D. Dubois, H. Prade, F. Touazi (coopéra7on avec A. Hadjali, S. Kaci)

Transcription

1 D. Dubois, H. Prade, F. Touazi (coopéra7on avec A. Hadjali, S. Kaci)

2 Mo7va7on Two ways of expressing qualitative preferences Logical statements with priorities (possibilistic logic) Conditional preference networks (CP-nets) Compared advantages The logical approach looks less constrained but suffers from drowning effects CP-nets may be found more convenient to a user for expressing preference but the format is rigid.

3 Ques7ons Is the translation of CP-nets in possibilistic logic possible? Previous results suggested that yes. Here we question these results and suggest that no. We study the difficulties of translation

4 Background on CP-nets A CP-net N is a directed acyclic graph relating variables A CP-net preference is of the form u ij : x i > x i (or u ij : x i >x i ) where u ij is called a context j (instances of parent variables of x i ) The CP-table T i attached to x i is a collection of CP-net preferences for all contexts of x i

5 Background on CP-nets CP-net preferences are understoood ceteris paribus: u ij : x i > x i should be understood as u ij x i y k > u ij x i y k for all instantiations y k of the remaining variables Two solutions x and x are comparable if there is a sequence of one-variable flips transforming one to the other. Father nodes Pa(x i ) seem to be more important than children nodes x i

6 Example 1: Jackets pants and shirt

7 Encoding of CP-nets in possibilistic logic u: x > x becomes ( u x, α) where α is a symbolic necessity weight The CP table in each node i is encoded as a single pair (φ i, α i ) = (( u i x i ) (u i x i ) α i ) If x i is a father of x j, α i > α j is assumed (incomparable otherwise) One can associate a possibilis7c base to a CPnet with one formula per node.

8 Ordering solu7ons for a Π base To each base Σ and each interpreta7on w, define a vector v w (Σ) with as many components as formulas (φ i, α i ) ( = CP net nodes). for component i : 1 > α i si w = φ i, and 1 α i otherwise Ranking solu7ons = ranking vectors v w (Σ).

9 Order relations over vectors Leximin Discrimin Symmetric Pareto : Means refined by Min Pareto

10 Results Contrary to what is claimed in previous papers (Kaci, Hadjali) neither symmetric Pareto nor Leximin orderings can encode Cp net orderings symmetric Pareto is not discriminant enough; and leximin is too discriminant (counter examples) We have strong evidence that the CP net ordering can be bracketed by these two rela7ons symmetric Pareto ordering captures the CP net ordering if father nodes have only one child node. The alleged father node priority in CP nets does not seem to be transi7ve (counter examples).

11 Perspec7ves Complete formal proofs for these conjectures. An apparently unsolved issue: Is a solu7on viola7ng a subset of CP net preferences violated by another solu7on be[er (in the CP net sense) than the la[er. Extend results to more general graphical preference representa7ons. Connec7on with the study of the logic of par7ally ordered logical bases (with C. Cayrol, Touazi) Connec7ons to other logical preference languages. Applica7ons à l interroga7on de BD

12 Publica7ons A. HadjAli, S. Kaci, H. Prade: Database preference queries a possibilis7c logic approach with symbolic priori7es. Ann. Math. Ar5f. Intell. 63(3 4): (2011) D. Dubois, H. Prade, F. Touazi. Condi7onal Preference Nets, Possibilis7c Logic, and the Transi7vity of Priori7es. Dans : SGAI Interna5onal Conference on Innova5ve Techniques and Applica5ons of Ar5ficial Intelligence, Cambridge, UK, 10/12/ /12/2013, Max Bramer, Miltos Petridis (Eds.), Springer, p , D. Dubois, H. Prade, F. Touazi. A Possibilis7c Logic Approach to Condi7onal Preference Queries Dans : Flexible Query Answering (FQAS 2013), Granada, Spain, 18/09/ /09/2013, Springer, Lecture Notes in Computer Science 8132, p , D. Dubois, H. Prade, F. Touazi. Condi7onal Preference Nets and Possibilis7c Logic Dans : European Conference on Symbolic and Quan7ta7ve Approaches to Reasoning with Uncertainty (ECSQARU 2013), Utrecht, 08/07/ /07/2013, Linda Van der Gaag (Eds.), Springer, Lecture Notes in Computer Science 7958, p , 2013.