Product-based Causal Networks and Quantitative Possibilistic Bases

In possibility theory, there are two kinds of possibilistic causal networks depending if possibilistic conditioning is based on the minimum or on the product operator. Similarly there are also two kinds of possibilistic logic: standard (min-based) possibilistic logic and quantitative (product-based) possibilistic logic. Recently, several equivalent transformations between standard possibilistic logic and min-based causal networks have been proposed. This paper goes one step further and shows that product-based causal networks can be encoded in product-based knowledge bases. The converse transformation is also provided. Introduction Generally, uncertain pieces of information or flexible constraints can be represented in different equivalent formats. In possibility theory, possible formats can be: • graphical-based representations, viewed as counterparts of probabilistic Bayesian networks [11,12], and • logical-based representations which are simple extensions of classical logic. In graphical representations [1,9,10], uncertain information is encoded by means of possibilistic causal networks which are composed of Directed Acyclic Graph (DAG) and conditional possibility distributions. In logical representations [7], uncertain information is encoded by means of possibilistic knowledge bases which are sets of weighted formulas having the form (φi, αi) where φi is a propositional formula and αi is a positive real number belonging to the unit interval [0,1]. Each possibilistic causal network (resp. each possibilistic knowledge base) induces a ranking between possible interpretations of a language, called a possibility distribution. The possibility degree associated with an interpretation is obtained by combining the satisfaction degrees of this interpretation with respect to each weighted formula of the knowledge base, or with respect to each conditional possibility degree of the causal network. Two combination operators have been used [7]: minimum operator and product operator. Therefore, there are two Copyright c © 2004, American Association for Artificial Intelligence (www.aaai.org). All rights reserved. kinds of causal networks: min-based possibilistic networks and product-based possibilistic networks. Similarly, two kinds of possibilistic knowledge bases are defined: min-based possibilistic logic (standard possibilistic logic) and product-based possibilistic logic called also quantitative possibilistic logic. In the rest of this paper, we only focus on product-based possibilistic causal networks and on quantitative possibilistic logic. Even if graphical or logical representation can encode same pieces of uncertain information, they in general use different inference tools. For instance, some inference tools in possibilistic causal networks are simple adaptations of probabilistic propagation algorithms [9,10]. In possibilistic logic, the inference tools are based on SAT provers (satisfiability test of propositional formulas). Hence, it is very important to have equivalent transformations from one representation format to another in order to take advantage of these different inference tools. Another need of these transformations is when we fuse uncertain information given in different formats provided by different sources. Indeed, existing fusion modes assume that all information is represented in a same format, which is not always the case in practice. Having transformations algorithms between different representations allow the use of existing fusion modes even if the information is represented in different formats. In [2,5] equivalent transformations have been provided between min-based possibilistic knowledge bases and minbased causal networks. This paper goes one step further. It provides an encoding of product-based possibilistic causal networks into quantitative possibilistic knowledge bases, and conversely. The rest of this paper is organised as follows. Next section gives a background on possibilistic logic and posibilistic causal networks. Section 3 studies the transformations between product-based graphs and quantitative possibilistic knowledge bases. Section 4 gives the converse transformation. Section 5 concludes the paper.