Prime Normal Forms in Belief Merging

The aim of Belief Merging is to aggregate possibly conflicting pieces of information issued from different sources. The quality of the resulting set is usually considered in terms of a closeness criterion between the resulting belief set and the initial belief sets. The notion of distance between belief sets is thus a crucial issue when we face the merging problem. The aim of this paper is twofold: introducing a syntactical way to calculate distances and proposing the use of a distance based on prime implicants and prime implicates that considers the importance of each propositional symbol in the belief set. Introduction The goal of Belief Merging is to aggregate in a consistent way multiple beliefs usually represented as sets of logical statements so as to obtain a new set of statements (Liberatore and Schaerf 1998). Such process is applied in knowledge based systems when a cognitive agent receives a set of contradictory pieces of information from many sources, e.g. other agents or its own sensors or “mental states”. Belief Merging process is based on three main components (Konieczny, Lang, and Marquis 2002): a notion of distance between propositional models, a function to aggregate distances and a procedure to select the closest eligible resulting sets w.r.t. the aggregation stage. Usually, the Hamming distance is adopted in order to calculate the distance between models, where a propositional symbol is considered as minimal change unit (Dalal 1988). This distance has been widely considered in belief revision (Dalal 1988; Satoh 1988), belief update (Forbus 1989; Winslett 1988) and belief merging (Konieczny, Lang, and Marquis 2002). However, this consideration can not be minimal (Marchi, Bittencourt, and Perrussel 2010; Bittencourt, Perrussel, and Marchi 2004), because changing one propositional symbol truth value may lead to significant changes if this symbol frequently appears in the formulas of the initial belief bases. Thus, the common notion of minimal change is biased by the structure of the belief bases. In that context, two main problems can be identified: first, the belief merging process is performed over set of models; and second, the minimal change unit may promote significant changes on the belief base. These issues have already Copyright c © 2011, Association for the Advancement of Artificial Intelligence (www.aaai.org). All rights reserved. been partially addressed in (Perrussel, Marchi, and Bittencourt 2008) where it was proposed a merging process that can satisfy the postulates characterizing belief merging proposed in (Konieczny, Lang, and Marquis 2002) in a partial way by choosing implicants (or implicates) among all the belief sets. As proposed, the merging process entails that some privileges are given to some specific formulas and these privileges are not compatible with the fairness principle which is a key one in belief merging: fairness states that all beliefs and all belief sets should be equally considered. In this paper, we propose a new way to merge beliefs that avoid this problem by proposing a new way to define the implicants/implicates which will belong to the resulting base; and it is not required that these statements should belong to one of the initial belief sets. Hence, in that way, we take care of the fairness issue. We also avoid this limitation about postulates satisfaction. This paper focuses on a syntactical way to perform the merging process where the belief bases are represented as sets of prime implicants, and by considering another minimal change distance based on prime implicants and prime implicates representation. These two representations enable us to correlate symbols in terms of models (prime implicants) and also in terms of structure (prime implicates). The key results are that the proposed framework (i) is fully compliant with the definition of belief merging and (ii) gives a new perspective on the closeness criterion for setting proximity between belief bases. The paper is organized as follows: we first give the formal definitions which set the notion of prime forms (implicate and implicant) and we detail the quantum notation introduced by (Bittencourt 1998) which aims at relating implicate and implicant. Next, we formally present the concept of belief merging and present the set of associated postulates proposed by Konieczny and Péres (Konieczny and Pérez 2002). We also remind the definition of majority (Lin and Mendelzon 1999) and arbitration operators (Liberatore and Schaerf 1998). The next section presents the syntactical approach of the belief merging process. Next, we introduce the new minimal change unit and present its application on belief merging processes. We finally conclude the paper by presenting some considerations and future work. 51 Proceedings of the Twenty-Fourth International Florida Artificial Intelligence Research Society Conference