A Deductive Question Answering System on Relational Data Bases

T h i s paper d e s c r i b e s a new f o r m a l i z a t i o n of a d e d u c t i v e q u e s t i o n a n s w e r i n g system on a r e l a t i o n a l d a t a base u s i n g a theorem p r o v i n g t e c h n i q u e . A theorem p r o v i n g p rocedu re f o r a f i n i t e domain i s i n v e s t i g a t e d and a d i r e c t p r o o f p r o c e d u r e based o n s u b s t i t u t i o n s o f e q u i v a l e n t f o r m u l a s w h i c h employs t h e b r e a d t h f i r s t sea rch i s i n t o r u d c e d . The sea rch s t r a t e g y i s t h e n expanded t o se t o p e r a t i o n s o f t h e r e l a t i o n a l a l g e b r a wh ich a re i n c o r p o r a t e d i n t o t h e p r o o f p rocedu re i n o r d e r t o i n c r e a s e t h e d a t a base search e f f i c i e n c y . V i r t u a l r e l a t i o n s a r c r e a l i z e d b y means o f i n t r o d u c i n g s e v e r a l ax ioms and u t i l i z i n g t h e d e d u c t i v e c a p a b i l i t y o f t he l o g i c a l sys tem. F u r t h e r m o r e , a c o n d i t i o n a l domain is, i n t r o d u c e d as one o f t h e v i r t u a l domains and i s used to g i v e a r e l a t i o n a l v i e w t o a pseudo r e l a t i o n a l d a t a base w h i c h can r e p r e s e n t e x c e p t i o n a l cases u s i n g some 1i nk i n f o r m a t i o n . A que ry t r a n s f o r m a t i o n system c a l l e d DBAP (Da ta Base Access P l a n n e r ) wh i ch embodies t h o s e f e a t u r e s i s implemented i n QJJSP. 1 . I n t r o d u c t i o n Many r e s e a r c h g roups i n the a r t i f i c i a l i n t e l l i g e n c e f i e l d have been c o n c e n t r a t i n g t h e i r e f f o r t s on how to r e p r e s e n t knowledge1 and how to p e r f o r m l o g i c a l i n f e r e n c e a n d / o r common sense r e a s o n i n g . The knowledge d a t a bases a r e o r g a n i z e d i n ve ry c o m p l i c a t e d ways i n o r d e r t o r e a l i z e those v e r y h i g h l e v e l f u n c t i o n s . These s t r u c t u r a l and o p e r a t i o n a l c o m p l e x i t i e s have been p r e v e n t i n g us f rom expand ing them to ve ry l a r g e knowledge d a t a bases . On t h e o t h e r h a n d , t h e r e have been many p r o j e c t s t o d e v e l o p v e r y l a r g e commerc ia l d a t a bases i n t h e d a t a base r e s e a r c h a r e a . T h i s k i n d o f d a t a base i s assumed to be used i n a r e l a t i v e l y s i m p l e manner and c o n s e q u e n t l y has s i m p l e s t r u c t u r e s : . E f f i c i e n t sea rch a l g o r i t h m s f o r such s i m p l e s t r u c t u r e s have been deve loped e x t e n s i v e l y and some s p e c i a l purpose hardware systems w i t h p a r a l l e l s e a r c h i n g c a p a b i l i t y a r e b e i n g deve loped i n many p l a c e s . Our c u r r e n t r e s e a r c h g o a l i s t o combine t h e s e two s e p a r a t e e f f o r t s t o b u i l d u p a v e r y l a r g e d a t a base w i t h t h e d e d u c t i v e c a p a b i l i t y [ 8 ] , [ l l ] . Codd, E . F . [ 2 ] i n t r o d u c e d an a l g o r i t h m t o c o n v e r t any que ry w r i t t e n i n a r e l a t i o n a l sub language t o a sequence o f r e l a t i o n a l a l g e b r a i c , o p e r a t i o n s i n o r d e r t o show t h e r e l a t i o n a l comp le teness o f t h e r e l a t i o n a l a l g e b r a . H i s a l g o r i t h m can be c o n s i d e r e d as a f o r m a l q u e s t i o n a n s w e r i n g (QA) p r o c e d u r e on a r e l a t i o n a l d a t a b a s e . On t h e o t h e r h a n d , G reen , C. and R a p h a e l , B. \k] f o r m a l i z e d a d e d u c t i v e QA system based o n f i r s t o r d e r l o g i c . The e s s e n t i a l p o i n t o V t h e i r f o r m a l i s m i s t h a t knowledge i s r e p r e s e n t e d by a s e t o f ax ioms and t h e answer o f t h e q u e s t i o n i s e x t r a c t e d f rom t h e r e f u t a t i o n p r o o f o f t h a t q u e s t i o n . I n t h i s , p a p e r , t h e s e two f o r m a l i s m s a re combined by i n t r o d u c i n g a p r o o f p r o c e d u r e f o r a f i n i t e s e t , where l o g i c a l e x p r e s s i o n s a re i n t e r p r e t e d as s e t o p e r a t i o n s on t h e s e t . A p r o o f p r o c e d u r e f o r q u e r i e s wh i ch r e q u i r e a l l answers s a t i s f y i n g t h e g i v e n s p e c i f i c o. t i o n i s p r e s e n t e d. I t i s a d i r e c t p r o o f p r o c e d u r e based o n s u b s t i t u t i o n s o f e q u i v a l e n t f o r m u l a s . As an i n t e r m e d i a t e r e s u l t o f t h e d i r e c t p r o o f , t he system g e n e r a t e s an access p l a n t o t h e d a t a base , and t h e n t h e p l a n i s execu ted t o ge t t h e a l l answers s a t i s f y i n g t h e s p e c i f i c a t i o n . The se t o p e r a t i o n s o f t h e r e l a t i o n a l a l g e b r a a re c o n s i d e r e d as expanded n o t i o n s o f t h e b r e a d t h f i r s t sea rch s t r a t e g y and a re i n c o r p o r a t e d i n t o t h e p r o o f p r o c e d u r e t o exp ress t h e accees p l a n . S t o n e b r a k e r , M . [ 1 0 ] i n t r o d u c e d the n o t i o n o f v iews (we c a l l them v i r t u a l r e l a t i o n s ) i n o r d e r t o p r o v i d e use rs w i t h t h e d e d u c t i v e c a p a b i l i t y , and r e a l i z e d them by means o f query m o d i f i c a t i o n . I n t h i s p a p e r , v i r t u a l r e l a t i o n s a r e c o n s i d e r e d t o p r o v i d e a semant i c model o f t he base r e l a t i o n s and a re d e f i n e d by a se t o f non -g round ax i oms . The query m o d i f i c a t i o n p rocess can be c o n s i d e r e d as s u b s t i t u t i o n p rocess o f a f o r m u l a by an e q u i v a l e n t f o r m u l a , t h e r u l e o f wh i ch i s g i v e n b y t h e a s s o c i a t e d a x i o m . An ax iom c a l l e d a c o n d i t i o n a l domain ax iom i s p a r t i c u l a r l y i n t e r e s t i n g . I t i s used to g i v e a r e l a t i o n a l v iew to a pseudo r e l a t i o n a l d a t a base wh ich can r e p r e s e n t e x c e p t i o n a l cases u s i n g some l i n k i n f o r m a t i o n . I n a d d i t i o n , some c o n s i d e r a t i o n s o n d e l e t i o n o f r e d u n d a n c i e s w i l l b e p r e s e n t e d . O p t i m i z a t i o n o f t h e access p l a n w i l l a l s o b e e o n e i d e r e d . The imp lemented query t r a n s f o r m a t i o n system DBAP w i l l b e b r i e f l y e x p l a i n e d . I n t h e l a s t s e c t i o n , t h e c o n c l u s i o n and some f u t u r e r e s e a r c h works to be done w i l l b e d e s c r i b e d . 2. Fo r m a l i z at i on G e n e r a l l y , a f o r m a l QA system c o n s i s t s of a s e t o f ax ioms and a theo rem p r o v e r to ge t answers f o r a g i v e n q u e r y . F i g . 1 shows t h e c o n f i g u r a t i o n o f ou r sys tem i n t e rms o f t h e f o r m a l i s m . I n a f o rma l s y s t e m , each datum i n t h e d a t a base has to be exp ressed by a g round c l a u s e (a c l a u s e w h i c h does n o t c o n t a i n any v a r i a b l e s ) . There a r e two t y p i c a l r e p r e s e n t a t i o n s : n a m e l y , t h e t u p l e w i s e r e p r e s e n t a t i o n and t h e d o m a i n w i s e N a t u r a l Lanr :uar f» -3 : Furukawa 50 p u t t i n g a p r e f i x symbol and read the expression as 'F ind a i l ?x such tha t ( i ) . . . ' . The i n ten t i ona l f i l e consis ts of non-ground axioms which def ine users ' views or v i r t u a l r e l a t i o n s . The ob jec t i ve of in t roducing users ' views is to keep the query language independent of the l o g i c a l s t ruc tu re of the r e l a t i o n a l data base. Assume that we have a r e l a t i o n a l data hase which consists of the fo l l ow ing base r e l a t i o n s : EMP(NAME, DNAME SAL) DEPT(WAME MGR) where the domais DNAME in EMP and NAME in DEPT are both the set of departments. Assume also that a user wants to def ine a v i r t u a l r e a l ! i o n VEMP(NAME DNAME SAL MGR). In the v i r t u a l r e l a t i o n VEMP, the domain MGR belongs to the employee r e l a t i o n , but in fact i t belongs to the department r e l a t i o n . The domain MGR is considered to have been t ransfered from the department r e l a t i o n to the employee r e l a t i o n , and we c a l l t h i s k ind of v i r t u a l domain a t r a n s i t i v e domain. in terms of the VEMP r e l a t i o n , the fac t tha t the manager of an employee i is x is expressed as VEMP.MGR(i,x). The QA system has to transform t h i s expression to the f o l l ow ing conjunct ion of l i t e r a l s on the base r e l a t i o n s : Let us consider the query "Who is the manager of Mr. SMITH ?" . In terms of the v i r t u a l r e l a t i o n VEMP, t h i s quest ion is l o g i c a l l y expressed by N a t u r a l L a n r u a £ 0 3 : Furukawa 60 (?x)(i)(VEMP.NAME(i, 'SMITH' A VEMP.MGR(i,?x)). (2 ) By subst i tut ing the second term in (2) by the righthand expression of the equivalence sign = in ( l ) , we obtain the fol lowing expression: ( a ?x) (3 i ) (a j ) (3y) (VEMP.NAME(i,'SMITH' ) A EMP. DMME( i , y) A DEPT.NAME(j,y) A DEFT.MGR( ,1 ,?x )) . So fo r , we have obtained the expression in terms of the base relat ions except the underlined l i t e r a l . This l i t e r a l is transformed to the corresponding base re la t ion l i t e r a l by the fol lowing axiom: (i)(x)(VEMP.NAME(i,x) i EMP.NAME(i,x)). This type of axiom is called a simple domain axiom, and a query which does not include any v i r t u a l re la t ion l i t e r a l s is called a base query. It is obvious that any query which is specified in terras of v i r t ua l relat ions is translated to an equivalent base query by logical inference. However, the trnasfromation by the resolution rule which is based on the modus ponens is insuf f i c ien t if we want to get a l l answers which sat isfy the given speci f icat ion. We can prove it easi ly . Denote a query by F[?x] and the required answers by {?x| F[?x]} . If we obtain a base query G[YxJ by applying the resolution r ul e s , then G [ ? x ] ~ F [ ? x ] . There fore, {?x| G[?xll £ {?x| F f?x ] l , where the equality holds only if G[?x] ^ F[VxJ. So fa r , these transformations can be realized by the query modif ication technique [10]. As far as the control structure is concerned, it is equivalent to the input resolution in the GL-resolution which is known to be val id only for a horn set [13] . But there exist more complicated axioms which require the whole inference capabi l i ty including the ancester resolut ion. We w i l l introduce a few such axioms later on. A v i r t ua l domain can be defined in terms of other predefined v i r t ua l domains. The axioms for such domains transform a l i t e r a l to a conjunction of l i t e r a l s some of which are not the base r e1at ion 1i t er a ls . In this paper, we consider only ex is tent ia l ly quanti f ied queries. It is easily shown that the resul t ing base queries after applying the transformations are also only exs is tent ia l ly quant i f ied. Therefore, we further simpli fy the notation for queries by omitt ing a l l quant i f iers . 3. Deletion of Redundancies A base query which is obtained so far may have some redundancies. Let us consider the base re la t ions: EMP(NAME DNAME); DEPT(NAME MGR LOG), and the v i r t ua l re la t ions: VEMP(NAME DNAME LOG); VDEPT(NAME MGR). Note that the LOG domain is t rans i t i ve . Assume that the following query is given: t inquires the manager of the department y to which the employee i located at 1*1*2 belongs. The expression (3) is transformed to the fol lowing