P r o v i n g theorems in a multi-source environment* Laurence Cholvy ONERA-CERT 2 avenue Edouard Belin, 31055 Toulouse, France email :
[email protected]
Abstract T h i s paper describes a logic for reasoning in a m u l t i - s o u r c e e n v i r o n m e n t and a theorem prover for t h i s logic. We assume the existence of several sources of i n f o r m a t i o n ( d a t a / k n o w l e d g e bases), each of t h e m p r o v i d i n g i n f o r m a t i o n . T h e m a i n p r o b l e m dealt w i t h here is the problem of the consistency of the i n f o r m a t i o n : even if each separate source is consistent, the global set of i n f o r m a t i o n may be inconsistent. In our a p p r o a c h , we assume t h a t the different sources are t o t a l l y ordered, according to their r e l i a b i l ity. T h i s order is then used in order to avoid inconsistency. T h e logic we define for reasoning in this case is based on a classical logic augmented w i t h pseudo-modalities. Its semantic is first d e t a i l e d . T h e n a sound and complete a x i o m a t i c is given. F i n a l l y , a t h e o r e m prover is specified at the meta-level. We prove t h a t it is correct w i t h regard to the logic. We then i m p l e m e n t it in a P R O L O G - l i k e language.
1
Introduction
More and m o r e , c o m p u t e r science applications need to use i n f o r m a t i o n w h i c h is not provided by a single source of i n f o r m a t i o n b u t by several. T h i s is the case, for instance, when one wants to use several expert systems, each of t h e m dealing w i t h a part i c u l a r p a r t of the g l o b a l p r o b l e m to be solved. In this case, the knowledge c o m i n g f r o m each expert system must be c o m b i n e d . T h i s c o m b i n a t i o n is not necessarily physical and the knowledge may r e m a i n d i s t r i b u t e d a m o n g the different systems. However, the set of k n o w l edge necessary to solve the global p r o b l e m is obtained by v i r t u a l l y g r o u p i n g the knowledge of the different systems. It is also the case of d i s t r i b u t e d databases. Each database stores i n f o r m a t i o n concerning a p a r t i c u l a r app l i c a t i o n d o m a i n . W h e n considering a larger d o m a i n , one has to federate different databases. Here a g a i n , the g r o u p i n g is v i r t u a l since, very o f t e n , the differentdatabases are locally managed by other people. A n o t h e r *This work has been financed by the CNRS and the French Ministry of Research
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case of collection of i n f o r m a t i o n p r o v i d e d by different sources can be f o u n d in the g r o u p i n g of beliefs. T h e classic example is t h a t of a police inspector [C B a r a l and S u b r a h m a n i a n , 1992) w h o questions different witnesses. Each witness has his own beliefs concerning the crime and the inspector has to collect all of t h e m in order to find the clue. Several problems arise when m e r g i n g different inform a t i o n sources [ C h o l v y , 1992a]. F i r s t of a l l , there are problems of language : the sources do not necessarily share a c o m m o n language for describing i n f o r m a t i o n . In databases for instance, t h i s happens when the different databases to be merged do not have the same schema. In such a case, t r a n s l a t i o n s are necessary [ E l m a g a r m i d and P u , ]. There also are problems of redundancy : a source may provide i n f o r m a t i o n w h i c h subsumes other i n f o r m a t i o n p r o v i d e d elsewhere. In this case, p r u n i n g is necessary. F i n a l l y , there are problems of consistency. Even if each source is consistent, the global set of inf o r m a t i o n may be c o n t r a d i c t o r y . T h i s is augmented by the delocalised management of the different i n f o r m a t i o n sources. People w h o need to gather i n f o r m a t i o n are not those who manage the different bases a n d , due to the d i s t r i b u t i o n , the different sources are managed by different people. So the n o t i o n of global consistency does not exist, since the sources are independently developed. However, when g r o u p i n g all of t h e m , the p r o b l e m of consistency arises.
O u r work concerns the last p r o b l e m : we are interested in defining a logical f r a m e w o r k w h i c h allows us to reason w i t h i n f o r m a t i o n p r o v i d e d by difTerent sources and w h i c h does not collapse in case of inconsistency. Classical logic cannot be used d i r e c t l y in this case since everyt h i n g is deducible f r o m a c o n t r a d i c t i o n (one can say it collapses in case of inconsistency). In the next section, we w i l l present different studies w h i c h dealt w i t h this p r o b l e m . We w i l l also present our a p p r o a c h . In section 3, we w i l l present the logic we have defined in order to reason in a m u l t i - s o u r c e e n v i r o n m e n t . In section 4, we define a theorem prover which i m p l e m e n t s our approach. It is described as a m e t a - p r o g r a m and we r u n it under a P R O L O G - l i k e language. Correctness of the theorem prover is proved.
2
Some related w o r k and our proposals
Work dealing with the problem of reasoning in the presence of inconsistency can be divided into two groups. • In the first group, we find studies which use classical logics but which eliminate inconsistency in order not to collapse. The main way to discard inconsistency is to manage maximal consistent subsets. Work by [C Baral and Subrahmanian, 1992] belongs to this group. They define several types of theory combinations, mainly based on the maximal consistent subsets of the global theory. Our past work [Cholvy, 1990], [Bauval and Cholvy, 1991], also belongs to this group. In the related domain of database updates, work by [Fagin et al., 1983] and [Kupper et al., 1984] also privilege maximal consistent subsets, as do [Gardenfors, 1988], [Nebel, 1989] in belief revision area, and [Ginsberg and Smith, 1988] in the problem of reasoning about actions. • In the second group, we find studies which do not use classical logics. For instance, paraconsistent logics are defined in such a way that some classical theorems (which allow us to derive everything from a contradiction) are not deducible in these logics [Besnard, 1990]. For example, is not a theorem in paraconsistent. logics. So a contradiction, say A A, cannot be used to derive anything, say B. Another example of non-classical logic which can manage with inconsistency is the modal logic for reasoning about updates defined in [Cerro and Herzig, 1986] and [Farinas and Herzig, 1992]. Even if the update is contradictory with the current database, the logic does not collapse. Besides this classification, another classification could be made : indeed, in each previous group, most of researchers have studied two cases. In a first step, they consider that all the data are equal with regard to the problem of inconsistency. In a second step, they studied the case where data are associated with extrainformation (tags, labels, degrees of priority, actions to be performed ...) which are used to restore consistency. For instance, [Fagin et al, 1983], [C Baral and Subrahmanian, 1992], [Besnard, 1990], [D. Dubois and Prades, 1992], [Gabbay and Hunter, 1991] present different types of extra-information. Finally, the previous works could be grouped according to another classification : most of them adopt a constructive approach in the sense that avoiding inconsistency leads to construct a new consistent base. Only work by [Cerro and Herzig, 1986] and [Farinas and Herzig, 1992] adopts a hypothetical approach : the logic they define, called assume, allows us to reason with hypothetical updates. The user can then deduce theorems of the form : such a formula will be true if I assume such other formula. The state after the update is never constructed. As for us, we have defined two logics for reasoning in a multi-source environment which do not collapse under inconsistency, [Cholvy, 1992a] [Cholvy, 1992b]. Both of them allow the user to assume that the different sources
of information are totally ordered according to their reliability. Our approach is then a hypothetical one. The two logics differ on the attitude they modelize : • The first logic, called FUSION-S, modelizes a suspicious attitude : it consists in suspecting all the information provided by a source if this source contradicts a more reliable source. • The second logic, called FUSION-T, modelizes a trusting attitude : if an information source contradicts a more reliable source, only the smallest set of contradictory information is suspected. Let us take the example of a police inspector who questions witnesses. Assume that a first witness, Bill, said that he saw a black car, while a second, John, said that he saw two men in a white car. An order may reflect the fact that the inspector himself has some conviction (which cannot be denied) or possesses some information which is true : for instance, he went to the meteorological station and he is sure that the crime was committed on a foggy day. So, he can assume that John is less reliable than Bill, since John was standing too far away from the scene of the crime and, because of the fog, he could not see well. So, the inspector may trust him less. In this case, the inspector will consider that : inspector If the inspector adopts the suspiscious attitude, he will conclude only that the car was black, i.e., he suspects all the information provided by John because John contradicts a more reliable witness. If he adopts the trusting attitude, he will conclude that they were two men in a black car. In fact, concerning the colour of the car, he trusts Bill more than John, so he can assume that the colour is black. Concerning the number of persons, John provides new information that does not contradict Bill's account. In the rest of this paper, we focus on the trusting attitude. The next section describes the logic which modelizes it. R e m a r k 1. As said previously, the trusting attitude suspects the smallest set of information which contradicts more reliable information. In this paper, smallest contradictory sets are pairs of literals : 1 and - 1. We can imagine a more "application-adapted" approach in which the notion of topics of information [Cazalens, 1992] [Cazalens and Demolombe, 1992], is taken into account in order to characterize the minimality. For example, assume a meeting in which a person listens to two people. The first is a teacher of logic, the second is a student. The conversation is technical. Implicitely, the person who listens trusts the teacher more than the student. Assume that during the conversation, the student affirms that "the first order logic is decidable". Immediately, the teacher denies it and reminds the student of the main result of undecidability of the first order logic. For the person, it is clear that the student is not reliable concerning the first order logic. If, later on, the student makes another affirmation about the first order logic, it is quite sure that the listener will not trust him. Indeed, because of the contradiction
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about the decidability, he rejects any affirmation of the student which "is about" the first order logic. In this case, works previously cited which try to formalize the notion of "is-about", could be used in order to reject a smallest set of information, defined by the domain of the first order logic. An attempt to use the notion of topics of information for a syntactical characterization of updates can be found in [Cholvy, 1993]. R e m a r k 2. In this paper, we assume the existence of a unique order on the different sources of information. Again, we can imagine a more "application-adapted" approach in which the sources are ordered according to several orders which are topic-dependent. Let us take the previous example and consider that the teacher of logic (who is an old-fashioned man) and the student are now speaking about pop music. It is quite sure that, regarding "pop music", he will trust the student more than the teacher, i.e., the student will be considered as more reliable than the teacher. So there will be two orders, one for the "logics" topic and one for the "pop music" topic. An attempt at a formalization of this idea can be found in [Cholvy, 1993].
3
A logic for reasoning in a multi-source e n v i r o n m e n t
In this section, we present a logic which implements the trusting attitude previously introduced. 3.1
T h e language
We assume that our language L is a finite set of propositional variables : p 1 , p 2 , ... PL- We note 1 , 2 . . n the n information sources we reason with. We will also say the databases to be merged. We assume that the sources (or databases) are finite, satisfiable but not necessarily complete, sets of literals of L. The logic we define, called FUSION-T(l..n), is obtained from the propositional logic, augmented with pseudo-modalities i.e., marks on formulas. These pseudo-modalities are :
F will mean that, when considering the total order on F is true in the database obtained by virtually merging database i1 and .. database i m . Notice that the general form of these pseudomodalities allows us to represent the particular case : [i] F, i = l . . n , which means that F is true in database i. 3.2
Semantics
The semantics of F U S I O N - T ( l . n ) is the following : a model of FUSION-T(L.n) is a pair : where : • W is the finite set of all the interpretations of the underlying propositional language L • r is a finite set of equivalence relations between interpretations in W.
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G e n e r a l i s a t i o n . Merging n databases 1,2 ..n, according to the trusting attitude and given an order : conies down to updating D B i n with the result of the update of D B i n _ ! with the result of the update .... of DBi2 with D B i i .
4
A t h e o r e m p r o v e r for t h i s logic
In this section, we deal with the implementation aspects. Our aim is to define a theorem prover for this logic which allows us to answer questions of the form : is formula F true in the merged information sources if the order is 0 ? i.e., to prove theorem [ 0 ] F . We suggest using a PROLOG-like language form implementing such a theorem prover, in order to reuse its facilities (unification, negation as failure, strategy ...) In subsection 4.1, we describe a first meta-program which describes a prover for proving formula if 1 is a literal. This meta-program is a set of definite Hornclauses (where negation is explicitely managed with positive literals) and which can easily be run on a PROLOGlike interpreter. In subsection 4.2, we give an optimisation where the ncgation-as-failure of PROLOG is used to manage the negation. Finally, in section 4.3, we extend this prover in order to prove formula [0] F, where F is any propositional formula (written under the conjunctive normal form). 4.1
T h e t h e o r e m p r o v e r as a m e t a - p r o g r a m : first version
Let us consider a meta-language M L , based on language L, defined by : • constants of ML are literals of L, plus a constant rioted nil • a function rioted sents the term :
By convention,
repre-
• predicat symbols of ML are : LFUSION, nonLFUSION, NIL and nonNIL. The intuitive semantics of the predicates is the following : - LFUS10N(0,1) means that it is the case that literal 1 is true in the merged databases if the order is O. - nonLFUSION(0,l) means that it is not the case that literal 1 is true in the merged database if the order is 0 - NIL(O) is true only for nil - nonNIL(O) is true except if O is nil. 4.1.1 The meta-program Let us consider M E T A 1 , the following set of the ML formulas :
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5
Conclusion
In this paper, we have dealt with the problem of reasoning in a multi-source environment by focusing on the global consistency of information. We have shown that this problem is a particular case of reasoning with inconsistency. For discarding inconsistency, we have suggested considering the relative reliability of the different information sources. This comes down to considering a total order on the sources. There are different ways to use this order to avoid the inconsistency but in this paper we have focused on one attitude, called trusting : when two sources are contradictory, it consists in rejecting the minimal contradictory information which is provided by the less reliable source. Our aim was to define a logic that would allow the user to reason in a multi-source environment according to a hypothetical approach : the merging of the different sources is never done, i.e., the database obtained
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b y m e r g i n g t h e i n f o r m a t i o n c o m i n g f r o m the different sources is never c o n s t r u t e d . T h e user o n l y assumes the order of the sources a n d tries to derive t r u e f o r m u l a in t h i s assumed database. As far as we k n o w , o n l y the w o r k cited in section 2 for database updates adopts such a hypothetical approach. T h e semantics we have attached to t h i s a t t i t u d e is in t e r m s of possible models a n d is an extension of the semantics defined for belief base updates. T h e n o t i o n of "nearest" models is defined in terms of c o m p l e m e n t a r y l i t e r a l s . T h e semantics is a p p r o p r i a t e even if i n f o r m a t i o n is extended to clauses [ C h o l v y , 1993]. U n f o r t u n a t e l y , the a x i o m a t i c s we have g i v e n , and the t h e o r e m prover we have defined, are o n l y adequate for l i t e r a l i n f o r m a t i o n . In a d d i t i o n , one can wonder w h a t happens if the order on the sources is n o t t o t a l . In this case, we find again the p r o b l e m o f reasoning w i t h inconsistency w i t h n o e x t r a i n f o r m a t i o n to restore consistency. Solutions briefly described in section 2 could be adapted here leading to a f o r m a l i s m w h i c h mixes our logics and a mechanism to avoid inconsistency ( m a n a g e m e n t of m a x i m a l consistent subsets for instance). B u t this needs to be studied more carefully. F i n a l l y , in section 2, we have shown t h a t another sem a n t i c s could be attached to the reasoning in a m u l t i source e n v i r o n m e n t . It consists in t a k i n g i n t o account n o t i o n s like topics of i n f o r m a t i o n . T h i s a p p l i c a t i o n oriented n o t i o n could be used to give a new d e f i n i t i o n of m i n i m a l i t y as well as an extension to several orders on sources of i n f o r m a t i o n .
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[Cazalens, 1992] S. Cazalens. F o r m a l i s a t i o n en logique n o n - s t a n d a r d de certaines methodes de raisonnement pour f o r u n i r des reponses cooperatives, dans des systemes de bases de donnees et de connaissances. P h D thesis, Universite Paul Sabatier, Toulouse, 1992. [Cerro and Herzig, 1986] L. Farinas Del Cerro and A. H e r z i g . Reasoning a b o u t database updates. In Jack M i n k e r , e d i t o r , Workshop of Foundations of deductive databases and logic p r o g r a m m i n g , 1986. [ C h o l v y , 1990] L. C h o l v y . Q u e r y i n g an inconsistent database. I n N o r t h H o l l a n d , e d i t o r , Proc o f A r t i f i c i a l
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