From: FLAIRS-02 Proceedings. Copyright © 2002, AAAI (www.aaai.org). All rights reserved.
Fusion of Possibilistic Knowledge Bases from a Postulate Point of View Salem Benferhat and Souhila Kaci Institut de Recherche en Informatique de Toulouse (I.R.I.T.)–C.N.R.S. Universit´e Paul Sabatier, 118 route de Narbonne 31062 T OULOUSE Cedex 4, F RANCE E-mail: benferhat, kaci @irit.fr
Abstract This paper proposes a postulate-based analysis of the fusion of possibilistic logic bases which are made of pieces of information expressing knowledge associated with certainty degrees. We propose two main sets of postulates: one focuses only on plausible conclusions, while the other considers both plausible conclusions and the certainty degrees associated with them. For each rational postulate, the class of operators that satisfies it is identified. The existence of weights associated with the pieces of information considerably enriches the postulates-based analysis, and leads to a refined classification of combination operators.
Introduction The fusion of pieces of information originated from different sources can be considered in various representations frameworks. Ranked propositional logic bases offers a reasonably expressive framework for representing pieces of information with their levels of reliability. Possibilistic logic [4] is a logic of weighted formulas, equipped with a semantics, where the weights can be understood as lower bounds of necessity degrees in the sense of possibility theory. Possibilistic logic inference associates the smallest weights of the formulas, involved in a proof, with the consequences. This is well in agreement with the idea of keeping track of the levels of reliability of the pieces of information used for deriving a conclusion. Recent works [3,2] have developed syntactic counterparts of the pointwise combination of possibility distributions expressing the semantics of possibilistic logic basis, and have shown [1] how the fusion of propositional logic bases [5,8,9,10,11,12] can be fully recovered with possibilistic logic framework. The present paper offers a characterization of possibilistic fusion modes in terms of postulates. It is in spirit of previous works on postulates for fusing non-prioritized propositional knowledge bases [5,8]. Clearly, the presence of the weights raise new issues. In [2], the authors have proposed an introductory analysis of the immediate adaptation of propositional postulates to the possibilistic case. However, this adaptation is not fully satisfactory. Indeed, from the classification of operators that Copyright c 2002, American Association for Artificial Intelligence (www.aaai.org). All rights reserved.
they proposed, there is no operator which satisfies all postulates. The reason is that some postulates which are natural in propositional logic cannot be directly used in the ranked models framework. After a short background on possibilistic logic [4,3] and propositional fusion postulates (Sections 2 and 3), the rest of the paper is organized into two main parts according to the consequence operator used, which may or not, associate a weight with the plausible consequence of knowledge bases.
Background Let be a finite propositionnal language. denotes the classical consequence relation. A possibilistic knowledge base is a set of weighted formulas where is a propositional formula and, belongs to the interval [0,1] and represents the level of certainty or priority attached to . Definition 1 Let be a possibilistic base, and a . We call the a-cut (resp. strict a-cut) of , denoted by a (resp. a ), the set of propositional formulas in having a certainty degree at least equal to a (resp. strictly greater than a). is inconsistent The expression denotes the inconsistency degree of . When is consistent, we have . Definition 2 and are said to be equivalent, denoted , iff , by where is the classical logic equivalence. There are two possible definitions of the inference process in possibilistic logic framework depending if we take into account the weights associated with conclusions or not: Definition 3 A formula is said to be a plausible consequence of , denoted by , iff . A possibilistic formula is said to be a possibilistic , iff consequence of , denoted by is consistent, , and . We define the plausible (resp. possibilistic) closure of , (resp. ), as follows: denoted by (resp.
).
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Note that if we ignore the weights in .
we simply get
Merging possibilistic knowledge bases A possibilistic merging operator, denoted by , is a function from to . Intuitively, will be used to merge the certainty degrees associated with pieces of information provided by different experts. More precisely, let be a set of consistent possibilistic bases. Our aim is to merge the bases of into a new one, denoted by , using the merging operator . is a set of weighted formulas such that is a B1 . . . . . . . . . . . . . . Bn
(a1, . . . . , an))
Figure 1: Merging possibilistic bases. propositional formula, and is the result of combining the certainty degrees associated with in each base of . More formally,(see also Figure 1) . Two natural properties for are: If
, then
. The first property says that if an information is not an explicit conclusion of any base, then it should not be an explicit conclusion of the result of merging. The second property is simply the monotonicity property which means that if all the experts say that a formula is more plausible than another formula , then the result of merging should confirm this preference. When we only have two bases and , then is equivalent to: and
.
Example 1 Let Let be a merging operator defined by Then,
and . .
. is also supposed commutative In the rest of this paper, and associative. We now define some classes of operators useful for the rest of the paper: is said to be: Definition 4 excessively optimistic if s.t. and .
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regular if it is neither excessively optimistic nor excessively pessimistic. strictly monotone iff , : if and .
, with , then
Postulates for propositional merging
(φ , a1) . . . . . . . . . . (φ φ , an)
(φ ,
excessively pessimistic if s.t. and . Excessively optimistic operators are such that merging not completely certain formulas can lead to totally certain beliefs. Excessively pessimistic formulas mean that merging some what certain information can lead to a completely uncertain formula.
We briefly give main postulates of merging propositional be a multi-set bases [5]. Let of propositional bases to be merged. is called an information set. (resp. ) denotes the conjunction (resp. disjunction) of the propositional bases of . The symbol denotes the union on multi-sets. For the sake of simplicity, if and are propositional bases and is an information set we simply write and instead of and respectively. We will denote the multi-set of size . A propositional merging operator is a function applied on and which returns a propositional base, denoted by . In [5], the authors have proposed a set of basic properties that a merging operator has to satisfy: is consistent, A : Consistency A : Information complementarity If is consistent, then A : Syntax independence , then , If where means that there exists a bijection from to such that . A : Cautiousness If is inconsistent, then In the case of bases, can be generalized as follows: If is inconsistent then , . A : Conjunction primacy , A : Recovering conjunction If is consistent, then Two classes of merging operators have been particulary analyzed in the literature: majority and arbitration operators [5] defined respectively by: Maj: Majority Arb: Arbitration
, and .
Plausible inference point of view The first adaptation focuses only on the set of plausible conclusions without taking care of degrees of inferences. The counterpart of the propositional postulates A will be denoted by P (P for plausible). Consistency Possibilistic logic, contrary to propositional logic, does not entail everything in presence of inconsistency. Therefore, rather than to require that the resulting base is consistent we require that the conclusions obtained from the fused bases are consistent, namely the adaptation of A is: is consistent. P : Hence, we have the following trivial result: Proposition 1 All operators satisfy P . Information complementarity is consistent, then P If . P says that the result of merging should recover all the information provided by the sources if they do not conflict. Such requirement can be satisfied if the merging operator guarantees that each formula which is entailed from at least one base, should also be explicitely present in the result of the merging. This can be captured by conjunctive operators defined by: Definition 5
is called a conjunctive operator if when for some , .
Proposition 2 A merging operator conjunctive operator.
satisfies P iff
is a
Syntax independence A has an immediate counterpart in the possibilistic logic setting, namely: then , where P If is defined as follows: such that and such that . Proposition 3 All operators satisfy P . Cautiousness The idea in the propositional postulate A is that when two bases are conflicting, the result of merging should not give preference to any base. This requirement is natural in propositional logic since formulas are flat and have the same reliability level. However, this cannot be right away in possibilistic logic framework. Indeed, Proposition 4 There is no strictly monotone operator which satisfies the immediate adpatation of A (Namely, if and is inconsistent, then and ). Note that strictly monotone operators are necessarily conjunctive. The converse is false. For example the max operator defined by is conjunctive but not strictly monotone. The following counter-example illustrates Proposition 4: be s.t. Counter-example 1 Let and where and . Let
be a strictly monotone operator. By construction, we have . We and since is strictly monohave tone and . We also have since and is strictly monotone. Hence, from which is entailed. There is a class of operators which satisfies the immediate adaptation of A (however, fails to satisfy P ). They are called disjunctive operators defined as follows: Definition 6
is called a disjunctive operator if when s.t. .
Note that disjunctive operators cannot be strictly monotone. Since formulas are weighted in possibilistic logic framework, a natural adaptation of A in this setting is to say that if and are conflicting and if they are equally prioritized, then the result of fusion should neither infer nor . The question now is ”how to express in possibilistic logic that two bases are equally prioritized ?”. Let us first introduce the notion of a certainty degree of a subbase: Definition 7 Let be a subbase of . We define its certainty degree, denoted by , by . is equal to the degree This definition means that of the least certain formulas in which belong to . Now, we define the priority between two bases as follows: Definition 8 each conflict
is said to be more prioritized than in , we have .
if for
is more prioritized than if the least formulas Namely, in each conflict between and are in . and are said to be equally prioritized if for each conflict in , the least prioritized formulas in belong to both and . and be the two following possibilisExample 2 Let tic bases defined as follows: and . : There are two conflicts in and . We have and . and are equally prioritized. Then, However if , is prioritized than since then , and . We now give a suitable adaptation of A in possibilistic logic framework: and is inconsistent and, and are equally P If prioritized, then and . Then we have: Proposition 5 All operators satisfy P . The remaining postulates have immediate adaptations: let .
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Possibilistic inference point of view
P P
If
is consistent, then .
Proposition 6 All operators satisfy is a strictly monotone operator.
.
satisfies
iff
Majority . P Intuitively, majority is related to the idea of reinforcement, namely if a same formula is believed to a degree by two agents, it should be believed with a larger weight in the result of merging. Reinforcement operators are defined as follows: Definition 9
is said to be a reinforcement operator if when . And, if for some , .
Note that reinforcement operators are also conjunctive. However, reinforcement and strictly monotone operators are unrelated. For example is strictly monotone but not a reinforcement operator. Note that reinforcement property is not enough to satisfy P . An example of a reinforcement operator which does is . To satisfy P , not satisfy P should also be strictly monotone. satisfies P if Proposition 7 1 and a reinforcement operator .
is a strictly monotone
Arbitration P . Arbitration postulate means that ignores the redundancy of information. Formally, we should have: . This requirement can be obtained by operators called idempotent operators and defined as follows: Definition 10 Proposition 8 ator.
is called an idempotent operator if . satisfies P
iff
is an idempotent oper-
To summarize the above results, we have: 1.
satisfies P P P P P P if and only if is a strictly monotone operator. An example of such operators . is the probabilistic sum, i.e.
2.
satisfies P P P P P P P if and only if is a strictly monotone and reinforcement operator. An example of such operators is the probabilistic sum.
3. If is an idempotent operator then it satisfies P P P P P . Examples of such operators are and . 1
provided that
does not conflict completely sure formulas of
.
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In this section, we briefly give the adaptation of propositional postulates, denoted by W (W for weighted) in possibilistic logic framework when we keep track of the weights attached to conclusions. Therefore we are interested in analysing the possibilistic closure . We recall that this set is defined by: Postulates W , W and W are the direct counterparts of P , P and P . Namely, is consistent. W , then . W If W If and is inconsistent and, and are equally prioritized, then and . Then, we have the following result: Proposition 9 All operators satisfy W , W and W . Information complementarity The idea in A is that the result of merging should recover classical conjunction when the bases are agreeing. When the weights attached to formulas are considered, recovering the complementarity between the bases means that, if a formula is entailed from at least one base then it should , with a weight at least equal also be entailed from to . Hence, we have the following adaptation of A : is consistent, then W If if then , with . We have shown in the first adaptation (when only considering plausible conclusions) that such requirement is obtained by operators called conjunctive operators and satisfying the following property: if . However, it may exist conjunctive operators which do not satisfy A . and be two possibilistic bases such Example 3 Let and which that are together consistent. Let be the mean operator defined by: . Then, which does not entail neither nor those entailed from . formulas entailed from Indeed, W excludes some conjunctive operators. Note that W is stronger than P . To satisfy the requirement expressed by W , we define a sub-class of conjunctive operators called strongly conjunctive, defined by: Definition 11 A merging operator is called strongly conjunctive if Hence, we have the following result: Proposition 10 satisfies W .
is a strongly conjunctive operator iff
Conjunction primacy The adaptation of A needs a discussion. A possible adaptation would be the following: let . . This adaptation does not reflect faithfully the idea behind is inconsistent. Indeed, A , when
in the propositional postulate A when is inconsistent, the postulate is trivially satisfied. But the above adaptation is not trivially satisfied since possibilistic inference does not entail everything in presence of inconsistency. Hence, it is more natural to restrict to the case of consistency. Another point is how to fix the definition of conjunction and in the sense of Definition 9 used between . A stronger reinforcement operator, defined below, is appropriate. Hence, we have: is consistent, then W If , is defined as follows: where Let and . Then, if or otherwise. The use of a parametrized operator means that combining two formulas whose certainty is above the inconsistency level, get the maximal weight. Proposition 11 All operators satisfy W . Conjunction recovering Contrary to A , A refers to a decomposition of one group into two subgroups. Then, an idempotent conjunction (which leads to the union of knowledge bases) is enough to adapt A : is consistent, then W If . We have shown in the first adaptation that the operator should be strictly monotone in order to recover the conjunction. However when we consider the weights attached to formulas, this is not enough. Indeed: satisfies W iff is a strongly conjuncProposition 12 tive and strictly monotone operator. As a summary of this adaptation, we get the following result: satisfies W W W W W W if and only if is a strongly conjunctive and a strictly monotone operator. Note that this adaptation is stronger than the plausible adaptation, which only requires to be strictly monotone to satify P . P
Conclusion In this paper two adaptations of propositional fusion postulates to the possibilistic framework has been given. We have pointed out the existence of several classes of merging operators. In particular different kinds of conjunctive operators have been defined, due to the information complementarity postulates. Among them associative strictly monotone (and also strongly conjunctive in the second adaptation) operators are particularly attractive, for knowledge fusion because they indeed satisfy all the basic postulates. A futur work will be to investigate the integration of integrity constraints and see how postulates given in [6] can be extented to the possibilistic logic framework. Integrity constraints are requirements that the resulting base should
satisfy. In possibilistic logic, such constraints are represented by means of fully certain formulas (with weight ). Another futur work is to see to what extent postulates of social choice [7] can be used for merging knowledge bases. Acknowledgment The authors would like to thank D. Dubois and H. Prade for their help on the first version of this paper.
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