (see [19]). Then, in 1973, Zadeh proposed an algorithm called Composition Rule of Inference (CRI) (see [19]) to solve FMP. By CRI the conclusion of FMP can be written as:
B ðyÞ ¼ supfA ðxÞ ^ RðAðxÞ; BðyÞÞjx 2 Xg;
y 2 Y;
ð5Þ
where R:[0, 1]2 ? [0, 1] is an implication operator, such as Zadeh’s operator RZ, Łukasiewicz’s operator RŁ, Gödel’s operator RG, etc., a ^ b = min{a, b}. The well-known Triple I method is another algorithm to solve the problem of FMP proposed in [12,16], such that B* is the smallest fuzzy subset of Y satisfying
8x 2 X;
8y 2 Y;
ðAðxÞ ! BðyÞÞ ! ðA ðxÞ ! B ðyÞÞ ¼ 1:
It has been proven in [15,18] that the Triple I conclusion B* of FMP always exists and
B ðyÞ ¼ supfA ðxÞ ðAðxÞ ! BðyÞÞjx 2 Xg;
y 2 Y;
ð6Þ
where is a t-norm. Formulas like (5) and (6) are called numerical solutions of GMP. Many numerical solutions were proposed under different assumptions by different authors (see [7,9,11]). A more reasonable solution may be constructed from the syntactical point of view. One typical example is available in Wang’s work in classical logic (see [12]), where he proved that every finite theory has a root, and a numerical solution can be deduced based on this root. In this paper, from the syntactic point of view, we are going to discuss the logical foundation of GMP/CGMP in the four logics mentioned above. A generalized root of a theory in these logics is defined, and its properties are carefully discussed, in order to draw some inference rules, which can help us find algorithms for determining their numerical solutions. This paper is organized as follows: In the first section we discuss the aim of our work. Section 2 introduces preliminaries of this paper. The definition of a generalized root and its properties are given in Section 3, and in Section 4 we explain the concept of syntactical solutions of GMP and CGMP, and discuss GMP and CGMP’s generalized inference rule (syntactic-form solution) and some algorithms to compute numerical solutions. 2. Preliminaries 2.1. Logic systems: Łuk, Göd, P and L (NM) A t-norm is a binary operation on [0,1] satisfying the following conditions: (i) is commutative and associative, i.e., for all x, y, z 2 [0, 1], x y = y x, (x y) z = x (y z); (ii) is nondecreasing in both arguments, i.e., x1 6 x2 implies x1 y 6 x2 y, y1 6 y2 implies x y1 6 x y2; (iii) 1 x = x and 0 x = 0 for all x 2 [0, 1]. is a continuous t-norm if it is a t-norm and is a continuous mapping of [0, 1]2 into [0, 1]. An implication operator is a function R:[0, 1]2 ? [0, 1] satisfying the following conditions: (i) R(x, y) is non-increasing w.r.t. x and is non-decreasing w.r.t. y. (ii) R(x, y) = 1 iff x 6 y. (iii) R(x, y) is left continuous w.r.t. x and is right continuous w.r.t. y if the corresponding t-norm is continuous. It is well known that different implication operators and valuation lattices, L (i.e. the set of truth degrees for fuzzy logic), determine different logic systems (see [1,5,6,21]). In this paper, an implication operator R needs to fulfill the condition that there exists a t-norm such that:
x y 6 z iff
x 6 Rðy; zÞ:
In that case (, R) is called an adjoint pair.
ð7Þ
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Four popularly used implication operators and the corresponding t-norms are defined as follows:
( RL ðx; yÞ ¼
1;
ð1 x þ yÞ; x > y (
1; x 6 y
RG ðx; yÞ ¼
y; (
RP ðx; yÞ ¼ ( R0 ðx; yÞ ¼
x6y
x>y
x>y
x L y ¼ maxð0; x þ y 1Þ; x; y 2 L;
;
x G y ¼ minðx; yÞ; x; y 2 L;
;
x P y ¼ x y; x; y 2 L;
1; x 6 y y ; x
;
1;
(
x6y
ð1 xÞ _ y; x > y
;
x n y ¼
minðx; yÞ; x þ y > 1 0;
xþy61
;
x; y 2 L;
where n is called the standard nilpotent minimum t-norm (see [2]). These four implication operators are called Łukasiewicz implication operator RŁ, Gödel implication operator RG, Product implication operator RP and R0-implication operator R0, respectively. In 1995, Fodor [2,3] discovered a kind of new t-norm n (nilpotent minimum) and corresponding fuzzy implications based on discussion of the so-called contrapositive symmetry. He wrote, ‘‘By these propitious characteristics, the nilpotent minimum can be admitted into investigations of many theoretical and practical problems soon.” Since 1997, Wang has published some papers on fuzzy logic and fuzzy reasoning (see [12,14–17]). In [14], he proposed a fuzzy implication operator, called R0 implication, and constructed a new formal system L for fuzzy propositional calculus. We see that R0 implication is indeed a particular implication based on the standard nilpotent minimum. If we fix a t-norm , then a propositional calculus (whose set of truth values is L) is fixed: is taken for the truth function of the strong conjunction &, the residuum R of becomes the truth function of the implication operator and R (., 0) is the truth function of the negation. For more detail, we use the following definitions: Definition 1. [5,6]. The propositional calculus PC() given by a t-norm has the set S of propositional variables p1, p2, . . . and connectives :; &; !. The set F(S) of well-formed formulas in PC() is defined inductively as follows: each propositional variable is a formula; if A, B are formulas, then :A; A&B and A ? B are all formulas. Definition 2. [1,5,6,13]. The formal deductive systems of PC() given by corresponding to RL, RG, RP and R0, are called Łukasiewicz fuzzy logic Łuk, Gödel fuzzy logic Göd, Product fuzzy logic P and R0-fuzzy logic L (the nilpotent minimum logic NM), respectively. 1 A logic system is called an n-valued logic system if the valuation lattice L ¼ 0; n1 ; . . . ; n2 n1 ; 1 . A logic system is called a fuzzy logic system if the valuation lattice L = [0, 1]. The inference rule of each logic system above is Modus Ponens (MP): from A and A ? B infer B. The following formulas are axioms of the basic logic BL [6]:
ðA1Þ ðu ! wÞ ! ððw ! vÞ ! ðu ! vÞÞ; ðA2Þ ðu&wÞ ! u; ðA3Þ ðu&wÞ ! ðw&uÞ; ðA4Þ ðu&ðu ! wÞÞ ! ðw&ðw ! uÞÞ; ðA5aÞ ðu ! ðw ! vÞÞ ! ððu&wÞ ! vÞ; ðA5bÞ ððu&wÞ ! vÞ ! ðu ! ðw ! vÞÞ; ðA6Þ ðu ! wÞ ! vÞ ! ðððw ! uÞ ! vÞ ! vÞ; ! u: ðA7Þ 0 The following are Lukasiewicz’s axioms [6]: (Ł1) u ? (w ? u), (Ł2) (u ? w) ? ((w ? v) ? ((w ? u) ? u).
(u ? v)), (Ł3) ð:u ! :wÞ ! ðw ! uÞ, (Ł4) ((u ? w) ? w) ?
The axioms of P are those of BL plus [6] (P1) ::v ! ððu v ! w vÞ ! ðu ! wÞÞ; ðP2Þ u ^ :u ! 0. The axiom system of Gödel logic is the extension of the axiom system of BL by the single axiom (G) u ? (u&u) stating (together with (u&u) ? u) the idempotence of &.
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Axioms of R0 fuzzy logic L are [14]:
ðL 1Þu ! ðw ! uÞ; ðL 2Þð:u ! :wÞ ! ðw ! uÞ; ðL 3Þðu ! ðw ! vÞÞ ! ðw ! ðu ! vÞÞ; ðL 4Þðw ! vÞ ! ððu ! wÞ ! ðu ! vÞÞ; ðL 5Þu ! ::u; ðL 6Þu ! u _ w; ðL 7Þu _ w ! w _ u; ðL 8Þðu ! vÞ ^ ðw ! vÞ ! ðu _ w ! vÞ; ðL 9ðu ^ w ! vÞ ! ðu ! vÞ _ ðw ! vÞ; ðL 10Þ ðu ! wÞ _ ððu ! wÞ ! :u _ wÞ; where u ^ w ¼ :ð:u _ :wÞ: ´jek’s work on the basic logic and Höhle’s work on the monoidal logic, proposed a In 2001, Esteva and Godo, based on Ha new propositional calculus, called the monoidal t-norm based logic MTL and its two schematic extensions WNM and IMTL, called the weak nilpotent minimum logic and the involutive monoidal t-norm based logic, respectively. Furthermore, they obtained a natural common schematic extension NM of WNM and IMTL, called the nilpotent minimum logic NM (see [1]). Obviously, the system NM and the system L are strongly related. In fact, it is easy to verify that two systems are equivalent, and related algebras (NM-algebras and R0-algebras) are essentially the same kind of algebraic structures (see [10]). Remark 1. (i) One can further define connectives ^ and _ in these four logic systems, but the definition of ^ and _ in L are different from those in the other three logic systems: in L ; u _ w is an abbreviation of :ððððu ! ðu ! wÞÞ ! uÞ ! uÞ ! :ððu ! wÞ ! wÞÞ; u ^ w is that of :ð:u _ :wÞ. While in Łuk, Göd and P, ^ and _ can be defined uniformly as follows [6]: u ^ w is an abbreviation of u&(u ? w), and u _ w is that of ((u ? w) ? w) ^ ((w ? u) ? u). (ii) Define on L an unary operator and two binary operators as follows:
x ¼ Rðx; 0Þ;
x&y ¼ x y;
x ) y ¼ Rðx; yÞ;
x; y 2 L:
ð8Þ
where (, R) is an adjoint pair on L. From Remark 1 (ii), the valuation lattice L becomes an algebra of type (, , )). Defined in the above-mentioned logic systems
Am :¼ A&A& &A |fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl ffl};
ð9Þ
m
and in the corresponding algebras L
aðmÞ :¼ |fflfflfflfflfflfflfflfflfflfflffl aa ffla}; ffl{zfflfflfflfflfflfflfflfflfflfflffl
ð10Þ
m
where is the t-norm defined on L. Remark 2. [6,13]. It is easy to verify that the following assertions are true for every m 2 N: (i) (ii) (iii) (iv)
in in in in
Göd, a(m) = a. L ; aðmÞ ¼ að2Þ ðm P 2Þ. Łuk, a(m) = (ma (m 1)) _ 0. P, a(m) = am.
Definition 3. [6,13]. (i) A homomorphism v:F(S) ? L of type ð:; &; !Þ from F(S) into the valuation lattice L, that is, v ð:AÞ ¼ v ðAÞ, v(A&B) = v(A) v(B), v(A ? B) = v(A) ) v(B), is called an R-valuation of F(S). The set of all R-valuations will be denoted by XR. (ii) A formula A 2 F(S) is called a tautology with respect to R if "v 2 XR, v(A) = 1 holds. (iii) Let A,B 2 F(S). If t(A) = t(B) for every t 2 XR, then A and B are called logically equivalent, denoted A B.
Remark 3. [6,13]. It is not difficult to verify in the above-mentioned four logic systems that t(A _ B) = max {t(A), t(B)}, and t(A ^ B) = min{t(A), t(B)} for every valuation t 2 XR. Moreover, one can check in Łuk and L* that A&B and :ðA ! :BÞ are logically equivalent. Definition 4. [6]. (i) A subset of F(S) is called a theory.
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(ii) Let C be a theory and L is a logic system, A 2 F(S). A deduction of A from C in logic system L, in symbols, C ‘ LA, is a finite sequence of formulas A1, . . . , Am = A such that for each 1 6 i 6 m, Ai is an axiom of L, or Ai 2 C, or there are j, k 2 {1, . . . , i 1} such that Ai follows from Aj and Ak by MP. Equivalently, we say that A is a conclusion of C (or C-conclusion). The set of all conclusions of C is denoted by D(C). By a proof of A we shall henceforth mean a deduction of A from the empty set. We shall also write ‘LA in place of ;‘LA and call A a theorem. (iii) Let A, B 2 F(S). If ‘LA ? B and ‘LB ? A hold, then A and B are called provably equivalent in logic system L, denoted as A B. To simplify, we always replace symbol ‘LA with ‘A in this paper. Theorem 1. [1,6,13]. Completeness theorem holds in every above-mentioned logic system, i.e. "A 2 F(S), A is a theorem in Łuk, Göd, P and L (NM) iff A is a tautology in Łuk, Göd, P and L (NM), respectively. Theorem 1 points out that semantics and syntax in these four logic systems are in perfect harmony. 2.2. Generalized deduction theorems in Łuk, Göd, P and L (NM)
Theorem 2. [1,5,6]. Suppose that C is a theory, A, B 2 F(S), then in Łuk, Göd, P and L (NM), C [ {A} ‘ B iff $m 2 N, s.t. C ‘ Am ? B. Theorem 3. [13]. Suppose that C is a theory, A, B 2 F(S), then (i) in L* (NM) the generalized deduction theorem holds, i.e. C [ {A} ‘ B iff C ‘ A2 ? B. (ii) in Göd, deduction theorem holds, i.e. C [ {A} ‘ B iff C ‘ A ? B It is easy for the reader to check the following Lemma: Lemma 1. Let C be a theory and A 2 F(S). If C ‘ A, then there exists a finite subset of C, say, {A1, . . . , Am} such that {A1, . . . , Am} ‘ A. Note that A&B ? C and A ? (B ? C) are provably equivalent by the definition of the deduction theorem and the generalized deduction theorem. It is easy for the reader to check the following Lemma: Lemma 2. Suppose that C = {A1, . . . , Am} is a finite theory, A 2 F(S). Then (i) in Łuk, Göd, P and L (NM), C ‘ A iff $n1, . . . , nm 2 N such that ‘ ðAn11 & &Anmm Þ ! A. (ii) in Göd, C ‘ A iff ‘(A1& &Am) ? A. (iii) in L ; C ‘ A iff ‘ ðA21 & &A2m Þ ! A. The following Lemma can be obtained via Lemma 2.2.8 of [6,13]: Lemma 3. Let C be a theory and let A, B, C, D be formulas. In Łuk, Göd, P and L (NM), the following conclusions hold: (i) (ii) (iii) (iv) (v) (vi) (vii)
If C ‘ A, C ‘ B, then C ‘ A&B. If C ‘ A ? B, C ‘ C ? D, then C ‘ (A&C) ? (B&D). If C ‘ A ? B, then C ‘ (A&C) ? (B&C). ‘(A&B) ? B. If C ‘ A, C ‘ B, then C ‘ A ^ B. C ‘ ((A ? C) ^ (B ? C)) ? ((A _ B) ? C). C ‘ (A&B) ? B.
3. Generalized roots of theories 3.1. Basic definitions and properties First let us briefly review the necessary concepts (see [12]). (A1) In a logic system L, define a binary relation on F(S) as follows: A B iff ‘A ? B, A, B 2 F(S). Obviously is reflective and transitive and hence is a pre-order on F(S).
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(A2) In a logic system L, A is said to be a root of C iff A 2 D(C) and "B 2 D(C) A B, where D(C) is the set consisting of all conclusions of C. Reference [12] discusses some syntactical conclusions B* on problems like GMP and CGMP in classical logic, and gives definitions for solutions of GMP and a root of a theory C. Based on the root’s definition and its properties, reference [12] obtains the result that the Triple I solution B* of GMP is a root of C = {A ? B, A*}. That is, the Triple I solution B* of GMP is one of the smallest formula in (F(S), ) satisfying the condition that ‘(A ? B) ? (A* ? B*). This tells us that the definition of a root, as well as its properties, plays an important role in solving problems like GMP. Unfortunately, it has been proven that a theory C in Łukasiewicz propositional logic has no roots in general (see [20]). This section defines another form of root called a generalized root in Łukasiewicz propositional logic, Gödel propositional logic, Product propositional logic, and R0-propositional logic. We will show that the generalized root and the root defined in [12] have some properties in common. In addition to generalized root properties, we will also discuss the structure of the set of all the conclusions of C, called D(C), and the relations, specifically, inclusion and equality =, between two conclusion sets, D(C1) and D(C2), when different theories C1, C2 are given. Definition 5. Suppose that C is a theory and A 2 F(S). If A 2 D(C) and " B 2 D(C), $ m 2 N such that ‘Am ? B, then A is called a generalized root of C. T Definition 6. Suppose that Ck F(S),k = 1, 2, . . . , l. The members of lk¼1 DðCk Þ are called common conclusions of C1, . . . , Cl, Tl T and A is said to be a generalized common root of C1, . . . , Cl if A 2 k¼1 DðCk Þ, and 8B 2 lk¼1 DðCk Þ9m 2 N such that ‘ Am ? B. The following propositions hold in the four logic systems of concern: Proposition 1. Suppose that C is a theory and A 2 F(S). If A is a generalized root of C, then D(C) = D(A), where D(A) is an abbreviation for D({A}).
Proof. Suppose that A is a generalized root of C. Then " B 2 D(C), it follows from Definition 5 that $m 2 N such that ‘Am ? B holds. Since {A} ‘ Am by Lemma 3, it follows from the inference rule MP that {A} ‘ B. This means that B 2 D(A), hence D(C) # D(A). For the converse, " B 2 D(A), that is, {A} ‘ B, it follows from the generalized deduction theorem that $m 2 N such that ‘ Am ? B. C ‘ Am follows from the assumption C ‘ A. Therefore we have C ‘ B by MP, and D(A) # D(C). h Proposition 2. Suppose that A 2 F(S). Then D(A) = {B 2 F(S)j $m 2 N, ‘Am ? B}. Proof. Let V = {B 2 F(S)j$m 2 N, ‘ Am ? B}." C 2 D(A), it follows from the generalized deduction theorem that $ m 2 N such that ‘Am ? C. Thus C 2 V and D(A) # V. For the converse, "B 2 V, $m 2 N such that ‘ Am ? B. It is easy to prove that {A} ‘ Am and it follows from the inference rule MP that {A} ‘ B holds, that is, B 2 D(A). Hence V # D(A) and so D(A) = V. h Theorem 4. Every finite theory has a generalized root. More precisely, A1& &As is a generalized root of theory C = {A1, . . . , As}. Proof. It is only necessary to prove that A1& &As is a generalized root of C. By Lemma 3, A1& &As 2 D(C). "B 2 D(C), it follows from Lemma 2 that there exists n1, . . . , ns 2 N such that ‘ ðAn11 & &Ans s Þ ! B. Let m = max{n1, . . . , ns}. We have ‘ ðA1 & &As Þm ! ðAn11 & &Ans s Þ by Lemma 3. Hence by Hypothetical Syllogism, we get ‘(A1& &As)m ? B. Thus A1& & As is a generalized root of C. h Proposition 3. Suppose that Ck # F(S) and Ak is a generalized root of Ck, k = 1, . . . , l. Then A1& &Al, is a generalized root of Sl k¼1 Ck .
C¼
Proof. It is easy to prove that C ‘ Ai, i = 1, . . . , l by assumption. Hence by Lemma 3, we get C ‘ A1& &Al, that is, mk 1 A1& &Al 2 D(C). "B 2 D(C), there exists B1, . . . , Bk 2 C and m1, . . . , mk 2 N such that ‘ ðBm 1 & &Bk Þ ! B by Lemmas 1 m s si 1 s1 and 2. " Bi (i = 1, . . . , k), there exist ji 2 {1, . . . , l} and si 2 N such that ‘ Aji ! Bi , where Bi 2 Cji . ‘ ðAm & &Ajk k k Þ ! j1 mk dl mk sk d1 m1 s1 m1 ðB1 & &Bk ) by Lemma 3. Clearly, there exists d1, . . . , dl 2 N such that ‘ ðA1 & &Al Þ ! ðAj1 & ; &Ajk Þ. Let mk 1 d = max{d1, . . . , dl}. By Lemma 3 and Hypothetical Syllogism, we have ‘ ðA1 & &Al Þd ! ðBm 1 & &Bk Þ. Also by Hypothetical k Syllogism ‘(A1& &Al) ? B is yielded. This shows that A1& &Al is a generalized root of C. h Proposition 4. Suppose that A and B are generalized roots of C1 and C2, respectively. Then D(C1) # D(C2) iff $ m 2 N, such that ‘Bm ? A.
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Proof. The proposition above directly follows from Definition 5, Proposition 2 and Lemma 3. Corollary 1. Suppose that ‘(B1& &Bt)m ? (A1& &As).
C1 = {A1, . . . , As}
and
C2 = {B1, . . . , Bt}.
Then
D(C1) # D(C2)
h iff
$m 2 N,
such
that
Proof. By Theorem 4 and Proposition 4, it is easy to check the corollary above. h 3.2. Results in the n-valued Łukasiewicz logic It has been proven that the semantics and syntax of the n-valued Łukasiewicz logic are in perfect harmony (see [4]). It is then easy to prove the following propositions: Proposition 5. Suppose that m,n 2 N. If m P n 1, then "a 2 L, a(m) = a(n1) holds.
Proposition 6. Suppose that A 2 F(S). If m P n 1, then Am An1. Theorem 5. Suppose that C is a theory and A 2 F(S). If A is a generalized root of C, then D(C) = {B 2 F(S)j‘An1 ? B}. Proof. The theorem above directly follows from Propositions 2 and 6. h Theorem 6. Suppose that Ai is a generalized root of Ci (i = 1, 2, . . . , k), then
Wk
n1 i¼1 Ai
is a generalized common root of C1, . . . , Ck.
W ! ki¼1 An1 ðt ¼ 1; . . . ; kÞ and An1 2 DðCt Þ, it follows from the inference rule MP that Proof. By Lemma 3, ‘ An1 t i t Wk n1 T 2 DðCt Þði ¼ 1; . . . ; kÞ. Since 8B 2 ki¼1 DðCi Þ; ‘ An1 ! Bði ¼ 1; . . . ; kÞ by Theorem 5, then we have ‘ ðAn1 ! BÞ i¼1 Ai i 1 n1 n1 n1 n1 n1 n1 ^ ^ ðAn1 ! BÞ. Since ðA ! BÞ ^ ^ ðA ! BÞ ðA _ _ A ! BÞ. Obviously ‘ A _ _ A ! B holds, k 1 k 1 k 1 k therefore An1 _ _ An1 is a generalized common root of C1, . . . , Ck. h 1 k 3.3. Results in the Gödel fuzzy logic
Proposition 7. Suppose that A is a generalized root of C. Then D(C) = {B 2 F(S)j‘A ? B}.
Proof. The proposition above directly follows from the definition of the generalized root and the deduction theorem. h Proposition 8. Suppose that A and B are generalized roots of C1 and C2, respectively. Then D(C1) # D(C2) iff ‘B ? A. Proof. The proposition above directly follows from Propositions 4 and 7. h Theorem 7. Suppose that Ck # F(S) and Ak is a generalized root of Ck(k = 1, 2, . . . , m). Then C1, . . . , Cm.
Wm
Proof. The proof is analogous to that of Theorem 6. h 3.4. Results in the R0- fuzzy logic L
Lemma 4. [13]. Suppose that A 2 F(S). Then Am A2, m = 2, 3, . . .
Proposition 9. Suppose that A is a generalized root of C. Then D(C) = {B 2 F(S)j‘A2 ? B}. Proof. The proof is analogous to that of Theorem 5. h
i¼1 Ai
is a generalized common root of
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Proposition 10. Suppose that A and B are generalized roots of C1 and C2, respectively. Then D(C1) # D(C2) iff ‘B2 ? A2. Proof. The proposition above directly follows from Lemmas 3, 4 and Proposition 9. h Theorem 8. Suppose that Ck # F(S) and Ak is a generalized root of Ck(k = 1, 2, . . . , m). Then of C1, . . . , Cm.
Wm
2 i¼1 Ai
is a generalized common root
Proof. The proof is analogous to that of Theorem 6. h 4. Definitions and expressions of the solutions of GMP and CGMP in Łuk, Göd, P and L (NM) Now we turn to the problems of GMP and CGMP. Our aim is to find a suitable conclusion B* of the prerequisites A ? B and A*. A ? B, A* and B* should certainly satisfy the following condition:
fA ! B; A g ‘ B :
ð11Þ
Obviously there are too many B* that satisfy (11). For example, (11) holds if B* is a tautology (theorem), but it is clearly not what we need since a tautology is a conclusion of any A ? B and A* is given as prerequisites. Hence, a tautology is not a suitable conclusion for GMP. By (A1), it is reasonable to ask for B* in GMP to be as small as possible. Definition 7. Let A, B, A*, B* 2 F(S). B* is called a Triple I2 solution of GMP if it satisfies the following conditions: (G1) {A ? B, A*} ‘ B*; (G2) if C satisfies (G1), then B* C (C 2 F(S)).
Theorem 9. Let C = {A ? B, A*}. Then Triple I2 solution of GMP B* is a generalized root of C. Proof. By Definition 7 (G1), B* 2 D(C). Since A*&(A ? B) is a generalized root of C by Theorem 4, then, by Definition 5, "C 2 D(C), and there is an m such that ‘((A ? B)&A*)m ? C and (A ? B)&A* 2 D(C). ‘B* ? ((A ? B)&A*) by Definition 7 (G2), thus, ‘ (B*)m ? ((A ? B)&A*)m by Lemma 3. It follows from Hypothetical Syllogism that ‘ (B*)m ? C, that is, B* is a generalized root of C. h Theorem 10. Suppose that C = {A ? B, A*}, A, B, A* 2 F(S). Then (i) in the n-valued Łukasiewicz logic Łn, the generalized root B* = (A*)n1&(A ? B)n1 of C is a Triple I2 solution of GMP. (ii) in L*, the generalized root B* = (A*)2&(A ? B)2 of C is a Triple I2 solution of GMP. (iii) in Gödel, the generalized root B* = A*&(A ? B) of C is a Triple I2 solution of GMP.
Proof. Because the proofs of (i), (ii), and (iii) are all simiar, here we only prove (i). In the n-valued Łukasiewicz logic Ln, since the deduction theorem means that
C [ A ‘ B iff C ‘ An1 ! B;
C FðSÞ;
A;
B 2 FðSÞ:
ð12Þ
Moreover, it is well known in Ln that
A ! ðB ! CÞ A&B ! C;
A;
B;
C 2 FðSÞ;
where is the logical equivalence relation. Since ‘ (A*) that the following assertions are also true:
ð13Þ n1
&(A ? B)
n1
? B* is clearly true, and it follows from (12) and (13)
‘ ðA Þn1 ! ððA ! BÞn1 ! B Þ; fA g ‘ ðA ! BÞn1 ! B ; fA ! B; A g ‘ B : Hence B* satisfies condition (G1). Assume that C satisfies (G1), that is, {A ? B, A*} ‘ C, C 2 F(S). Then it follows from the deduction theorem of Ln that
fA g ‘ ðA ! BÞn1 ! C; ‘ ðA Þn1 ! ððA ! BÞn1 ! CÞ: Therefore it follows from (13) that ‘(A*)n1&(A ? B)n1 ? C and ‘B* C, i.e. B* satisfies condition (G2). The proof of Theorem 10 is complete. h
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Definition 8. Let Ci ¼ fAi ! B; Ai g and Ai ; B; Ai ; B 2 FðSÞði ¼ 1; 2; . . . ; mÞ. B* is called a Triple I2 solution of CGMP if it satisfies the following conditions: (D1) Ci ‘ B*(i = 1, 2, . . . , m); T (D2) if C 2 m k¼1 DðCk Þ, then B* C, C 2 F(S).
Remark 4. It is easy to verify that if B* is a Triple I2 solution of CGMP, then B* is a generalized common root of C1, . . . , Cl by Definitions 5 and 8. Theorem 11. Suppose that Ci ¼ fAi ! B; Ai g; Ai ; B; Ai 2 FðSÞ; i ¼ 1; 2; . . . ; m . Then (i) in the n-valued Łukasiewicz logic Ln,
ððA1 Þn1 &ðA1 ! BÞn1 Þ _ _ ððAm Þn1 &ðAm ! BÞn1 Þ
ð14Þ
is a Triple I2 solution of CGMP. (ii) in L*,
ððA1 Þ2 &ðA1 ! BÞ2 Þ _ _ ððAm Þ2 &ðAm ! BÞ2 Þ
ð15Þ
is a Triple I2 solution of CGMP. (iii) in Gödel,
ðA1 &ðA1 ! BÞÞ _ _ ðAm &ðAm ! BÞÞ
ð16Þ
is a Triple I2 solution of CGMP.
Proof. Because the proofs of (i) (ii), and (iii) are all similar, here we only prove (i). In the n-valued Łukasiewicz logic Łn, Ai &ðAi ! BÞ is a generalized root of Ci by Theorem 4, and ððA1 Þn1 &ðA1 ! BÞn1 Þ _ _ ððAm Þn1 &ðAm ! BÞn1 Þ is a generalized common root of C1, . . . , Cm by Theorem 6. Then ððA1 Þn1 &ðA1 ! BÞn1 Þ _ _ ððAm Þn1 &ðAm ! BÞn1 Þ satisfies condition (D1). T 8C 2 m we have ‘ ðAi Þn1 &ðAi ! BÞn1 ! C (i = 1, 2, . . . , m) by Theorem 5, thus ‘ ðððA1 Þn1 & k¼1 DðCk Þ, n1 ðA1 ! BÞ Þ _ _ ððAm Þn1 &ðAm ! BÞn1 ÞÞ ! C , then ððA1 Þn1 &ðA1 ! BÞn1 Þ _ _ ððAm Þn1 &ðAm ! BÞn1 Þ C . The proof of Theorem 11 (i) is complete. h 5. Semantic type Triple I2 solutions of GMP and CGMP
Remark 5. [12]. Suppose that XR is the set consisting of all valuations from F(S) to L (see Definition 3), where L is the set of truth degrees for fuzzy logic and F(S) is the set of well-formed formulas (see Definition 1). Suppose that t 2 XR and A 2 F(S), then t(A) is the value of t at A. Let us also agree to use the symbol A(t) to denote t(A). A formula A induces a mapping A:XR ? L (denoted by A, itself) naturally defined by "t 2 XR, A(t) = t(A) where t(A) is the value of t at A. By Theorems 9, 10 and Remark 5, the following algorithms are obvious. Lemma 5. The FMP-type Triple I2 solution B* of GMP can be computed as follows: (i) in the n-valued Łukasiewicz logic Ln, B*(v) = sup{(A*(u))n1 (A(u) ? B(v))n1ju 2 X}, (ii) in L*, B*(v) = sup{(A*(u))2 (A(u) ? B(v))2}, v 2 Y; (iii) in Gödel, B*(v) = sup{A*(u) (A(u) ? B(v))}, v 2 Y.
v 2 Y;
Lemma 6. The FMP-type Triple I2 solution B* of CGMP can be computed as follows: (i) in the n-valued Łukasiewicz logic Ln, B ðv Þ ¼ supfððA1 ðuÞÞn1 ðA1 ðuÞ ! Bðv ÞÞn1 Þ _ _ ððAm ðuÞÞn1 ðAm ðuÞ ! Bðv ÞÞn1 Þju 2 Xg; v 2 Y ; (ii) in L*, B ðv Þ ¼ supfððA1 ðuÞÞ2 ðA1 ðuÞ ! Bðv ÞÞ2 Þ _ _ ððAm ðuÞÞ2 ðAm ðuÞ ! Bðv ÞÞ2 Þju 2 Xg; v 2 Y; (iii) in Gödel, B ðv Þ ¼ supfððA1 ðuÞ ðA1 ðuÞ ! Bðv ÞÞÞ _ _ ðAm ðuÞ ðAm ðuÞ ! Bðv ÞÞÞju 2 Xg; v 2 Y.
Example 1. Suppose that X = Y = [0, 1], A, A* 2 F(X), B 2 F(Y) are as follows: AðuÞ ¼ uþ2 3 ; Bðv Þ ¼ 1 v ; A ðuÞ ¼ 1 u and u, v 2 [1, 0]. In the 7-valued Łukasiewicz logic L7, compute the FMP-type Triple I2 solution B* of GMP by using Lemma 5.
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Solution. Since A ðuÞ ðAðuÞ ! Bðv ÞÞ ¼ ð1 uÞ and the corresponding t-norm in this logic that
uþ2 3
! ð1 v Þ , it follows from the definition of implication operator
(
4u3v ; v > 1u uþ2 3 3 : þ1v ^1¼ AðuÞ ! Bðv Þ ¼ 1 3 1; v 6 1u 3 8 96u18v 1u ; 3 < v < 32u > 3 6 < 6 ðAðuÞ ! Bðv ÞÞ ¼ 1; : v 6 1u 3 > : 32u 0; vP 6 ( 1 1 6u; u 6 6 : ðA ðuÞÞ6 ¼ 0; u P 16 8 1u 38u 1 > < 3 8u 6v ; 3 < v < 6 ; u < 6 6 6 ðA ðuÞÞ ðAðuÞ ! Bðv ÞÞ ¼ 1 6u; : v 6 1u ; u < 16 3 > : 0; otherwise 8 v 6 13 > < 1; 6 6 B ðv Þ ¼ supfðA ðuÞÞ ðAðuÞ ! Bðv ÞÞ ju 2 ½0; 1 g ¼ 3 6v ; 13 < v < 12 : > : 0; otherwise
Example 2. Suppose that X, Y, A, A*, and B are the same as those in Example 1. In L*, compute the FMP-type Triple I2 solution B* of GMP by using Lemma 5. Solution. Since
(
2
ðA ðuÞÞ ¼
1 u; u < 12 ; 0;
8 > 3 ; 3 > : 1 v ; 1u < v 6 2þu : 3 3 8 1u v6 3 ; 1; > > > < 0; v > 2þu ; 3 ðAðuÞ ! Bðv ÞÞ2 ¼ 1u > 1 v ; 3 < v 6 2þu ; v < 12 ; > 3 > : 1u 2þu 0; < v 6 3 ; v P 12 : 83 1u 2þu 1 > < 1 u; 3 < v 6 3 ; u < 2 ; v 6 u; 2 2 ðA ðuÞÞ ðAðuÞ ! Bðv ÞÞ ¼ 1 v ; v 6 12 ; u < v ; 1u < v 6 2þu ; 3 3 > 8 > < 1; B ðv Þ ¼ 1 v ; > : 0;
:
0;
or u < 12 ; v 6 1u ; 3
otherwise:
v 6 13 ; 1 < v 6 12 ; 3 v P 12 :
Example 3. Suppose that X, Y, A, A*, and B are the same as those in Example 1. Compute the FMP-solutions B* by using Zadeh’s CRI method, and the conclusion of FMP is as follows[13]:
B ðv Þ ¼ supfA ðuÞ ^ ðAðuÞ ! Bðv ÞÞju 2 Xg;
v 2 Y;
where the implication operator is a Łukasiewicz’s implication operator. Solution. Since A ðuÞ ^ ðAðuÞ ! Bðv ÞÞ ¼ ð1 uÞ ^ uþ2 ! ð1 v Þ , it follows from the definition of implication operator 3 that
( 4u3v ; uþ2 3 þ 1 vÞ ^ 1 ¼ AðuÞ ! Bðv Þ ¼ ð1 3 1; ( 4u3v 1þ2u ; v> 3 ; 3 A ðuÞ ^ ðAðuÞ ! Bðv ÞÞ ¼ 1 u; v 6 1þ2u : 3 Therefore we have B ðv Þ ¼
4
v; 1;
3
v > 13 ; v 6 13 :
v > 1u ; 3 1u v6 3 :
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6. Concluding remarks This paper proposes a new concept called a generalized root of a theory, C, in Łukasiewicz propositional fuzzy logic Łuk, Gödel propositional fuzzy logic Göd, Product propositional fuzzy logic P, and the nilpotent minimum logic NM (the R0-propositional fuzzy logic L ). This concept is based on the generalized deduction theorem and the completeness theorem. Then it is proved that D(C) is completely determined by its generalized root whenever C has a generalized root. In Łukasiewicz propositional fuzzy logic, the n-valued Łukasiewicz propositional logic, Gödel propositional fuzzy logic, Product propositional fuzzy logic and the nilpotent minimum logic (the R0-propositional fuzzy logic), every finite theory has a generalized root. The conditions for the sets D(C) of two different theories being equal and included are given. Finally, syntactic-form conclusions of problems like GMP, CGMP were drawn in Łuk, Göd, P and NM ðL Þ, via the generalized root of a theory, C, and their numerical versions are provided as new algorithms for solving GMP and CGMP. Acknowledgment The authors would like to express their sincere thanks to the anonymous referee for his or her valuable suggestions. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21]
F. Esteva, L. Godo, Monoidal t-norm based logic: towards a logic for left-continuous t-norms, Fuzzy Sets and Systems 124 (2001) 271–288. J. Fodor, Nilpotent minimum and related connectives for fuzzy logic, in: Proceedings of FUZZ-IEEE’95, 1995, pp.2077–2082. J. Fodor, Contrapositive symmetry of fuzzy implications, Fuzzy Sets and Systems 69 (1995) 141–156. R.S. Grigolia, Algébraicˇéskij analiz n-znc˘nych logic˘éskich sistém Łukasiewicza-Tarskiego, in: Proceedings of Tbilisi Universtity A 6-7 (149-150) (1973), Mathematical and Natural Sciences, Tbilisi, pp. 121-132. S. Gottwald, A Treatise on Many-Valued Logics, Studies in Logic and Computation, Research Studies Press, Baldock, 2001. P. Hájek, Metamathematics of Fuzzy Logic, Kluwer Academic Publishers., Dordrecht, 1998. H.W. Liu, G.J. Wang, Unified forms of fully implicational restriction methods for fuzzy reasoning, Information Sciences 177 (3) (2007) 956–966. B. Bouchon-Meunier, V. Kreinovich, Fuzzy Modus Ponens as calculus of logical modifiers: towards Zadeh’s vision of implication calculus, Information Sciences 116 (1999) 219–227. D.W. Pei, On optimal solutions of GMP and GMT, Fuzzy systems and Mathematics 17 (2) (2003) 1–9. D.W. Pei, On the equivalent forms of fuzzy logic systems NM and IMTL, Fuzzy Sets and Systems 138 (2003) 187–195. D.W. Pei, Unified full implication algorithms of fuzzy reasoning, Information Sciences 178 (2) (2008) 520–530. G.J. Wang, H. Wang, Non-fuzzy versions of fuzzy reasoning in classical logics, Information Sciences 138 (2001) 211–236. G.J. Wang, Introduction to Mathematical Logic and Resolution Principle, Science in China Press, Beijing, 2003. in Chinese. G.J. Wang, A formal deductive system for fuzzy propositional calculus, Chinese Science Bulletin 42 (1997) 1521–1525. G.J. Wang, On the logic foundation of fuzzy reasoning, Information Sciences 117 (1999) 47–88. G.J. Wang, Theory of Non-classical Mathematical Logic and Approximate Reasoning, Science in China Press, Beijing, 2000. in Chinese. G.J. Wang, L. Fu, Unified forms of Triple I method, Computers and Mathematics with Applications 49 (2005) 923–932. G.J. Wang, H.J. Zhou, Quantitative logic, Information Sciences 179 (3) (2009) 226–247. L.A. Zadeh, Outline of a new approach to the analysis of complex systems and decision processes, IEEE Transactions on Systems Man Cybernetics 3 (1973) 28–44. J.C. Zhang, Some properties of the roots of theories in propositional logic systems, Computers and Mathematics with Applications 55 (2008) 2086– 2093. H.J. Zhou, G.J. Wang, Generalized consistency degrees of theories w.r.t. formulas in several standard complete logic systems, Fuzzy Sets and Systems 157 (2006) 2058–2073.
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Contents lists available at ScienceDirect
Information Sciences journal homepage: www.elsevier.com/locate/ins
Some properties of fuzzy reasoning in propositional fuzzy logic systems q Jiancheng Zhang *, Xiyang Yang Department of Mathematics, Quanzhou Normal University, Fujian 362000, China
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 8 January 2010 Received in revised form 15 July 2010 Accepted 29 July 2010
In order to analyze the logical foundation of fuzzy reasoning, this paper first introduces the concept of generalized roots of theories in Łukasiewicz propositional fuzzy logic Łuk, Gödel propositional fuzzy logic Göd, Product propositional fuzzy logic P, and nilpotent minimum logic NM (the R0-propositional fuzzy logic L ). Next, it is proved that all consequences of a theory C, named D(C), are completely determined by its generalized root whenever C has a generalized root. Moreover, it is proved that every finite theory C has a generalized root, which can be expressed by a specific formula. Finally, we demonstrate the existence of a non-fuzzy version of Fuzzy Modus Ponens (FMP) in Łuk, Göd, P and NM ðL Þ, and we provide its numerical version as a new algorithm for solving FMP. 2010 Elsevier Inc. All rights reserved.
Keywords: Fuzzy reasoning Algorithm Propositional fuzzy logic Generalized root Deduction theorem
1. Introduction Fuzzy reasoning is the theoretical foundation of fuzzy control. As fuzzy control became more widely applied and different kinds of fuzzy reasoning methods were introduced, researchers became more interested in the logical foundation of fuzzy reasoning. One of the main deduction rules in propositional logic is Modus Ponens (MP), which can be expressed as follows:
from A ! B and A infer B:
ð1Þ
Where A, B are two propositions. A and A ? B are called premises and B is called a conclusion. In cases where A ? B and a proposition A* that is close to A are given, a conclusion B* close to B must be drawn. This deduction can be written as:
from A ! B and A
infer B :
ð2Þ
For example, if we know that young people (A) are usually physically fit (B), and that some person is almost young (A*), then we can reasonably conclude that this person (A*) is probably nearly physically fit (B*). Examples like this can be found everywhere in common life. In what follows, we call the inference model (2) Generalized Modus Ponens (GMP). GMP is much more general and useful than deduction form (1), because A* is allowed to be different from A, thus many different propositions A* adapt to (2) (see [8]). More generally, consider the following inference model:
from Ai ! B and Ai
ði ¼ 1; 2; . . . ; mÞ infer B ;
ð3Þ
where the major premises Ai ? B(i = 1, . . . , m) and the minor premise Ai ði ¼ 1; . . . ; mÞ are given and a conclusion B* is to be collectively determined by the pairs fAi ! B; Ai gði ¼ 1; . . . ; mÞ. We call this inference model (3) Collective Generalized Modus Ponens (CGMP). q
Projected by the Natural Science Foundation of Fujian Province of China (No. 2006J0221).
* Corresponding author. Tel.: +86 059522983326; fax: +86 059522917084. E-mail address: [email protected] (J. Zhang). 0020-0255/$ - see front matter 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.ins.2010.07.035
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GMP and CGMP have a lot of applications, therefore it is necessary to put forward a program to find a reasonable B* in (2) and (3). It is Zadeh who first investigated the GMP-like problem where the propositions A, A* and B, B* appeared in (2) were regarded as fuzzy subsets on universes X and Y, respectively. In this framework, (2) turns out to be
from AðxÞ ! BðyÞ and A ðxÞ; x 2 X; y 2 Y;
calculate B ðyÞ:
ð4Þ
We call (4) Fuzzy Modus Ponens (FMP) (see [19]). Then, in 1973, Zadeh proposed an algorithm called Composition Rule of Inference (CRI) (see [19]) to solve FMP. By CRI the conclusion of FMP can be written as:
B ðyÞ ¼ supfA ðxÞ ^ RðAðxÞ; BðyÞÞjx 2 Xg;
y 2 Y;
ð5Þ
where R:[0, 1]2 ? [0, 1] is an implication operator, such as Zadeh’s operator RZ, Łukasiewicz’s operator RŁ, Gödel’s operator RG, etc., a ^ b = min{a, b}. The well-known Triple I method is another algorithm to solve the problem of FMP proposed in [12,16], such that B* is the smallest fuzzy subset of Y satisfying
8x 2 X;
8y 2 Y;
ðAðxÞ ! BðyÞÞ ! ðA ðxÞ ! B ðyÞÞ ¼ 1:
It has been proven in [15,18] that the Triple I conclusion B* of FMP always exists and
B ðyÞ ¼ supfA ðxÞ ðAðxÞ ! BðyÞÞjx 2 Xg;
y 2 Y;
ð6Þ
where is a t-norm. Formulas like (5) and (6) are called numerical solutions of GMP. Many numerical solutions were proposed under different assumptions by different authors (see [7,9,11]). A more reasonable solution may be constructed from the syntactical point of view. One typical example is available in Wang’s work in classical logic (see [12]), where he proved that every finite theory has a root, and a numerical solution can be deduced based on this root. In this paper, from the syntactic point of view, we are going to discuss the logical foundation of GMP/CGMP in the four logics mentioned above. A generalized root of a theory in these logics is defined, and its properties are carefully discussed, in order to draw some inference rules, which can help us find algorithms for determining their numerical solutions. This paper is organized as follows: In the first section we discuss the aim of our work. Section 2 introduces preliminaries of this paper. The definition of a generalized root and its properties are given in Section 3, and in Section 4 we explain the concept of syntactical solutions of GMP and CGMP, and discuss GMP and CGMP’s generalized inference rule (syntactic-form solution) and some algorithms to compute numerical solutions. 2. Preliminaries 2.1. Logic systems: Łuk, Göd, P and L (NM) A t-norm is a binary operation on [0,1] satisfying the following conditions: (i) is commutative and associative, i.e., for all x, y, z 2 [0, 1], x y = y x, (x y) z = x (y z); (ii) is nondecreasing in both arguments, i.e., x1 6 x2 implies x1 y 6 x2 y, y1 6 y2 implies x y1 6 x y2; (iii) 1 x = x and 0 x = 0 for all x 2 [0, 1]. is a continuous t-norm if it is a t-norm and is a continuous mapping of [0, 1]2 into [0, 1]. An implication operator is a function R:[0, 1]2 ? [0, 1] satisfying the following conditions: (i) R(x, y) is non-increasing w.r.t. x and is non-decreasing w.r.t. y. (ii) R(x, y) = 1 iff x 6 y. (iii) R(x, y) is left continuous w.r.t. x and is right continuous w.r.t. y if the corresponding t-norm is continuous. It is well known that different implication operators and valuation lattices, L (i.e. the set of truth degrees for fuzzy logic), determine different logic systems (see [1,5,6,21]). In this paper, an implication operator R needs to fulfill the condition that there exists a t-norm such that:
x y 6 z iff
x 6 Rðy; zÞ:
In that case (, R) is called an adjoint pair.
ð7Þ
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Four popularly used implication operators and the corresponding t-norms are defined as follows:
( RL ðx; yÞ ¼
1;
ð1 x þ yÞ; x > y (
1; x 6 y
RG ðx; yÞ ¼
y; (
RP ðx; yÞ ¼ ( R0 ðx; yÞ ¼
x6y
x>y
x>y
x L y ¼ maxð0; x þ y 1Þ; x; y 2 L;
;
x G y ¼ minðx; yÞ; x; y 2 L;
;
x P y ¼ x y; x; y 2 L;
1; x 6 y y ; x
;
1;
(
x6y
ð1 xÞ _ y; x > y
;
x n y ¼
minðx; yÞ; x þ y > 1 0;
xþy61
;
x; y 2 L;
where n is called the standard nilpotent minimum t-norm (see [2]). These four implication operators are called Łukasiewicz implication operator RŁ, Gödel implication operator RG, Product implication operator RP and R0-implication operator R0, respectively. In 1995, Fodor [2,3] discovered a kind of new t-norm n (nilpotent minimum) and corresponding fuzzy implications based on discussion of the so-called contrapositive symmetry. He wrote, ‘‘By these propitious characteristics, the nilpotent minimum can be admitted into investigations of many theoretical and practical problems soon.” Since 1997, Wang has published some papers on fuzzy logic and fuzzy reasoning (see [12,14–17]). In [14], he proposed a fuzzy implication operator, called R0 implication, and constructed a new formal system L for fuzzy propositional calculus. We see that R0 implication is indeed a particular implication based on the standard nilpotent minimum. If we fix a t-norm , then a propositional calculus (whose set of truth values is L) is fixed: is taken for the truth function of the strong conjunction &, the residuum R of becomes the truth function of the implication operator and R (., 0) is the truth function of the negation. For more detail, we use the following definitions: Definition 1. [5,6]. The propositional calculus PC() given by a t-norm has the set S of propositional variables p1, p2, . . . and connectives :; &; !. The set F(S) of well-formed formulas in PC() is defined inductively as follows: each propositional variable is a formula; if A, B are formulas, then :A; A&B and A ? B are all formulas. Definition 2. [1,5,6,13]. The formal deductive systems of PC() given by corresponding to RL, RG, RP and R0, are called Łukasiewicz fuzzy logic Łuk, Gödel fuzzy logic Göd, Product fuzzy logic P and R0-fuzzy logic L (the nilpotent minimum logic NM), respectively. 1 A logic system is called an n-valued logic system if the valuation lattice L ¼ 0; n1 ; . . . ; n2 n1 ; 1 . A logic system is called a fuzzy logic system if the valuation lattice L = [0, 1]. The inference rule of each logic system above is Modus Ponens (MP): from A and A ? B infer B. The following formulas are axioms of the basic logic BL [6]:
ðA1Þ ðu ! wÞ ! ððw ! vÞ ! ðu ! vÞÞ; ðA2Þ ðu&wÞ ! u; ðA3Þ ðu&wÞ ! ðw&uÞ; ðA4Þ ðu&ðu ! wÞÞ ! ðw&ðw ! uÞÞ; ðA5aÞ ðu ! ðw ! vÞÞ ! ððu&wÞ ! vÞ; ðA5bÞ ððu&wÞ ! vÞ ! ðu ! ðw ! vÞÞ; ðA6Þ ðu ! wÞ ! vÞ ! ðððw ! uÞ ! vÞ ! vÞ; ! u: ðA7Þ 0 The following are Lukasiewicz’s axioms [6]: (Ł1) u ? (w ? u), (Ł2) (u ? w) ? ((w ? v) ? ((w ? u) ? u).
(u ? v)), (Ł3) ð:u ! :wÞ ! ðw ! uÞ, (Ł4) ((u ? w) ? w) ?
The axioms of P are those of BL plus [6] (P1) ::v ! ððu v ! w vÞ ! ðu ! wÞÞ; ðP2Þ u ^ :u ! 0. The axiom system of Gödel logic is the extension of the axiom system of BL by the single axiom (G) u ? (u&u) stating (together with (u&u) ? u) the idempotence of &.
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Axioms of R0 fuzzy logic L are [14]:
ðL 1Þu ! ðw ! uÞ; ðL 2Þð:u ! :wÞ ! ðw ! uÞ; ðL 3Þðu ! ðw ! vÞÞ ! ðw ! ðu ! vÞÞ; ðL 4Þðw ! vÞ ! ððu ! wÞ ! ðu ! vÞÞ; ðL 5Þu ! ::u; ðL 6Þu ! u _ w; ðL 7Þu _ w ! w _ u; ðL 8Þðu ! vÞ ^ ðw ! vÞ ! ðu _ w ! vÞ; ðL 9ðu ^ w ! vÞ ! ðu ! vÞ _ ðw ! vÞ; ðL 10Þ ðu ! wÞ _ ððu ! wÞ ! :u _ wÞ; where u ^ w ¼ :ð:u _ :wÞ: ´jek’s work on the basic logic and Höhle’s work on the monoidal logic, proposed a In 2001, Esteva and Godo, based on Ha new propositional calculus, called the monoidal t-norm based logic MTL and its two schematic extensions WNM and IMTL, called the weak nilpotent minimum logic and the involutive monoidal t-norm based logic, respectively. Furthermore, they obtained a natural common schematic extension NM of WNM and IMTL, called the nilpotent minimum logic NM (see [1]). Obviously, the system NM and the system L are strongly related. In fact, it is easy to verify that two systems are equivalent, and related algebras (NM-algebras and R0-algebras) are essentially the same kind of algebraic structures (see [10]). Remark 1. (i) One can further define connectives ^ and _ in these four logic systems, but the definition of ^ and _ in L are different from those in the other three logic systems: in L ; u _ w is an abbreviation of :ððððu ! ðu ! wÞÞ ! uÞ ! uÞ ! :ððu ! wÞ ! wÞÞ; u ^ w is that of :ð:u _ :wÞ. While in Łuk, Göd and P, ^ and _ can be defined uniformly as follows [6]: u ^ w is an abbreviation of u&(u ? w), and u _ w is that of ((u ? w) ? w) ^ ((w ? u) ? u). (ii) Define on L an unary operator and two binary operators as follows:
x ¼ Rðx; 0Þ;
x&y ¼ x y;
x ) y ¼ Rðx; yÞ;
x; y 2 L:
ð8Þ
where (, R) is an adjoint pair on L. From Remark 1 (ii), the valuation lattice L becomes an algebra of type (, , )). Defined in the above-mentioned logic systems
Am :¼ A&A& &A |fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl ffl};
ð9Þ
m
and in the corresponding algebras L
aðmÞ :¼ |fflfflfflfflfflfflfflfflfflfflffl aa ffla}; ffl{zfflfflfflfflfflfflfflfflfflfflffl
ð10Þ
m
where is the t-norm defined on L. Remark 2. [6,13]. It is easy to verify that the following assertions are true for every m 2 N: (i) (ii) (iii) (iv)
in in in in
Göd, a(m) = a. L ; aðmÞ ¼ að2Þ ðm P 2Þ. Łuk, a(m) = (ma (m 1)) _ 0. P, a(m) = am.
Definition 3. [6,13]. (i) A homomorphism v:F(S) ? L of type ð:; &; !Þ from F(S) into the valuation lattice L, that is, v ð:AÞ ¼ v ðAÞ, v(A&B) = v(A) v(B), v(A ? B) = v(A) ) v(B), is called an R-valuation of F(S). The set of all R-valuations will be denoted by XR. (ii) A formula A 2 F(S) is called a tautology with respect to R if "v 2 XR, v(A) = 1 holds. (iii) Let A,B 2 F(S). If t(A) = t(B) for every t 2 XR, then A and B are called logically equivalent, denoted A B.
Remark 3. [6,13]. It is not difficult to verify in the above-mentioned four logic systems that t(A _ B) = max {t(A), t(B)}, and t(A ^ B) = min{t(A), t(B)} for every valuation t 2 XR. Moreover, one can check in Łuk and L* that A&B and :ðA ! :BÞ are logically equivalent. Definition 4. [6]. (i) A subset of F(S) is called a theory.
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(ii) Let C be a theory and L is a logic system, A 2 F(S). A deduction of A from C in logic system L, in symbols, C ‘ LA, is a finite sequence of formulas A1, . . . , Am = A such that for each 1 6 i 6 m, Ai is an axiom of L, or Ai 2 C, or there are j, k 2 {1, . . . , i 1} such that Ai follows from Aj and Ak by MP. Equivalently, we say that A is a conclusion of C (or C-conclusion). The set of all conclusions of C is denoted by D(C). By a proof of A we shall henceforth mean a deduction of A from the empty set. We shall also write ‘LA in place of ;‘LA and call A a theorem. (iii) Let A, B 2 F(S). If ‘LA ? B and ‘LB ? A hold, then A and B are called provably equivalent in logic system L, denoted as A B. To simplify, we always replace symbol ‘LA with ‘A in this paper. Theorem 1. [1,6,13]. Completeness theorem holds in every above-mentioned logic system, i.e. "A 2 F(S), A is a theorem in Łuk, Göd, P and L (NM) iff A is a tautology in Łuk, Göd, P and L (NM), respectively. Theorem 1 points out that semantics and syntax in these four logic systems are in perfect harmony. 2.2. Generalized deduction theorems in Łuk, Göd, P and L (NM)
Theorem 2. [1,5,6]. Suppose that C is a theory, A, B 2 F(S), then in Łuk, Göd, P and L (NM), C [ {A} ‘ B iff $m 2 N, s.t. C ‘ Am ? B. Theorem 3. [13]. Suppose that C is a theory, A, B 2 F(S), then (i) in L* (NM) the generalized deduction theorem holds, i.e. C [ {A} ‘ B iff C ‘ A2 ? B. (ii) in Göd, deduction theorem holds, i.e. C [ {A} ‘ B iff C ‘ A ? B It is easy for the reader to check the following Lemma: Lemma 1. Let C be a theory and A 2 F(S). If C ‘ A, then there exists a finite subset of C, say, {A1, . . . , Am} such that {A1, . . . , Am} ‘ A. Note that A&B ? C and A ? (B ? C) are provably equivalent by the definition of the deduction theorem and the generalized deduction theorem. It is easy for the reader to check the following Lemma: Lemma 2. Suppose that C = {A1, . . . , Am} is a finite theory, A 2 F(S). Then (i) in Łuk, Göd, P and L (NM), C ‘ A iff $n1, . . . , nm 2 N such that ‘ ðAn11 & &Anmm Þ ! A. (ii) in Göd, C ‘ A iff ‘(A1& &Am) ? A. (iii) in L ; C ‘ A iff ‘ ðA21 & &A2m Þ ! A. The following Lemma can be obtained via Lemma 2.2.8 of [6,13]: Lemma 3. Let C be a theory and let A, B, C, D be formulas. In Łuk, Göd, P and L (NM), the following conclusions hold: (i) (ii) (iii) (iv) (v) (vi) (vii)
If C ‘ A, C ‘ B, then C ‘ A&B. If C ‘ A ? B, C ‘ C ? D, then C ‘ (A&C) ? (B&D). If C ‘ A ? B, then C ‘ (A&C) ? (B&C). ‘(A&B) ? B. If C ‘ A, C ‘ B, then C ‘ A ^ B. C ‘ ((A ? C) ^ (B ? C)) ? ((A _ B) ? C). C ‘ (A&B) ? B.
3. Generalized roots of theories 3.1. Basic definitions and properties First let us briefly review the necessary concepts (see [12]). (A1) In a logic system L, define a binary relation on F(S) as follows: A B iff ‘A ? B, A, B 2 F(S). Obviously is reflective and transitive and hence is a pre-order on F(S).
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(A2) In a logic system L, A is said to be a root of C iff A 2 D(C) and "B 2 D(C) A B, where D(C) is the set consisting of all conclusions of C. Reference [12] discusses some syntactical conclusions B* on problems like GMP and CGMP in classical logic, and gives definitions for solutions of GMP and a root of a theory C. Based on the root’s definition and its properties, reference [12] obtains the result that the Triple I solution B* of GMP is a root of C = {A ? B, A*}. That is, the Triple I solution B* of GMP is one of the smallest formula in (F(S), ) satisfying the condition that ‘(A ? B) ? (A* ? B*). This tells us that the definition of a root, as well as its properties, plays an important role in solving problems like GMP. Unfortunately, it has been proven that a theory C in Łukasiewicz propositional logic has no roots in general (see [20]). This section defines another form of root called a generalized root in Łukasiewicz propositional logic, Gödel propositional logic, Product propositional logic, and R0-propositional logic. We will show that the generalized root and the root defined in [12] have some properties in common. In addition to generalized root properties, we will also discuss the structure of the set of all the conclusions of C, called D(C), and the relations, specifically, inclusion and equality =, between two conclusion sets, D(C1) and D(C2), when different theories C1, C2 are given. Definition 5. Suppose that C is a theory and A 2 F(S). If A 2 D(C) and " B 2 D(C), $ m 2 N such that ‘Am ? B, then A is called a generalized root of C. T Definition 6. Suppose that Ck F(S),k = 1, 2, . . . , l. The members of lk¼1 DðCk Þ are called common conclusions of C1, . . . , Cl, Tl T and A is said to be a generalized common root of C1, . . . , Cl if A 2 k¼1 DðCk Þ, and 8B 2 lk¼1 DðCk Þ9m 2 N such that ‘ Am ? B. The following propositions hold in the four logic systems of concern: Proposition 1. Suppose that C is a theory and A 2 F(S). If A is a generalized root of C, then D(C) = D(A), where D(A) is an abbreviation for D({A}).
Proof. Suppose that A is a generalized root of C. Then " B 2 D(C), it follows from Definition 5 that $m 2 N such that ‘Am ? B holds. Since {A} ‘ Am by Lemma 3, it follows from the inference rule MP that {A} ‘ B. This means that B 2 D(A), hence D(C) # D(A). For the converse, " B 2 D(A), that is, {A} ‘ B, it follows from the generalized deduction theorem that $m 2 N such that ‘ Am ? B. C ‘ Am follows from the assumption C ‘ A. Therefore we have C ‘ B by MP, and D(A) # D(C). h Proposition 2. Suppose that A 2 F(S). Then D(A) = {B 2 F(S)j $m 2 N, ‘Am ? B}. Proof. Let V = {B 2 F(S)j$m 2 N, ‘ Am ? B}." C 2 D(A), it follows from the generalized deduction theorem that $ m 2 N such that ‘Am ? C. Thus C 2 V and D(A) # V. For the converse, "B 2 V, $m 2 N such that ‘ Am ? B. It is easy to prove that {A} ‘ Am and it follows from the inference rule MP that {A} ‘ B holds, that is, B 2 D(A). Hence V # D(A) and so D(A) = V. h Theorem 4. Every finite theory has a generalized root. More precisely, A1& &As is a generalized root of theory C = {A1, . . . , As}. Proof. It is only necessary to prove that A1& &As is a generalized root of C. By Lemma 3, A1& &As 2 D(C). "B 2 D(C), it follows from Lemma 2 that there exists n1, . . . , ns 2 N such that ‘ ðAn11 & &Ans s Þ ! B. Let m = max{n1, . . . , ns}. We have ‘ ðA1 & &As Þm ! ðAn11 & &Ans s Þ by Lemma 3. Hence by Hypothetical Syllogism, we get ‘(A1& &As)m ? B. Thus A1& & As is a generalized root of C. h Proposition 3. Suppose that Ck # F(S) and Ak is a generalized root of Ck, k = 1, . . . , l. Then A1& &Al, is a generalized root of Sl k¼1 Ck .
C¼
Proof. It is easy to prove that C ‘ Ai, i = 1, . . . , l by assumption. Hence by Lemma 3, we get C ‘ A1& &Al, that is, mk 1 A1& &Al 2 D(C). "B 2 D(C), there exists B1, . . . , Bk 2 C and m1, . . . , mk 2 N such that ‘ ðBm 1 & &Bk Þ ! B by Lemmas 1 m s si 1 s1 and 2. " Bi (i = 1, . . . , k), there exist ji 2 {1, . . . , l} and si 2 N such that ‘ Aji ! Bi , where Bi 2 Cji . ‘ ðAm & &Ajk k k Þ ! j1 mk dl mk sk d1 m1 s1 m1 ðB1 & &Bk ) by Lemma 3. Clearly, there exists d1, . . . , dl 2 N such that ‘ ðA1 & &Al Þ ! ðAj1 & ; &Ajk Þ. Let mk 1 d = max{d1, . . . , dl}. By Lemma 3 and Hypothetical Syllogism, we have ‘ ðA1 & &Al Þd ! ðBm 1 & &Bk Þ. Also by Hypothetical k Syllogism ‘(A1& &Al) ? B is yielded. This shows that A1& &Al is a generalized root of C. h Proposition 4. Suppose that A and B are generalized roots of C1 and C2, respectively. Then D(C1) # D(C2) iff $ m 2 N, such that ‘Bm ? A.
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Proof. The proposition above directly follows from Definition 5, Proposition 2 and Lemma 3. Corollary 1. Suppose that ‘(B1& &Bt)m ? (A1& &As).
C1 = {A1, . . . , As}
and
C2 = {B1, . . . , Bt}.
Then
D(C1) # D(C2)
h iff
$m 2 N,
such
that
Proof. By Theorem 4 and Proposition 4, it is easy to check the corollary above. h 3.2. Results in the n-valued Łukasiewicz logic It has been proven that the semantics and syntax of the n-valued Łukasiewicz logic are in perfect harmony (see [4]). It is then easy to prove the following propositions: Proposition 5. Suppose that m,n 2 N. If m P n 1, then "a 2 L, a(m) = a(n1) holds.
Proposition 6. Suppose that A 2 F(S). If m P n 1, then Am An1. Theorem 5. Suppose that C is a theory and A 2 F(S). If A is a generalized root of C, then D(C) = {B 2 F(S)j‘An1 ? B}. Proof. The theorem above directly follows from Propositions 2 and 6. h Theorem 6. Suppose that Ai is a generalized root of Ci (i = 1, 2, . . . , k), then
Wk
n1 i¼1 Ai
is a generalized common root of C1, . . . , Ck.
W ! ki¼1 An1 ðt ¼ 1; . . . ; kÞ and An1 2 DðCt Þ, it follows from the inference rule MP that Proof. By Lemma 3, ‘ An1 t i t Wk n1 T 2 DðCt Þði ¼ 1; . . . ; kÞ. Since 8B 2 ki¼1 DðCi Þ; ‘ An1 ! Bði ¼ 1; . . . ; kÞ by Theorem 5, then we have ‘ ðAn1 ! BÞ i¼1 Ai i 1 n1 n1 n1 n1 n1 n1 ^ ^ ðAn1 ! BÞ. Since ðA ! BÞ ^ ^ ðA ! BÞ ðA _ _ A ! BÞ. Obviously ‘ A _ _ A ! B holds, k 1 k 1 k 1 k therefore An1 _ _ An1 is a generalized common root of C1, . . . , Ck. h 1 k 3.3. Results in the Gödel fuzzy logic
Proposition 7. Suppose that A is a generalized root of C. Then D(C) = {B 2 F(S)j‘A ? B}.
Proof. The proposition above directly follows from the definition of the generalized root and the deduction theorem. h Proposition 8. Suppose that A and B are generalized roots of C1 and C2, respectively. Then D(C1) # D(C2) iff ‘B ? A. Proof. The proposition above directly follows from Propositions 4 and 7. h Theorem 7. Suppose that Ck # F(S) and Ak is a generalized root of Ck(k = 1, 2, . . . , m). Then C1, . . . , Cm.
Wm
Proof. The proof is analogous to that of Theorem 6. h 3.4. Results in the R0- fuzzy logic L
Lemma 4. [13]. Suppose that A 2 F(S). Then Am A2, m = 2, 3, . . .
Proposition 9. Suppose that A is a generalized root of C. Then D(C) = {B 2 F(S)j‘A2 ? B}. Proof. The proof is analogous to that of Theorem 5. h
i¼1 Ai
is a generalized common root of
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Proposition 10. Suppose that A and B are generalized roots of C1 and C2, respectively. Then D(C1) # D(C2) iff ‘B2 ? A2. Proof. The proposition above directly follows from Lemmas 3, 4 and Proposition 9. h Theorem 8. Suppose that Ck # F(S) and Ak is a generalized root of Ck(k = 1, 2, . . . , m). Then of C1, . . . , Cm.
Wm
2 i¼1 Ai
is a generalized common root
Proof. The proof is analogous to that of Theorem 6. h 4. Definitions and expressions of the solutions of GMP and CGMP in Łuk, Göd, P and L (NM) Now we turn to the problems of GMP and CGMP. Our aim is to find a suitable conclusion B* of the prerequisites A ? B and A*. A ? B, A* and B* should certainly satisfy the following condition:
fA ! B; A g ‘ B :
ð11Þ
Obviously there are too many B* that satisfy (11). For example, (11) holds if B* is a tautology (theorem), but it is clearly not what we need since a tautology is a conclusion of any A ? B and A* is given as prerequisites. Hence, a tautology is not a suitable conclusion for GMP. By (A1), it is reasonable to ask for B* in GMP to be as small as possible. Definition 7. Let A, B, A*, B* 2 F(S). B* is called a Triple I2 solution of GMP if it satisfies the following conditions: (G1) {A ? B, A*} ‘ B*; (G2) if C satisfies (G1), then B* C (C 2 F(S)).
Theorem 9. Let C = {A ? B, A*}. Then Triple I2 solution of GMP B* is a generalized root of C. Proof. By Definition 7 (G1), B* 2 D(C). Since A*&(A ? B) is a generalized root of C by Theorem 4, then, by Definition 5, "C 2 D(C), and there is an m such that ‘((A ? B)&A*)m ? C and (A ? B)&A* 2 D(C). ‘B* ? ((A ? B)&A*) by Definition 7 (G2), thus, ‘ (B*)m ? ((A ? B)&A*)m by Lemma 3. It follows from Hypothetical Syllogism that ‘ (B*)m ? C, that is, B* is a generalized root of C. h Theorem 10. Suppose that C = {A ? B, A*}, A, B, A* 2 F(S). Then (i) in the n-valued Łukasiewicz logic Łn, the generalized root B* = (A*)n1&(A ? B)n1 of C is a Triple I2 solution of GMP. (ii) in L*, the generalized root B* = (A*)2&(A ? B)2 of C is a Triple I2 solution of GMP. (iii) in Gödel, the generalized root B* = A*&(A ? B) of C is a Triple I2 solution of GMP.
Proof. Because the proofs of (i), (ii), and (iii) are all simiar, here we only prove (i). In the n-valued Łukasiewicz logic Ln, since the deduction theorem means that
C [ A ‘ B iff C ‘ An1 ! B;
C FðSÞ;
A;
B 2 FðSÞ:
ð12Þ
Moreover, it is well known in Ln that
A ! ðB ! CÞ A&B ! C;
A;
B;
C 2 FðSÞ;
where is the logical equivalence relation. Since ‘ (A*) that the following assertions are also true:
ð13Þ n1
&(A ? B)
n1
? B* is clearly true, and it follows from (12) and (13)
‘ ðA Þn1 ! ððA ! BÞn1 ! B Þ; fA g ‘ ðA ! BÞn1 ! B ; fA ! B; A g ‘ B : Hence B* satisfies condition (G1). Assume that C satisfies (G1), that is, {A ? B, A*} ‘ C, C 2 F(S). Then it follows from the deduction theorem of Ln that
fA g ‘ ðA ! BÞn1 ! C; ‘ ðA Þn1 ! ððA ! BÞn1 ! CÞ: Therefore it follows from (13) that ‘(A*)n1&(A ? B)n1 ? C and ‘B* C, i.e. B* satisfies condition (G2). The proof of Theorem 10 is complete. h
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Definition 8. Let Ci ¼ fAi ! B; Ai g and Ai ; B; Ai ; B 2 FðSÞði ¼ 1; 2; . . . ; mÞ. B* is called a Triple I2 solution of CGMP if it satisfies the following conditions: (D1) Ci ‘ B*(i = 1, 2, . . . , m); T (D2) if C 2 m k¼1 DðCk Þ, then B* C, C 2 F(S).
Remark 4. It is easy to verify that if B* is a Triple I2 solution of CGMP, then B* is a generalized common root of C1, . . . , Cl by Definitions 5 and 8. Theorem 11. Suppose that Ci ¼ fAi ! B; Ai g; Ai ; B; Ai 2 FðSÞ; i ¼ 1; 2; . . . ; m . Then (i) in the n-valued Łukasiewicz logic Ln,
ððA1 Þn1 &ðA1 ! BÞn1 Þ _ _ ððAm Þn1 &ðAm ! BÞn1 Þ
ð14Þ
is a Triple I2 solution of CGMP. (ii) in L*,
ððA1 Þ2 &ðA1 ! BÞ2 Þ _ _ ððAm Þ2 &ðAm ! BÞ2 Þ
ð15Þ
is a Triple I2 solution of CGMP. (iii) in Gödel,
ðA1 &ðA1 ! BÞÞ _ _ ðAm &ðAm ! BÞÞ
ð16Þ
is a Triple I2 solution of CGMP.
Proof. Because the proofs of (i) (ii), and (iii) are all similar, here we only prove (i). In the n-valued Łukasiewicz logic Łn, Ai &ðAi ! BÞ is a generalized root of Ci by Theorem 4, and ððA1 Þn1 &ðA1 ! BÞn1 Þ _ _ ððAm Þn1 &ðAm ! BÞn1 Þ is a generalized common root of C1, . . . , Cm by Theorem 6. Then ððA1 Þn1 &ðA1 ! BÞn1 Þ _ _ ððAm Þn1 &ðAm ! BÞn1 Þ satisfies condition (D1). T 8C 2 m we have ‘ ðAi Þn1 &ðAi ! BÞn1 ! C (i = 1, 2, . . . , m) by Theorem 5, thus ‘ ðððA1 Þn1 & k¼1 DðCk Þ, n1 ðA1 ! BÞ Þ _ _ ððAm Þn1 &ðAm ! BÞn1 ÞÞ ! C , then ððA1 Þn1 &ðA1 ! BÞn1 Þ _ _ ððAm Þn1 &ðAm ! BÞn1 Þ C . The proof of Theorem 11 (i) is complete. h 5. Semantic type Triple I2 solutions of GMP and CGMP
Remark 5. [12]. Suppose that XR is the set consisting of all valuations from F(S) to L (see Definition 3), where L is the set of truth degrees for fuzzy logic and F(S) is the set of well-formed formulas (see Definition 1). Suppose that t 2 XR and A 2 F(S), then t(A) is the value of t at A. Let us also agree to use the symbol A(t) to denote t(A). A formula A induces a mapping A:XR ? L (denoted by A, itself) naturally defined by "t 2 XR, A(t) = t(A) where t(A) is the value of t at A. By Theorems 9, 10 and Remark 5, the following algorithms are obvious. Lemma 5. The FMP-type Triple I2 solution B* of GMP can be computed as follows: (i) in the n-valued Łukasiewicz logic Ln, B*(v) = sup{(A*(u))n1 (A(u) ? B(v))n1ju 2 X}, (ii) in L*, B*(v) = sup{(A*(u))2 (A(u) ? B(v))2}, v 2 Y; (iii) in Gödel, B*(v) = sup{A*(u) (A(u) ? B(v))}, v 2 Y.
v 2 Y;
Lemma 6. The FMP-type Triple I2 solution B* of CGMP can be computed as follows: (i) in the n-valued Łukasiewicz logic Ln, B ðv Þ ¼ supfððA1 ðuÞÞn1 ðA1 ðuÞ ! Bðv ÞÞn1 Þ _ _ ððAm ðuÞÞn1 ðAm ðuÞ ! Bðv ÞÞn1 Þju 2 Xg; v 2 Y ; (ii) in L*, B ðv Þ ¼ supfððA1 ðuÞÞ2 ðA1 ðuÞ ! Bðv ÞÞ2 Þ _ _ ððAm ðuÞÞ2 ðAm ðuÞ ! Bðv ÞÞ2 Þju 2 Xg; v 2 Y; (iii) in Gödel, B ðv Þ ¼ supfððA1 ðuÞ ðA1 ðuÞ ! Bðv ÞÞÞ _ _ ðAm ðuÞ ðAm ðuÞ ! Bðv ÞÞÞju 2 Xg; v 2 Y.
Example 1. Suppose that X = Y = [0, 1], A, A* 2 F(X), B 2 F(Y) are as follows: AðuÞ ¼ uþ2 3 ; Bðv Þ ¼ 1 v ; A ðuÞ ¼ 1 u and u, v 2 [1, 0]. In the 7-valued Łukasiewicz logic L7, compute the FMP-type Triple I2 solution B* of GMP by using Lemma 5.
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Solution. Since A ðuÞ ðAðuÞ ! Bðv ÞÞ ¼ ð1 uÞ and the corresponding t-norm in this logic that
uþ2 3
! ð1 v Þ , it follows from the definition of implication operator
(
4u3v ; v > 1u uþ2 3 3 : þ1v ^1¼ AðuÞ ! Bðv Þ ¼ 1 3 1; v 6 1u 3 8 96u18v 1u ; 3 < v < 32u > 3 6 < 6 ðAðuÞ ! Bðv ÞÞ ¼ 1; : v 6 1u 3 > : 32u 0; vP 6 ( 1 1 6u; u 6 6 : ðA ðuÞÞ6 ¼ 0; u P 16 8 1u 38u 1 > < 3 8u 6v ; 3 < v < 6 ; u < 6 6 6 ðA ðuÞÞ ðAðuÞ ! Bðv ÞÞ ¼ 1 6u; : v 6 1u ; u < 16 3 > : 0; otherwise 8 v 6 13 > < 1; 6 6 B ðv Þ ¼ supfðA ðuÞÞ ðAðuÞ ! Bðv ÞÞ ju 2 ½0; 1 g ¼ 3 6v ; 13 < v < 12 : > : 0; otherwise
Example 2. Suppose that X, Y, A, A*, and B are the same as those in Example 1. In L*, compute the FMP-type Triple I2 solution B* of GMP by using Lemma 5. Solution. Since
(
2
ðA ðuÞÞ ¼
1 u; u < 12 ; 0;
8 > 3 ; 3 > : 1 v ; 1u < v 6 2þu : 3 3 8 1u v6 3 ; 1; > > > < 0; v > 2þu ; 3 ðAðuÞ ! Bðv ÞÞ2 ¼ 1u > 1 v ; 3 < v 6 2þu ; v < 12 ; > 3 > : 1u 2þu 0; < v 6 3 ; v P 12 : 83 1u 2þu 1 > < 1 u; 3 < v 6 3 ; u < 2 ; v 6 u; 2 2 ðA ðuÞÞ ðAðuÞ ! Bðv ÞÞ ¼ 1 v ; v 6 12 ; u < v ; 1u < v 6 2þu ; 3 3 > 8 > < 1; B ðv Þ ¼ 1 v ; > : 0;
:
0;
or u < 12 ; v 6 1u ; 3
otherwise:
v 6 13 ; 1 < v 6 12 ; 3 v P 12 :
Example 3. Suppose that X, Y, A, A*, and B are the same as those in Example 1. Compute the FMP-solutions B* by using Zadeh’s CRI method, and the conclusion of FMP is as follows[13]:
B ðv Þ ¼ supfA ðuÞ ^ ðAðuÞ ! Bðv ÞÞju 2 Xg;
v 2 Y;
where the implication operator is a Łukasiewicz’s implication operator. Solution. Since A ðuÞ ^ ðAðuÞ ! Bðv ÞÞ ¼ ð1 uÞ ^ uþ2 ! ð1 v Þ , it follows from the definition of implication operator 3 that
( 4u3v ; uþ2 3 þ 1 vÞ ^ 1 ¼ AðuÞ ! Bðv Þ ¼ ð1 3 1; ( 4u3v 1þ2u ; v> 3 ; 3 A ðuÞ ^ ðAðuÞ ! Bðv ÞÞ ¼ 1 u; v 6 1þ2u : 3 Therefore we have B ðv Þ ¼
4
v; 1;
3
v > 13 ; v 6 13 :
v > 1u ; 3 1u v6 3 :
Author's personal copy J. Zhang, X. Yang / Information Sciences 180 (2010) 4661–4671
4671
6. Concluding remarks This paper proposes a new concept called a generalized root of a theory, C, in Łukasiewicz propositional fuzzy logic Łuk, Gödel propositional fuzzy logic Göd, Product propositional fuzzy logic P, and the nilpotent minimum logic NM (the R0-propositional fuzzy logic L ). This concept is based on the generalized deduction theorem and the completeness theorem. Then it is proved that D(C) is completely determined by its generalized root whenever C has a generalized root. In Łukasiewicz propositional fuzzy logic, the n-valued Łukasiewicz propositional logic, Gödel propositional fuzzy logic, Product propositional fuzzy logic and the nilpotent minimum logic (the R0-propositional fuzzy logic), every finite theory has a generalized root. The conditions for the sets D(C) of two different theories being equal and included are given. Finally, syntactic-form conclusions of problems like GMP, CGMP were drawn in Łuk, Göd, P and NM ðL Þ, via the generalized root of a theory, C, and their numerical versions are provided as new algorithms for solving GMP and CGMP. Acknowledgment The authors would like to express their sincere thanks to the anonymous referee for his or her valuable suggestions. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21]
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