Reasoning with propositional knowledge based ... - RIT Scholar Works
Rochester Institute of Technology
RIT Scholar Works Articles
1996
Reasoning with propositional knowledge based on fuzzy neural logic Wangming Wu Hoon-Heng Teh Bo Yuan
Follow this and additional works at: http://scholarworks.rit.edu/article Recommended Citation Wu, Wangming; Teh, Hoon-Heng; and Yuan, Bo, "Reasoning with propositional knowledge based on fuzzy neural logic" (1996). International Journal of Intelligent Systems - John Wiley & Sons, Inc, Vol. 11 (), pp. 251-265. Accessed from http://scholarworks.rit.edu/article/993
This Article is brought to you for free and open access by RIT Scholar Works. It has been accepted for inclusion in Articles by an authorized administrator of RIT Scholar Works. For more information, please contact [email protected].
Reasoning with Propositional Knowledge Based on Fuzzy Neural Logic Wangming Wu and Hoon-Heng Teh
Institute of Systems Science, National University of Singapore, Singapore 0511 Bo Yuan
Department of Systems Science and Industrial Engineering, Thomas J. Watson School of Engineering, Binghamton University (SUNY), Binghamton, New York 13902
In this article, a new kind of reasoning for propositional knowledge, which is based on the fuzzy neural logic initialed by Teh, is introduced. A fundamental theorem is presented showing that any fuzzy neural logic network can be represented by operations: bounded sum, complement, and scalar product. Propositional calculus of fuzzy neural logic is also investigated. Linear programming problems risen from the propositional calculus of fuzzy neural logic show a great advantage in applying fuzzy neural logic to answer imprecise questions in knowledge-based systems. An example is reconsidered here to illustrate the theory. © 1996 John Wiley & Sons, Inc.
I. INTRODUCTION Fuzzy logic, attributable to Zadeh,' is basically a logic for approximate reasoning. As a generalization of classical two-valued logic, fuzzy logic extends the range of truth value from {0, to [0, 1]. The basic logical connectives AND, OR, and NOT in fuzzy logic are interpreted as t-norms, t-conorms, and complements (e.g., Wu 2). By such a generalization, a classical two-valued logic system becomes a special case of fuzzy logic systems. In recent years, a new kind of logic named fuzzy neural logic has been introduced by Teh. 3 Informally, a fuzzy neural logic is a kind of continuousvalued logic that includes the bounded weighted averages as its basic operation. Notice that a bounded weighted average (or convex combination) is an aggregate operation used in many practical problems. However, a bounded weighted average is not used to interpret AND or OR logical connectives, because they are not associative. This article shows that every operation in Kleene–Lukasiewicz fuzzy logic can be implemented by some simple fuzzy neural logic networks. This means Kleene–Lukasiewicz fuzzy logic is a special case of fuzzy neural logic discussed here. Fuzzy neural logic has a bounded weighted average as its addiINTERNATIONAL JOURNAL OF INTELLIGENT SYSTEMS, VOL. 11, 251-265 (1996) © 1996 John Wiley & Sons, Inc. CCC 0884-8173/96/050251-15
PROPOSITIONAL KNOWLEDGE
(a) Bounded sum
(c) Lukasiewicz implication
253
(b) Bounded product
(d) Complement
Figure 2. Basic logical operations implemented by neurons
threshold function f, which is defined by
(1) This function is also called the linear threshold function. Clearly, b E [0, 11. In particular, we can implement the bounded sum ED, the bounded product C), the Lukasiewicz implication and the complement c* by following neurons shown in Figure 2.
254
WU, TEH, AND YUAN
In general, if w i , . . . w„ are real numbers, then there exists a natural number N = 1 + Imax(Iwi I, • • • , Iwn1).1 such that w, = c,N, where [x] means the integer part of real number x and c, = w,IN E [-1, 1], i 1, . . . , n. Then we have f(E= I wi) = f (1\1 /7=1 aicil = fiN 17= 1 ai lci lsgn(c,)) E [0, 1]. Therefore, any neuron can be decomposed as a combination of scalar product operations and the simple neurons with connection weights either 1 or –1. (Fig. 4) Note that the slope threshold function f can be represented by f(x) = max(min(x, 1), 0) = min(max(x, 0), 1). Thus, we have
and Proof. We only prove Eq. 2 here. Equation 3 can be proved in a similar way. By Eq. 1, we have
256
WU, TEH, AND YUAN
Figure 5. The dual of a neuron
can also be represented by compositions of logical operations ED, (:), and c. Since a 1 C) a2 = (di' ED aZY , we have the following fundamental theorem. THEOREM 1. (the fundamental theorem). Every fuzzy neural logic network can be represented by the composition of logical operations ED , , and scalar product operations. Let us consider the dual of a neuron shown in Figure 5.
Figure 6. The equivalent network of a network in Figure 5
PROPOSITIONAL KNOWLEDGE
257
Figure 7. Two special cases of Theorem 2.1.
The networks in Figure 7 are two special cases of the dual theorem.
III. PROPOSITIONAL CALCULUS OF FUZZY NEURAL LOGIC SYSTEMS We are now ready to develop a propositional system of fuzzy neural logic. The symbols, with or without subscripts or primes, such as X, Y, that are used to denote propositions (i.e., sentences) are called atomic formulas, or atoms for short. From propositions, we can build compound propositions by using logical connectives. In fuzzy neural logic, we use five logical connectives: (negation), & (conjunction), Y (disjunction), —> (implication), (equivalence), and a class of scalar product connective po„ where a E [0, 1]. These connectives can be used to build compound propositions from atoms. More generally, we can construct more complicated compound propositions by applying them repeatedly. These compound propositions are called well-formed formulas. More formally, we give the following definition.
258
WU, TEH, AND YUAN
PROPOSITIONAL KNOWLEDGE
259
260
WU, TEH, AND YUAN
(A 1 , . . . , A n ) x/4_, if every valuation v such that v(A 1 ) = • • • = v(A n) = 1, the inequality v(A) holds. Whenever A = 1, we denote (A 1 , . . . A n) A instead of (11 1 , . . , A n) 1 A. THEOREM 5. If
& A n) —> A, then (A 1 , . . . , An )
(A i & •
A.
Proof. If (A 1 & • • • & An) --> A, then for every valuation v, v(A 1 & • • • & An) —> A) = (v(A l & • • & A n))` ED v(A) = (v(A 1 )` e • • • ED v(A n)` v(A)) A. Now, v is a valuation such that v(A i ) = • • • = v(A„) = 1. Thus, v((A 1 & • • • & An) ---> A) = 0 v(A) = v(A) A. This completes the proof. Remark 6. The inverse of Theorem 5 is not true. For example, let A 1 = X, A = X & X, where X is an atom. If v is a valuation such that v(A 1 ) = v(X) = 1, then v(A) = v(X & X) = v(X) C)v(X) = 1. Thus, (A 1 ) But A I ---> A is not a tautology (e.g., put v(X) = 0.5, then v(A 1 ----> A) 0.5 (0.5 Qx 0.5) = 0.5 1). THEOREM 6. Let A E [0, 1] and A, B, C, A 1 , . . . , A n , B 1 , . . . , B, be formulas, then
(15) (16) (17) (18)
(Modus ponens) (A, A --> B) B. (Modus tollens) (A --> B, (A --> B, B (Syllogism) C) A —> C. (Mean preserving)(A 1 --> B 1 , . . • An —> M,(.13 1 , . . . , B„).
Proof. By the Theorem 5, over, for any valuation v, v(((A
(15)
follows from (3).
Mn(A,, . . . , A„) —>
(16)
follows from (2). More-
B) & (B ----> C)) ---> (A —> C)) = {((v(A)) c 8) v(B)) C) ((v(B))` = (v(A)
v(C))}` ED ((v(A))c
(v(B))`) ED (v(B) ® (v(C))()
v(C))
((v(A))( ED v(C))
= ((v(A)) c V (v(B))`) El) (v(B) V v(C)) = (v(A) A v(B))( (v(B))` ED v(B) =
(v(B) V v(C)) 1.
Thus, we have (17). Finally, for any valuation v, let a, = v(A;), b, = v(B,), i = 1, . . . , n. & • • • & (A„--> Bn))--- > (WA] = ((d; ED b1) C) • • C) (an
, 4,) —› Mn(B I , .
b„))` El) ((a 1 + • • + a n)ln)c
13,)))
(b 1 + • • • + b„)In
= (a l C) b;) CD • • • ED (an C) b`r,') e (al/nED • • • ED (41 n) e (b i l n ED • • • ED bn l n) = (a 1 C)
di' I n
b i lne • • • e(a„® bc,) e a nc i n 8) bn l n
lln + • • • + 1/n = 1,
PROPOSITIONAL KNOWLEDGE
261
WU, TEH, AND YUAN
262
(ix) ),(p„XE)' iff (ax9' -X 0 (where x 1 = x, x° = xc). 1 (i.e., ax O py (x) ),(p,„X & poY) iff ax + /3y - A). A). (xi) x(jo,„X YpoY) iff ax + /3y - A 0 (i.e., ax py 1. (i.e., ax /3y A). ppY) if ax - /3y + (xii) (xiii) kx -> pp(-,Y)) iff ax - Ay + A :5- 1 - /3. (i.e., ((ax)` pyc)` ._ A). (2) Suppose A is a fuzzy clause, A = (p ai X19 8 ' V • • • (p„A•) 8 ", where e„ E {0, 1}, Xi are atoms, i = 1, . . . , n. Denote x, = v(X,), then + • • • +
iff
(xix)
-X
0.
Therefore, if A is a local implication formula, H.>21 is equivalent to a linear inequality on the fuzzy variables. (p„,,X,,n)'n, where 8i, (3) Suppose A is a fuzzy phrase, A = (pa ,X;0 8, Y • • • 8; E {0, 1}, X, are atoms, i = 1, . . . , n. Denote x i = v(Xi ), then (xx) xA iff (a lxVi + • • • + (a„4. ) 8,• -A
n - 1.
(4) Let w 1 , . . . , w„ E [0, 1] and w 1 + • • + w„ = 1; s i . . . , sm E [0, 1] and Si +• • • + sn, = 1. A = p„„Xl & • • • &p„,„X,, and B = p,Y„,. Then Y• • (xxi) A AA
B iff w i x t + • • • + wnx„ - s l Y i - • • - sm y„, + X
1•
Now, we give a main theorem of this section. X i . • , X ni . • . Y,n be atoms; Pi = (p„,X1'0 8•1 & • • (pa,,A-18-, where au, eu, 6ij E [0, 1], for i = 1, . . . , k, j = 1, . . . , n; Ci = (p0,Y1,)"'m • • • (pa,,,,rur)11,-, where 14, yii , E [0, 1], for i = 1, . . . , k,j = 1, . . . , m; Pi Ci is a local implication formula, for i = 1, . . . k; w 1 , . . . , W n E [0, 1] and w i + • • + wn = 1; s i , . . ,s„, E [0, 1] ands, + . . . + s u = 1. Then to find the maximum A E [0, 1] such that the whole implication formula A B is a A-logical consequence of the local implication formulas PI C 1 , . . . , Pk Ck is equivalent to find A = 1 - max(w i x i + . . . wnxn s l y ' - . • • - smY., 0), where (x i , . . . , ; Yi9 • • • y m ) is an optimal solution of the following linear programming problem:
THEOREM 7. Let
(LP) s .t.
max w i x i +" • • + W n Xn SIYI - • • • -SnzYm (a d 49 1 8 ' ' • + (a,„x0 1-8- + ([3, 1 y1. )Thi + • 1, . . . , k; i . . , n; 0 1, t = 0 yu 1, u = 1, , tn.
+ (/3mm7"" Y
Proof. By Definition 6, (P I --> CI, • • • P k -) Ck)(A -> B) means: for any valuation v such that v(P,---> C I ) = • • • = v(13 k Ck) = 1, v(A B) holds. Let x t = v(Xt), Yu = t = 1, . . , n; u = 1, . . . , m. Clearly, X t ' Yu E [0, 1], for t = 1, . . . , n, u = 1, . . . , m. Now, v(P,--> C,) = 1 if and only if (ail .4,9 1
+ • • • + (amxsin)'-'m n + (Ail Yin)Thr + •. • • + (PimY,),';'") Th'" 1, i = 1, . . . , k;
and v(A ---> B) wl xi
if and only if + • • • +
wn x„ - s l y] -
• • • - s mY m +
X
1.
Therefore, if A is the maximum such that the whole implication formula A -> B is a A-logical consequence of the local implication formulas P1 -->
PROPOSITIONAL KNOWLEDGE
263
IV. AN EXAMPLE In this section, we apply our approach to an example presented by Castro et al. 5 Here is their example: Suppose the knowledge base is stated as a set of statements in the propositional calculus: (1) If the bond market goes up and the interest rates decrease, either the stock market goes down or taxes are not raised; (2) If the political situation is unstable and the currency is devalued, the inflation increases; (3) If the interest rates increase and the bond market goes down, the stock goes up; (4) If the labor situation is unstable and the political situation is stable, the bond market goes up; (5) If the inflation decreases and the labor situation is unstable, the currency is devalued or political situation is stable; (6) If the stock market goes up, at least one favorable condition exists; (7) If the public deficit increases, and labor situation is unstable, the inflation increases; (8) If the public deficit increases and taxes are raised, the inflation increases; (9) If the political situation is unstable and the employed decrease, the stock market goes down; (10) If the public deficit decreases and taxes are not raised, the interest rate increases; (11) If currency is devalued then the public deficit increases or taxes are raised.
We can represent atoms and their negations in Table I. The statements (1)–(11) can be written as the following formulas:
264
WU, TEH, AND YUAN
Table 1. Atoms and their negations in the example. Economical Evolution The stock market goes up. The stock market goes down. The bond market goes up. The bond market goes down. The inflation decreases. The inflation increases. Currency is revalued. Currency is devalued. The interest rates increases. The interest rates decreases.
Country Situation Labor situation is stable. : Labor situation is unstable. Political situation is stable. : Political situation is unstable. Taxes are not raised. : Taxes are raised. The public deficit decreases. : The public deficit increases. The employed increases. The employed decreases. 5:
A positive economical evolution may be defined a, X5 ), so a negative economical evolution will be defined as Similarly, a favorable country situation may be defined as B and an unfavorable country situation as —B = —M5(171, „, Does a positive economical evolution imply a favorable country situation? In other words, does A —> B? It should be noted that A 1 , . . . , A 11 are all local implication formulas. But A —> B is a whole implication formula. To find the maximum X such that (A 1 , . . . , A 11 ) x(A B), by Theorem 7, is equivalent to solve the following linear programming problem (LP):
PROPOSITIONAL KNOWLEDGE
265
An optimal solution of this (LP) problem is The optimal value is 0.6. It follows that "a positive economic evolution implies a favorable country situation" is a 0.4logical consequence of the knowledge base. In Castro et al., 5 the authors applied fuzzy Boolean programming to solve the above problem. They used 20 Boolean variables based on classical twovalued logic. Our approach is based on fuzzy neuron logic and uses only 10 fuzzy variables. The methodology based on fuzzy neural networks and linear programming is more reasonable, because knowledge represented in a knowledge base may not be precise.
V. CONCLUSIONS In this article, based on the fundamental theorem of fuzzy neural logic, we establish the propositional calculus of fuzzy neural logic. Linear programming approach taken from the propositional calculus of fuzzy neural logic is a useful tool to answer imprecise questions in knowledge-based systems. It seems possible to use network approach to solve logical problems. We will investigate this interesting question in a future paper.
References 1. L.A. Zadeh, "Fuzzy logic and approximate reasoning," Synthese, 30, 407-442 (1975). 2. W.M. Wu, "A multivalued logic system with respect to T-norms," in M.M. Gupta and T. Yamakawa, Eds., Fuzzy Computing, North-Holland, Amsterdam, 1988, pp. 101-118. 3. Real World Computing Partnership Neuro ISS Laboratory, "A new class of neural networks called neural logic networks," Second Technical Report Preprint, Institute of Systems Science, National University of Singapore, 1994. 4. H.P. Williams, "Logical problems and integer programming," Bull. Inst. Math. Appl., 13, 18-20 (1977). 5. J.L. Castro, F. Herrera, and J.L. Verdegay, "Knowledge-based systems and fuzzy Boolean programming," International Journal of Intelligent Systems, 9, 211-225 (1994).
RIT Scholar Works Articles
1996
Reasoning with propositional knowledge based on fuzzy neural logic Wangming Wu Hoon-Heng Teh Bo Yuan
Follow this and additional works at: http://scholarworks.rit.edu/article Recommended Citation Wu, Wangming; Teh, Hoon-Heng; and Yuan, Bo, "Reasoning with propositional knowledge based on fuzzy neural logic" (1996). International Journal of Intelligent Systems - John Wiley & Sons, Inc, Vol. 11 (), pp. 251-265. Accessed from http://scholarworks.rit.edu/article/993
This Article is brought to you for free and open access by RIT Scholar Works. It has been accepted for inclusion in Articles by an authorized administrator of RIT Scholar Works. For more information, please contact [email protected].
Reasoning with Propositional Knowledge Based on Fuzzy Neural Logic Wangming Wu and Hoon-Heng Teh
Institute of Systems Science, National University of Singapore, Singapore 0511 Bo Yuan
Department of Systems Science and Industrial Engineering, Thomas J. Watson School of Engineering, Binghamton University (SUNY), Binghamton, New York 13902
In this article, a new kind of reasoning for propositional knowledge, which is based on the fuzzy neural logic initialed by Teh, is introduced. A fundamental theorem is presented showing that any fuzzy neural logic network can be represented by operations: bounded sum, complement, and scalar product. Propositional calculus of fuzzy neural logic is also investigated. Linear programming problems risen from the propositional calculus of fuzzy neural logic show a great advantage in applying fuzzy neural logic to answer imprecise questions in knowledge-based systems. An example is reconsidered here to illustrate the theory. © 1996 John Wiley & Sons, Inc.
I. INTRODUCTION Fuzzy logic, attributable to Zadeh,' is basically a logic for approximate reasoning. As a generalization of classical two-valued logic, fuzzy logic extends the range of truth value from {0, to [0, 1]. The basic logical connectives AND, OR, and NOT in fuzzy logic are interpreted as t-norms, t-conorms, and complements (e.g., Wu 2). By such a generalization, a classical two-valued logic system becomes a special case of fuzzy logic systems. In recent years, a new kind of logic named fuzzy neural logic has been introduced by Teh. 3 Informally, a fuzzy neural logic is a kind of continuousvalued logic that includes the bounded weighted averages as its basic operation. Notice that a bounded weighted average (or convex combination) is an aggregate operation used in many practical problems. However, a bounded weighted average is not used to interpret AND or OR logical connectives, because they are not associative. This article shows that every operation in Kleene–Lukasiewicz fuzzy logic can be implemented by some simple fuzzy neural logic networks. This means Kleene–Lukasiewicz fuzzy logic is a special case of fuzzy neural logic discussed here. Fuzzy neural logic has a bounded weighted average as its addiINTERNATIONAL JOURNAL OF INTELLIGENT SYSTEMS, VOL. 11, 251-265 (1996) © 1996 John Wiley & Sons, Inc. CCC 0884-8173/96/050251-15
PROPOSITIONAL KNOWLEDGE
(a) Bounded sum
(c) Lukasiewicz implication
253
(b) Bounded product
(d) Complement
Figure 2. Basic logical operations implemented by neurons
threshold function f, which is defined by
(1) This function is also called the linear threshold function. Clearly, b E [0, 11. In particular, we can implement the bounded sum ED, the bounded product C), the Lukasiewicz implication and the complement c* by following neurons shown in Figure 2.
254
WU, TEH, AND YUAN
In general, if w i , . . . w„ are real numbers, then there exists a natural number N = 1 + Imax(Iwi I, • • • , Iwn1).1 such that w, = c,N, where [x] means the integer part of real number x and c, = w,IN E [-1, 1], i 1, . . . , n. Then we have f(E= I wi) = f (1\1 /7=1 aicil = fiN 17= 1 ai lci lsgn(c,)) E [0, 1]. Therefore, any neuron can be decomposed as a combination of scalar product operations and the simple neurons with connection weights either 1 or –1. (Fig. 4) Note that the slope threshold function f can be represented by f(x) = max(min(x, 1), 0) = min(max(x, 0), 1). Thus, we have
and Proof. We only prove Eq. 2 here. Equation 3 can be proved in a similar way. By Eq. 1, we have
256
WU, TEH, AND YUAN
Figure 5. The dual of a neuron
can also be represented by compositions of logical operations ED, (:), and c. Since a 1 C) a2 = (di' ED aZY , we have the following fundamental theorem. THEOREM 1. (the fundamental theorem). Every fuzzy neural logic network can be represented by the composition of logical operations ED , , and scalar product operations. Let us consider the dual of a neuron shown in Figure 5.
Figure 6. The equivalent network of a network in Figure 5
PROPOSITIONAL KNOWLEDGE
257
Figure 7. Two special cases of Theorem 2.1.
The networks in Figure 7 are two special cases of the dual theorem.
III. PROPOSITIONAL CALCULUS OF FUZZY NEURAL LOGIC SYSTEMS We are now ready to develop a propositional system of fuzzy neural logic. The symbols, with or without subscripts or primes, such as X, Y, that are used to denote propositions (i.e., sentences) are called atomic formulas, or atoms for short. From propositions, we can build compound propositions by using logical connectives. In fuzzy neural logic, we use five logical connectives: (negation), & (conjunction), Y (disjunction), —> (implication), (equivalence), and a class of scalar product connective po„ where a E [0, 1]. These connectives can be used to build compound propositions from atoms. More generally, we can construct more complicated compound propositions by applying them repeatedly. These compound propositions are called well-formed formulas. More formally, we give the following definition.
258
WU, TEH, AND YUAN
PROPOSITIONAL KNOWLEDGE
259
260
WU, TEH, AND YUAN
(A 1 , . . . , A n ) x/4_, if every valuation v such that v(A 1 ) = • • • = v(A n) = 1, the inequality v(A) holds. Whenever A = 1, we denote (A 1 , . . . A n) A instead of (11 1 , . . , A n) 1 A. THEOREM 5. If
& A n) —> A, then (A 1 , . . . , An )
(A i & •
A.
Proof. If (A 1 & • • • & An) --> A, then for every valuation v, v(A 1 & • • • & An) —> A) = (v(A l & • • & A n))` ED v(A) = (v(A 1 )` e • • • ED v(A n)` v(A)) A. Now, v is a valuation such that v(A i ) = • • • = v(A„) = 1. Thus, v((A 1 & • • • & An) ---> A) = 0 v(A) = v(A) A. This completes the proof. Remark 6. The inverse of Theorem 5 is not true. For example, let A 1 = X, A = X & X, where X is an atom. If v is a valuation such that v(A 1 ) = v(X) = 1, then v(A) = v(X & X) = v(X) C)v(X) = 1. Thus, (A 1 ) But A I ---> A is not a tautology (e.g., put v(X) = 0.5, then v(A 1 ----> A) 0.5 (0.5 Qx 0.5) = 0.5 1). THEOREM 6. Let A E [0, 1] and A, B, C, A 1 , . . . , A n , B 1 , . . . , B, be formulas, then
(15) (16) (17) (18)
(Modus ponens) (A, A --> B) B. (Modus tollens) (A --> B, (A --> B, B (Syllogism) C) A —> C. (Mean preserving)(A 1 --> B 1 , . . • An —> M,(.13 1 , . . . , B„).
Proof. By the Theorem 5, over, for any valuation v, v(((A
(15)
follows from (3).
Mn(A,, . . . , A„) —>
(16)
follows from (2). More-
B) & (B ----> C)) ---> (A —> C)) = {((v(A)) c 8) v(B)) C) ((v(B))` = (v(A)
v(C))}` ED ((v(A))c
(v(B))`) ED (v(B) ® (v(C))()
v(C))
((v(A))( ED v(C))
= ((v(A)) c V (v(B))`) El) (v(B) V v(C)) = (v(A) A v(B))( (v(B))` ED v(B) =
(v(B) V v(C)) 1.
Thus, we have (17). Finally, for any valuation v, let a, = v(A;), b, = v(B,), i = 1, . . . , n. & • • • & (A„--> Bn))--- > (WA] = ((d; ED b1) C) • • C) (an
, 4,) —› Mn(B I , .
b„))` El) ((a 1 + • • + a n)ln)c
13,)))
(b 1 + • • • + b„)In
= (a l C) b;) CD • • • ED (an C) b`r,') e (al/nED • • • ED (41 n) e (b i l n ED • • • ED bn l n) = (a 1 C)
di' I n
b i lne • • • e(a„® bc,) e a nc i n 8) bn l n
lln + • • • + 1/n = 1,
PROPOSITIONAL KNOWLEDGE
261
WU, TEH, AND YUAN
262
(ix) ),(p„XE)' iff (ax9' -X 0 (where x 1 = x, x° = xc). 1 (i.e., ax O py (x) ),(p,„X & poY) iff ax + /3y - A). A). (xi) x(jo,„X YpoY) iff ax + /3y - A 0 (i.e., ax py 1. (i.e., ax /3y A). ppY) if ax - /3y + (xii) (xiii) kx -> pp(-,Y)) iff ax - Ay + A :5- 1 - /3. (i.e., ((ax)` pyc)` ._ A). (2) Suppose A is a fuzzy clause, A = (p ai X19 8 ' V • • • (p„A•) 8 ", where e„ E {0, 1}, Xi are atoms, i = 1, . . . , n. Denote x, = v(X,), then + • • • +
iff
(xix)
-X
0.
Therefore, if A is a local implication formula, H.>21 is equivalent to a linear inequality on the fuzzy variables. (p„,,X,,n)'n, where 8i, (3) Suppose A is a fuzzy phrase, A = (pa ,X;0 8, Y • • • 8; E {0, 1}, X, are atoms, i = 1, . . . , n. Denote x i = v(Xi ), then (xx) xA iff (a lxVi + • • • + (a„4. ) 8,• -A
n - 1.
(4) Let w 1 , . . . , w„ E [0, 1] and w 1 + • • + w„ = 1; s i . . . , sm E [0, 1] and Si +• • • + sn, = 1. A = p„„Xl & • • • &p„,„X,, and B = p,Y„,. Then Y• • (xxi) A AA
B iff w i x t + • • • + wnx„ - s l Y i - • • - sm y„, + X
1•
Now, we give a main theorem of this section. X i . • , X ni . • . Y,n be atoms; Pi = (p„,X1'0 8•1 & • • (pa,,A-18-, where au, eu, 6ij E [0, 1], for i = 1, . . . , k, j = 1, . . . , n; Ci = (p0,Y1,)"'m • • • (pa,,,,rur)11,-, where 14, yii , E [0, 1], for i = 1, . . . , k,j = 1, . . . , m; Pi Ci is a local implication formula, for i = 1, . . . k; w 1 , . . . , W n E [0, 1] and w i + • • + wn = 1; s i , . . ,s„, E [0, 1] ands, + . . . + s u = 1. Then to find the maximum A E [0, 1] such that the whole implication formula A B is a A-logical consequence of the local implication formulas PI C 1 , . . . , Pk Ck is equivalent to find A = 1 - max(w i x i + . . . wnxn s l y ' - . • • - smY., 0), where (x i , . . . , ; Yi9 • • • y m ) is an optimal solution of the following linear programming problem:
THEOREM 7. Let
(LP) s .t.
max w i x i +" • • + W n Xn SIYI - • • • -SnzYm (a d 49 1 8 ' ' • + (a,„x0 1-8- + ([3, 1 y1. )Thi + • 1, . . . , k; i . . , n; 0 1, t = 0 yu 1, u = 1, , tn.
+ (/3mm7"" Y
Proof. By Definition 6, (P I --> CI, • • • P k -) Ck)(A -> B) means: for any valuation v such that v(P,---> C I ) = • • • = v(13 k Ck) = 1, v(A B) holds. Let x t = v(Xt), Yu = t = 1, . . , n; u = 1, . . . , m. Clearly, X t ' Yu E [0, 1], for t = 1, . . . , n, u = 1, . . . , m. Now, v(P,--> C,) = 1 if and only if (ail .4,9 1
+ • • • + (amxsin)'-'m n + (Ail Yin)Thr + •. • • + (PimY,),';'") Th'" 1, i = 1, . . . , k;
and v(A ---> B) wl xi
if and only if + • • • +
wn x„ - s l y] -
• • • - s mY m +
X
1.
Therefore, if A is the maximum such that the whole implication formula A -> B is a A-logical consequence of the local implication formulas P1 -->
PROPOSITIONAL KNOWLEDGE
263
IV. AN EXAMPLE In this section, we apply our approach to an example presented by Castro et al. 5 Here is their example: Suppose the knowledge base is stated as a set of statements in the propositional calculus: (1) If the bond market goes up and the interest rates decrease, either the stock market goes down or taxes are not raised; (2) If the political situation is unstable and the currency is devalued, the inflation increases; (3) If the interest rates increase and the bond market goes down, the stock goes up; (4) If the labor situation is unstable and the political situation is stable, the bond market goes up; (5) If the inflation decreases and the labor situation is unstable, the currency is devalued or political situation is stable; (6) If the stock market goes up, at least one favorable condition exists; (7) If the public deficit increases, and labor situation is unstable, the inflation increases; (8) If the public deficit increases and taxes are raised, the inflation increases; (9) If the political situation is unstable and the employed decrease, the stock market goes down; (10) If the public deficit decreases and taxes are not raised, the interest rate increases; (11) If currency is devalued then the public deficit increases or taxes are raised.
We can represent atoms and their negations in Table I. The statements (1)–(11) can be written as the following formulas:
264
WU, TEH, AND YUAN
Table 1. Atoms and their negations in the example. Economical Evolution The stock market goes up. The stock market goes down. The bond market goes up. The bond market goes down. The inflation decreases. The inflation increases. Currency is revalued. Currency is devalued. The interest rates increases. The interest rates decreases.
Country Situation Labor situation is stable. : Labor situation is unstable. Political situation is stable. : Political situation is unstable. Taxes are not raised. : Taxes are raised. The public deficit decreases. : The public deficit increases. The employed increases. The employed decreases. 5:
A positive economical evolution may be defined a, X5 ), so a negative economical evolution will be defined as Similarly, a favorable country situation may be defined as B and an unfavorable country situation as —B = —M5(171, „, Does a positive economical evolution imply a favorable country situation? In other words, does A —> B? It should be noted that A 1 , . . . , A 11 are all local implication formulas. But A —> B is a whole implication formula. To find the maximum X such that (A 1 , . . . , A 11 ) x(A B), by Theorem 7, is equivalent to solve the following linear programming problem (LP):
PROPOSITIONAL KNOWLEDGE
265
An optimal solution of this (LP) problem is The optimal value is 0.6. It follows that "a positive economic evolution implies a favorable country situation" is a 0.4logical consequence of the knowledge base. In Castro et al., 5 the authors applied fuzzy Boolean programming to solve the above problem. They used 20 Boolean variables based on classical twovalued logic. Our approach is based on fuzzy neuron logic and uses only 10 fuzzy variables. The methodology based on fuzzy neural networks and linear programming is more reasonable, because knowledge represented in a knowledge base may not be precise.
V. CONCLUSIONS In this article, based on the fundamental theorem of fuzzy neural logic, we establish the propositional calculus of fuzzy neural logic. Linear programming approach taken from the propositional calculus of fuzzy neural logic is a useful tool to answer imprecise questions in knowledge-based systems. It seems possible to use network approach to solve logical problems. We will investigate this interesting question in a future paper.
References 1. L.A. Zadeh, "Fuzzy logic and approximate reasoning," Synthese, 30, 407-442 (1975). 2. W.M. Wu, "A multivalued logic system with respect to T-norms," in M.M. Gupta and T. Yamakawa, Eds., Fuzzy Computing, North-Holland, Amsterdam, 1988, pp. 101-118. 3. Real World Computing Partnership Neuro ISS Laboratory, "A new class of neural networks called neural logic networks," Second Technical Report Preprint, Institute of Systems Science, National University of Singapore, 1994. 4. H.P. Williams, "Logical problems and integer programming," Bull. Inst. Math. Appl., 13, 18-20 (1977). 5. J.L. Castro, F. Herrera, and J.L. Verdegay, "Knowledge-based systems and fuzzy Boolean programming," International Journal of Intelligent Systems, 9, 211-225 (1994).
