Preference Revision via Declarative Debugging Pierangelo Dell’Acqua Dept. of Science and Technology - ITN Linköping University, Sweden EPIA’05, Covilhã, PortugalDecember, 2005 Luís Moniz Pereira Centro de Inteligência Artificial - CENTRIA Universidade Nova de Lisboa, Portugal

Preference criteria are subject to be: modified when new information is brought to the knowledge of an individual, or aggregated when one needs to represent and reason about the simultaneous preferences of several individuals. Problem

Example: suppose you invite three friends Karin, Helena and Elisa to go and see a movie. - Karin prefers thrillers to action movies. - Helena, on the other hand, prefers action movies to thrillers. - Finally, Elisa is like Helena and prefers action movies to thrillers. Which movie do you choose?

Often, the resulting preference criteria may not satisfy the required properties (e.g., a strict partial order) and must therefore be revised.

The problem of combining preferences arises in several application domains. In computer science: database and information retrieval based on collaborative filtering, e.g. recommendation systems internet search and meta-search systems multi-media systems, e.g., adaptive radio but also in: economics: utility theory political science: work on voting or polling over the internet (electronic democracy) social science: work on social choice behaviour Applications

Preference aggregation has been studied from several perspectives: [J. Chomicki,03], [H. Andreka et al.,02] study the preservation of properties by different composition operators [Grosof,93] proposes a new method for preference aggregation that generalizes the lexicographic combination method [Rossi et al.,04] study the problem of fairness of preference aggregation systems [Yager,01] and [Rossi et al.,04] investigates the problem of preference aggregation in the context of MASs Proposed approaches

In contrast, we investigate how to reconcile (a posteriori) preference criteria once they are modified or aggregated. We consider preference criteria expressible by logic programs, and investigate the problem of revising them via declarative debugging. We employ an adapted version of the contradiction removal method defined for the class of normal logic programs plus integrity constraints proposed in [*]. [*] L. M. Pereira, C. Damásio, and J. J. Alferes, Debugging by Diagnosing Assumptions. In P. Fritzson (ed.), 1st Int. Ws. on Automatic Algorithmic Debugging, AADEBUG'93, LNCS 749, pp Preproceedings by Linköping Univ., 1993 Our approach

Let L be a first order language. A normal logic program P over L is a set of rules and integrity constraints: A  L 1,..., L n (n  0)   L 1,..., L n where A is an atom, every L i is a literal and  is an atom denoting contradiction. The meaning of P is given by Well-Founded Semantics. If a literal L belongs to the well-founded model of P, we write P ² L. P is contradictory if P ²  Language L

Given a set N, a preference relation  is any binary relation on N. a  b means that a is preferred to b. Every preference relation  induces an indifference relation ». a and b are indifferent a » b iff a ¨ b and b ¨ a. Preference relations

Typical properties of  include: - irreflexivity: 8 x. x ¨ x - asymmetry: 8 x 8 y. x  y ) y ¨ x - transitivity: 8 x 8 y 8 z. (x  y Æ y  z) ) x  z - negative transitivity: 8 x 8 y 8 z. (x ¨ y Æ y ¨ z) ) x ¨ z - connectivity: 8 x 8 y. x  y Ç y  x Ç x = y The relation  is: - a strict partial order if it is irreflexive and transitive (hence asymmetric); - a weak order if it is a negatively transitive strict partial order; - a total order if it is a connected strict partial order.

Given a contradictory program P, to revise its contradiction (  ) we modify P by adding and removing rules. In this framework, the diagnostic process reduces to finding such rules. Given a set C of predicate symbols of L, C induces a partition of P into two disjoint parts: P = P c [ P s P c : changeable part, P s : stable part Let D =  U, I  where U Å I = ;, U µ C and I µ P c. Then D is a diagnosis for P iff (P-I) [ U 2 . D =  U, I  is a minimal diagnosis if there exists no diagnosis D 2 =  U 2, I 2  for P such that (U 2 [ I 2 ) ½ (U [ I). Diagnoses

Given two preference relations  1 and  2, the prioritized composition  of  1 and  2 is defined as: x  y ´ x  1 y Ç ( x » 1 y Æ x  2 y ) x » 1 y ´ x ¨ 1 y Æ y ¨ 1 x Suppose that  is required to be strict partial order. Let:a  1 bb  2 c c  2 a  cond b  2 a Example: prioritized composition

a  b b  c c  a ab c | ab c | ab c | ab c | Suppose we want to revise only the preference relation  2. Three possible revisions : Then,  is not a strict partial order.

P is contradictory: M P = {..., p(a,b), p(b,c), p(c,a),  } P admits three minimal diagnoses: D 1 =  {p2(a,c) }, {p2(c,a)  cond} , D 2 =  { p2(c,b)}, {p2(b,c)}  D 3 =  { }, {p2(b,c), p2(c,a)  cond}    p(x,x)   p(x,y), p(y,x)   p(x,y), p(y,z), not p(x,z) p(x,y)  p1(x,y) p(x,y)  ind1(x,y), p2(x,y) ind1(x,y)  not p1(x,y), not p1(y,x) p1(a,b) cond This situation can be formalized as: p2(b,c) p2(c,a)  cond p2(b,a) PsPs PcPc

To compute the minimal diagnoses of a contradictory program P, we employ a contradiction removal method ( see [*] ) Based on the idea of revising (to false) some of the default atoms. A default atom not A can be revised to false by simply adding A to P. The default atoms not A that are allowed to change their truth value are exactly those for which there exists no rule in P defining A. Such literals are called revisables. A set Z of revisables is a revision of P iff P [ Z 2  Computing minimal diagnoses

Consider the program P = P c  P s with revisables { b, d, e, f }. P is contradictory since M P = { a, a’,  }. The revisions of P are {e}, {d,f}, {e,f}, and {d,e,f}, where the first two are minimal. Example a  not b, not c a’  not d c  e PcPc   a, a’   b   d, not f PsPs

The transformation  maps programs over L into equivalent programs that are suitable for contradiction removal. The transformation  that maps P into a program P ‘ =  ( P ) is obtained by applying to P the following two operations: Add not incorrect (A  Body) to the body of each rule A  Body in P c Add the rule: p(x 1,..., x n )  uncovered( p(x 1,..., x n ) ) for each predicate p with arity n in C, where x 1,..., x n are variables. Property: Let P be a program over L and L a literal. Then, P ² L iff  ( P ) ² L Transformation 

 ( P ) admits three minimal revisions wrt. the revisables of the form incorrect(.) and uncovered(.) : Z 1 = { uncovered(p2(a,c)), incorrect(p2(c,a)  cond) } Z 2 = { uncovered(p2(c,b)), incorrect(p2(b,c)) } Z 3 = { incorrect(p2(b,c)), incorrect(p2(c,a)  cond) } Example: prioritized composition (con’t)   p(x,x)   p(x,y), p(y,x)   p(x,y), p(y,z), not p(x,z) p(x,y)  p1(x,y) p(x,y)  ind1(x,y), p2(x,y) ind1(x,y)  not p1(x,y), not p1(y,x) p1(a,b) cond p2(b,c)  not incorrect(p2(b,c)) p2(c,a)  cond, not incorrect(p2(c,a)  cond) p2(b,a)  not incorrect(p2(b,a)) p2(x,y)  uncovered(p2(x,y))  ( P )

The following result relates the minimal diagnoses of P with the minimal revisions of  ( P ). Theorem: A pair D =  U, I  is a diagnosis for P iff Z = {uncovered(A): A  U} [ { incorrect( A  Body ): A  Body  I} is a revision of  ( P ), where the revisables are all the literals of the form incorrect(.) and uncovered(.). Furthermore, D is a minimal diagnosis iff Z is a minimal revision. To compute the minimal diagnosis of P we consider the transformed program  ( P ) and compute its minimal revisions. An algorithm for computing minimal revisions is given in [*]. Property

A preference revision problem typically has several minimal diagnoses. To select the best diagnoses, one can employ meta-preference information: - temporal information - weights associated to preferences - specificity of diagnoses - minimality wrt. the number of changes - fairness, e.g., from the MAS perspective Selecting minimal diagnosis

We have presented an approach to preference revision that is based on a declarative debugging technique. The proposed framework: - is flexible and general: it allows to express any preference criteria and methods for preference aggregation; - gives us an extra level of abstraction by permitting to select the best diagnosis for the problem at hand; - has a correct proof procedure. Conclusions