XuHua 435 - IJCAI
LOCK,
LINEAR
PARAMODULATION
Liu
IN OPERATOR FUZZY
LOGIC
XuHua
D e p t . of Comp. S c i e . J i l i n University Changchun, PRC.
ABSTRACT
The author proposed concepts of O p e r a t o r Fuzzy Logic and -Resolution in 1984. He and his cooperaters have obtained some theoretical results. This paper introduces -paramodulation to handle a set of clauses with the predicate of equality, and thus the equality substitution can be used in fuzzy reasoning. Then the - p a r a m o d u l a t i o n method is proved to be complete w i t h lock-semantic method. F i n a l l y , Yang F e n g j i e and I have i m p r o v e d the p a r a m o d u l a t i o n , and proved t h a t the linear paramodulation is complete too.
I .
T h i s p a p e r proposed a paramodulation to handle fuzzy equality. In conjunction with resolution, -paramodulation can be used to prove f u z z y theorems in OFL. We proved that paramodulation is complete for the inconsistent set of clauses in conjunction with resolution.
II .
FUNDAMENTAL
CONCEPTS AND PROPERTIES
From [ 1 - 3 ] , w e know t h a t O p e r a t o r Fuzzy L o g i c i s b u i l t o n o p e r a t o r l a t t i c e and i n t e r v a l [ 0 , 1 ] i s a n o p e r a t o r l a t t i c e if we d e f i n e :
INTRODUCTION for
The author proposed concepts of Operator Fuzzy Logic( o r OFL for short) a n d K~Resolution in 1984. During the past few years (1985-1987), he and H. Xiao, t h e n h e , K . Y . F a n g , C a r l . K . C h a n g and J e f f . J-P Tsai working together on the resolution method obtained a series of results. We proved that a r e s o l v e n t of two f u z z y c l a u s e s Ci and C2 is a l o g i c a l c o n c e q u e n c e of C1 and C2, and that resolution method is complete for inconsistent set of clauses. We explained the practical meaning of fuzzy reasoning based on r e s o l u t i o n method i n OFL. From t h e above results, we can see that some theorems which can not be proved w i t h the traditional logic, can b e proved f u z z i l y by using r e s o l u t i o n method in our l o g i c system. Equality relation is a very important relation in mathematics. This equality relation lias some important special properties: it is t r a n s i t i v e and can s u b s t i t u t e e q u a l s f o r e q u a l s . Thus it is natural that the transitive and substitutive properties of fuzzily equality are needed in proving fuzzy theorems.
any x , y [ 0 , 1 ] . in the following discussion, we assume that OFL is built on interval [0,1] and t h e amount of operator is finite . These concepts such as operator lattice, fuzzy literal, formula, interpretation, truth-value T1(G) of formula G under 1, complemented literal, identical literal, can be found in Refs. [ 1 - 3 ] . Definition 1. Choose arbitrarily Formula G is c a l l e d valid i f and only if f o r e v e r y I t h e r e e x i s t s T1 (G) G is called inconsistent i f and o n l y if for every 1, t h e r e e x i s t s T1(G) Obviously, Formula G is valid if and only if Formula is -inconsi stent . Definition 2. Let C1 and C2 be two clauses without the same variables and and be two l i t e r a l s of Ci and C2 r e s p e c t i v e l y . If L1 a n d L2 h a v e a Most G e n e r a l Unifier (MGU f o r short) and and are complemental, then .. is the binary -resolvent for d e n o t e d by R A ( C 1 , C2 ), w h e r e
C1
and
XuHua
C2,
435
waiting for publishing. omitted here.
The* f o l l o w i n g p r o p e r t i e s are simple, and so t h e i r proofs are o m i t t e d . Property 1. Let P be an atom, then Property 2.
Let P be an atom,
then
Henceforth, IP can be denoted by P in OFL. Property 3. Let P be an atom, then Property 4. Let P be an atom, then
Its
proof
is
III. Definition 6. An E - i n t e r p r e t a t i o n IE of set S of clauses is an i n t e r p r e t a t i o n of S satisfyingthe four following c o n d i t i o n s . Let and be any terms in the Herbrand universe of S, and l e t be any fuzzy l i t e r a l in S. Then
Property 5. Let G be formula, then « For example, Let G=0.3P, 1={P}, then
Property 6.
Let G be a formula,
then
Property 7.
Let G be a formula,
then
Property 8.
Let P be an atom,
Definition 7. Let S be a set of clauses, and Then the set KK of . e q u a l i t y axioms f o r S is the set c o n s i s t i n g of the f o l l o w i n g clauses: f o r any
then >
For example, 0.8(0.6P) (0.8*0.6)P=0.7P. Property 9. Let G be a formula, then ion 3. Let G and H be two I be any interpretation. If then i t i s said t h a t G - i m p l i e s H, denoted by Definition 4. Let G and H be two formulas and I be any i n t e r p r e t a t i o n . If then i t i s said that G -strong implies H, denoted by Property 10. Let A be a formula, then
for
any atom
occurring
in S,
De fin i t formulas,
Property 11. Then
Let A,B and C be formulas.
f o r any f u n c t i o n symbol occurring i n Definition 8. A set S of clauses is called inconsistent if and only if Ti E (S f o r any E - i n t e r p r e t a t i o n IE ; S is called - v a l i d i f and only if f o r any I E . 1f IE is an E - i n t e r p r e t a t i o n of S, we can o b t a i n r e s u l t s obviously as follows:
Property 12. Let Ci and C2 be two clauses and Rx(Ci , C2 ) be a resolvent of C1 and C2. Then because IE The proof of t h i s property w i l l be given by another paper ( t o appear). Property 13. Let A,B and C be three formulas. Then
Definition 5. A a -empty clause f o r any literal condition:
is
then and
therefore,
clause C is called (denoted by if it satisfies the
Property 14. Let is inconsistent if and a resolution deduction clause from S. This property is theorem in Ref. [1] and
436
an E - i n t e r p r e t a t i o n ,
Automated Deduction
A set of clauses only if there is of the empty an important another paper is
From the above d i s c u s s i o n , we can e a s i l y see t h a t where is a set of e q u a l i t y axioms of S.
If
as
I
i s a n i n t e r p r e t a t i o n o f S and w e can o b t a i n r e s u l t s o b v i o u s l y follows:
thus
is c a l l e d a b i n a r y paramodulant of C1 and C2 , and are called p a r a m o d u l a t e d l i t e r a l s where I ] denotes the result obtained by r e p l a c i n g one s i n g l e o c c u r r e n c e o f tr in
therefore, t h e r e must
be
3. L e t According to
4.
be
Suppose
there must S i m i l a r l y , we then
thus t h e r e must
t h e r e must
be can e a s i l y
see
that
if
be namely ,
From t h e above d i s c u s s i o n , we can e a s i l y see t h a t 1 i s a n E - i n t e r p r e t a t i o n o f S. T h e r e f o r e , we can o b t a i n a theorem as f o l l o w s :
Theorem 1 . Let S be a set of clauses be the set of -equality axioms for For any interpretation I of S, I is E-interpretation if and only if
and S. an
Theorem 2. Let S be a set of clauses be the set of equality axioms for Then S is -inconsistent if and only IS \- inConsistent.
and S. if
Definition 10. A p a r a m o d u l a n t of C1 and C2 is a b i n a r y p a r a m o d u l a n t of C1 or a x - f a c t o r of Ci and C2 or a f a c t o r of C2 , denoted by Definition 1 1 . L e t S be a s e t of c l a u s e s ; G and H be two c l a u s e s in S. [f when for every E-interpretation 1E of S, then it is said that G A E - s t r o n g l y i m p l i e s H, denoted by G=>H. Theorem 3. Let C1 and C2 be two clauses, >0- 5, and suppose paramodulant. Then Without l o s s of g e n e r a l i t y , we may assume that C1 is C2 is , is a binary paramodulant, w i t h . Let MGU o f t and r be , IE be any one o f E - i n t e r p r e t a t i o n such t h a t Then,
Obviously,
we have
( N o t i c e : every v a r i a b l e i n C i and C 2 i s c o n s i d e r e d to be governed by a u n i v e r s a l quantifer). Choosing an arbitrarily ground instance of c1 and a ground i n s t a n c e of C2 , we have the following considerations:
Suppose S i s inconsistent, but (SUKX ) is n o t inconsistent. Then there exists an interpretation I such that Thus From theorem 1, we know t h a t I must be an E - i n t e r p r e t a t i o n . From we know t h a t S i s n o t -inconsistent, which c o n t r a d i c t s the assumption t h a t S is -inconsistent. Suppose is inconsistent, but S is not -inconsistent. Then t h e r e e x i s t s an E-interpretation IF such that Clearly, Hence This contradicts the assumption that is inconsistent.
IV.PARAM0DULAT10N
Definition 9. L e t ^>O.5 and C1 , C2 be two c l a u s e s w i t h o u t any v a r i a b l e s i n common such t h a t Ki L[t]VC , XJ >x or x < 1 - x , x2 ( r = s )VC2 ' , x2>x, where x1L[t] is a fuzzy literal c o n t a i n i n g t h e term t and C1' and C2' a r e c l a u s e s . If t and r have MGU
LINEAR
PARAMODULATION
Liu
IN OPERATOR FUZZY
LOGIC
XuHua
D e p t . of Comp. S c i e . J i l i n University Changchun, PRC.
ABSTRACT
The author proposed concepts of O p e r a t o r Fuzzy Logic and -Resolution in 1984. He and his cooperaters have obtained some theoretical results. This paper introduces -paramodulation to handle a set of clauses with the predicate of equality, and thus the equality substitution can be used in fuzzy reasoning. Then the - p a r a m o d u l a t i o n method is proved to be complete w i t h lock-semantic method. F i n a l l y , Yang F e n g j i e and I have i m p r o v e d the p a r a m o d u l a t i o n , and proved t h a t the linear paramodulation is complete too.
I .
T h i s p a p e r proposed a paramodulation to handle fuzzy equality. In conjunction with resolution, -paramodulation can be used to prove f u z z y theorems in OFL. We proved that paramodulation is complete for the inconsistent set of clauses in conjunction with resolution.
II .
FUNDAMENTAL
CONCEPTS AND PROPERTIES
From [ 1 - 3 ] , w e know t h a t O p e r a t o r Fuzzy L o g i c i s b u i l t o n o p e r a t o r l a t t i c e and i n t e r v a l [ 0 , 1 ] i s a n o p e r a t o r l a t t i c e if we d e f i n e :
INTRODUCTION for
The author proposed concepts of Operator Fuzzy Logic( o r OFL for short) a n d K~Resolution in 1984. During the past few years (1985-1987), he and H. Xiao, t h e n h e , K . Y . F a n g , C a r l . K . C h a n g and J e f f . J-P Tsai working together on the resolution method obtained a series of results. We proved that a r e s o l v e n t of two f u z z y c l a u s e s Ci and C2 is a l o g i c a l c o n c e q u e n c e of C1 and C2, and that resolution method is complete for inconsistent set of clauses. We explained the practical meaning of fuzzy reasoning based on r e s o l u t i o n method i n OFL. From t h e above results, we can see that some theorems which can not be proved w i t h the traditional logic, can b e proved f u z z i l y by using r e s o l u t i o n method in our l o g i c system. Equality relation is a very important relation in mathematics. This equality relation lias some important special properties: it is t r a n s i t i v e and can s u b s t i t u t e e q u a l s f o r e q u a l s . Thus it is natural that the transitive and substitutive properties of fuzzily equality are needed in proving fuzzy theorems.
any x , y [ 0 , 1 ] . in the following discussion, we assume that OFL is built on interval [0,1] and t h e amount of operator is finite . These concepts such as operator lattice, fuzzy literal, formula, interpretation, truth-value T1(G) of formula G under 1, complemented literal, identical literal, can be found in Refs. [ 1 - 3 ] . Definition 1. Choose arbitrarily Formula G is c a l l e d valid i f and only if f o r e v e r y I t h e r e e x i s t s T1 (G) G is called inconsistent i f and o n l y if for every 1, t h e r e e x i s t s T1(G) Obviously, Formula G is valid if and only if Formula is -inconsi stent . Definition 2. Let C1 and C2 be two clauses without the same variables and and be two l i t e r a l s of Ci and C2 r e s p e c t i v e l y . If L1 a n d L2 h a v e a Most G e n e r a l Unifier (MGU f o r short) and and are complemental, then .. is the binary -resolvent for d e n o t e d by R A ( C 1 , C2 ), w h e r e
C1
and
XuHua
C2,
435
waiting for publishing. omitted here.
The* f o l l o w i n g p r o p e r t i e s are simple, and so t h e i r proofs are o m i t t e d . Property 1. Let P be an atom, then Property 2.
Let P be an atom,
then
Henceforth, IP can be denoted by P in OFL. Property 3. Let P be an atom, then Property 4. Let P be an atom, then
Its
proof
is
III. Definition 6. An E - i n t e r p r e t a t i o n IE of set S of clauses is an i n t e r p r e t a t i o n of S satisfyingthe four following c o n d i t i o n s . Let and be any terms in the Herbrand universe of S, and l e t be any fuzzy l i t e r a l in S. Then
Property 5. Let G be formula, then « For example, Let G=0.3P, 1={P}, then
Property 6.
Let G be a formula,
then
Property 7.
Let G be a formula,
then
Property 8.
Let P be an atom,
Definition 7. Let S be a set of clauses, and Then the set KK of . e q u a l i t y axioms f o r S is the set c o n s i s t i n g of the f o l l o w i n g clauses: f o r any
then >
For example, 0.8(0.6P) (0.8*0.6)P=0.7P. Property 9. Let G be a formula, then ion 3. Let G and H be two I be any interpretation. If then i t i s said t h a t G - i m p l i e s H, denoted by Definition 4. Let G and H be two formulas and I be any i n t e r p r e t a t i o n . If then i t i s said that G -strong implies H, denoted by Property 10. Let A be a formula, then
for
any atom
occurring
in S,
De fin i t formulas,
Property 11. Then
Let A,B and C be formulas.
f o r any f u n c t i o n symbol occurring i n Definition 8. A set S of clauses is called inconsistent if and only if Ti E (S f o r any E - i n t e r p r e t a t i o n IE ; S is called - v a l i d i f and only if f o r any I E . 1f IE is an E - i n t e r p r e t a t i o n of S, we can o b t a i n r e s u l t s obviously as follows:
Property 12. Let Ci and C2 be two clauses and Rx(Ci , C2 ) be a resolvent of C1 and C2. Then because IE The proof of t h i s property w i l l be given by another paper ( t o appear). Property 13. Let A,B and C be three formulas. Then
Definition 5. A a -empty clause f o r any literal condition:
is
then and
therefore,
clause C is called (denoted by if it satisfies the
Property 14. Let is inconsistent if and a resolution deduction clause from S. This property is theorem in Ref. [1] and
436
an E - i n t e r p r e t a t i o n ,
Automated Deduction
A set of clauses only if there is of the empty an important another paper is
From the above d i s c u s s i o n , we can e a s i l y see t h a t where is a set of e q u a l i t y axioms of S.
If
as
I
i s a n i n t e r p r e t a t i o n o f S and w e can o b t a i n r e s u l t s o b v i o u s l y follows:
thus
is c a l l e d a b i n a r y paramodulant of C1 and C2 , and are called p a r a m o d u l a t e d l i t e r a l s where I ] denotes the result obtained by r e p l a c i n g one s i n g l e o c c u r r e n c e o f tr in
therefore, t h e r e must
be
3. L e t According to
4.
be
Suppose
there must S i m i l a r l y , we then
thus t h e r e must
t h e r e must
be can e a s i l y
see
that
if
be namely ,
From t h e above d i s c u s s i o n , we can e a s i l y see t h a t 1 i s a n E - i n t e r p r e t a t i o n o f S. T h e r e f o r e , we can o b t a i n a theorem as f o l l o w s :
Theorem 1 . Let S be a set of clauses be the set of -equality axioms for For any interpretation I of S, I is E-interpretation if and only if
and S. an
Theorem 2. Let S be a set of clauses be the set of equality axioms for Then S is -inconsistent if and only IS \- inConsistent.
and S. if
Definition 10. A p a r a m o d u l a n t of C1 and C2 is a b i n a r y p a r a m o d u l a n t of C1 or a x - f a c t o r of Ci and C2 or a f a c t o r of C2 , denoted by Definition 1 1 . L e t S be a s e t of c l a u s e s ; G and H be two c l a u s e s in S. [f when for every E-interpretation 1E of S, then it is said that G A E - s t r o n g l y i m p l i e s H, denoted by G=>H. Theorem 3. Let C1 and C2 be two clauses, >0- 5, and suppose paramodulant. Then Without l o s s of g e n e r a l i t y , we may assume that C1 is C2 is , is a binary paramodulant, w i t h . Let MGU o f t and r be , IE be any one o f E - i n t e r p r e t a t i o n such t h a t Then,
Obviously,
we have
( N o t i c e : every v a r i a b l e i n C i and C 2 i s c o n s i d e r e d to be governed by a u n i v e r s a l quantifer). Choosing an arbitrarily ground instance of c1 and a ground i n s t a n c e of C2 , we have the following considerations:
Suppose S i s inconsistent, but (SUKX ) is n o t inconsistent. Then there exists an interpretation I such that Thus From theorem 1, we know t h a t I must be an E - i n t e r p r e t a t i o n . From we know t h a t S i s n o t -inconsistent, which c o n t r a d i c t s the assumption t h a t S is -inconsistent. Suppose is inconsistent, but S is not -inconsistent. Then t h e r e e x i s t s an E-interpretation IF such that Clearly, Hence This contradicts the assumption that is inconsistent.
IV.PARAM0DULAT10N
Definition 9. L e t ^>O.5 and C1 , C2 be two c l a u s e s w i t h o u t any v a r i a b l e s i n common such t h a t Ki L[t]VC , XJ >x or x < 1 - x , x2 ( r = s )VC2 ' , x2>x, where x1L[t] is a fuzzy literal c o n t a i n i n g t h e term t and C1' and C2' a r e c l a u s e s . If t and r have MGU
