How to Choose Weightings to Avoid Collisions in a Restricted Penalty Logic

Penalty Logic is a natural and commonsense Knowledge Representation technique to deal with potentially inconsistent beliefs. Penalty Logic allows some kind of compensation between different pieces of information. But one of the main and less studied flaws of Penalty Logic is the influence of the choice of weights on inference: the same pieces of information can provide extremely different results just by changing some weights. This paper concentrates on weightings and on the problem of collisions between interpretations which yield weak conclusions. It focuses more particularly on a family of weightings, the σ-weightings. We show that some of these weightings avoid collisions but that in the meanwhile they disable the mechanism of compensation (and so the interest) of Penalty Logic. We establish then that two of them are suitable for avoiding collisions and maintaining compensation. We obtain their logical characterizations while considering the weightings only and not the associated formulas. Finally, we propose an original weighting, the Paralex Weighting, that improves even more the previous weightings. Introduction Penalty Logic is a natural and commonsense Knowledge Representation technique to deal with potentially inconsistent beliefs. It has been proposed in (Pinkas 1991; 1995) and developed in (Dupin de Saint-Cyr, Lang, & Schiex 1994). Penalty logic provides an intuitive framework to deal with weighted formulas. A penalty is associated with each interpretation: this penalty is the sum of the weights of the formulas falsified by the interpretation. Thus, the main characteristic of this formalism is its ability to compensate by additivity of the weights: if the most preferred piece of information is falsified by an interpretation, the interpretation is not automatically rejected. However this formalism is well-known to be syntaxdependent as far as formulas are concerned. Moreover, one of the main and less studied flaws of Penalty Logic is the influence of the choice of weights on inference: the same pieces of information can provide extremely different results just by changing some weights. But an expert cannot take into account the processes of compensation and Copyright c © 2008, Association for the Advancement of Artificial Intelligence (www.aaai.org). All rights reserved. deduction of Penalty Logic when he encodes his beliefs: when they do not represent an additive measure (such as money), the weights he provides are often artificial. It is easier for an expert to represent a cost than the reliability of testimonies, opinions or judgments. Many qualitative methods have been provided to deal with prioritized information only (without any weight). Different strategies can be applied, like best-out ordering (Benferhat et al. 1993), discrimin ordering (Brewka 1989; Geffner 1992; Benferhat et al. 1993), leximin ordering (Lehman 1995; Benferhat et al. 1993) or linear ordering (Nebel 1994). But none of these formalisms allows compensation like Penalty Logic: falsifying several formulas of less importance can be equal to falsifying a formula of greater importance. Moreover, none of these methods treats the central problem of this paper: collision avoiding. This paper concentrates on weightings and on the problem of collisions between interpretations. Two interpretations are said to collide if their κ-values (the value of the κ-function associated with the interpretations) are equal. In this case, interpretations cannot be sorted out and the conclusion can be excessively cautious. If two interpretations falsify formulas of different importance, they should ideally have different κ-values in order to avoid weak conclusions. We show in this paper how to improve the results of Penalty Logic just by considering the choice of weightings. This paper provides a survey of different natural weightings that can automatically be generated from the initial weighted beliefs provided by the expert. We show that some of these weightings increase the risk of collision. Others avoid collisions but in the meanwhile they disable the mechanism of compensation (and so the interest) of Penalty Logic. We study more particularly a family of weightings, the σ-weightings. We establish then that two of them are suitable for avoiding collisions and maintaining compensation. We obtain their logical characterizations while considering these weightings only and not the associated formulas. But they compare interpretations with respect to the minimal (least important) formulas that they falsify. That is the reason why we propose finally an original weighting, called Paralex Weighting, that solves this problem. The first section provides some usual methods for dealing with ranked information and Penalty Logic. The second section presents the problem of collisions for Penalty ha l-0 08 66 55 6, v er si on 1 26 S ep 2 01 3 Author manuscript, published in "11th International Conference on Principles of Knowledge Representation and Reasoning (KR'08), Sydney : Australia (2008)"