Corrigendum to Database Preference Queries - A Possibilistic Logic Approach with Symbolic Priorities

This note corrects a claim made in the above-mentioned paper about the exact representation of a conditional preference network by means of a possibilistic logic base with partially ordered symbolic weights. We provide a counter-example that shows that the possibilistic logic representation is indeed not always exact. This is the basis of a short discussion on the difficulty of obtaining an exact representation. This note corrects a claim made in [6] about the representation of Conditional Preference networks (CP-nets for short) [1] by means of a possibilistic logic base [2], as well as a similar claim in [7,8]. A CP-net encodes a set of preference statements concerning the values of Boolean decision variables, conditioned on the values of other Boolean decision variables that influence the former. More formally, let V = {X1, · · · , Xn} be a set of Boolean variables. We denote by Ast(S) the set of interpretations of variables of S (⊆ V ). Definition 1 A CP-net N over V = {X1, · · · , Xn} is a directed graph with nodes X1, · · · , Xn, and there is a directed edge from Xi to Xj if the preference about the value Xj depends on the value of Xi. Each node Xi ∈ V is associated with a conditional preference table CPTi that associates a strict preference (xi > ¬xi or ¬xi > xi) with each possible instantiation uji ∈ Ast(Pa(Xi)) of the parents Pa(Xi) of Xi (if any). Each entry in a conditional preference table CPTi is of the form φ = u j i : ?jxi > ?j¬xi, where uji ∈ Ast(Pa(Xi)), ?j is blank if the preference is xi > ¬xi and is ¬ otherwise. Each interpretation ω, i.e., an instantiation of all variables in V , is understood as a solution to the decision problem described by the preference statements in the CP-net. A CP-net induces a partial preference ordering over solutions defined as follows. Each Allel Hadjali IRISA/ENSSAT, Université Rennes I, 6 rue de Kérampont 22305 Lannion Cedex, France, E-mail: [email protected] Didier Dubois · Henri Prade · Fayçal Touazi IRIT, Université Paul Sabatier, 118 route de Narbonne 31062 Toulouse Cedex 9, France E-mail: [email protected], [email protected], [email protected]