Preference Reasoning in Logic Programming

1
Preference Revision via Declarative Debugging
Pierangelo DellAcqua Dept. of Science and
Technology - ITN Linköping University, Sweden
Luís Moniz Pereira Centro de Inteligência
Artificial - CENTRIA Universidade Nova de Lisboa,
Portugal
EPIA05, Covilhã, Portugal
December, 2005
2
Problem

• 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.

3

• 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?

4

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

5
Applications

• 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

6
Proposed approaches

• 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

7
Our approach

• 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.
  58-74. Preproceedings by Linköping Univ., 1993

8
Language L

• Let L be a first order language.
• A normal logic program P over L is a set of rules
  and integrity constraints
• A ? L1 , . . . , Ln (n ? 0)
• ? ? L1 , . . . , Ln
• where A is an atom, every Li 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 ?

9
Preference relations

• 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
  .

10

• 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.

11
Diagnoses

• 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 Pc Ps
• Pc changeable part, Ps stable part
• Let D ?U, I? where U Å I , U µ C and
  I µ Pc.
• Then D is a diagnosis for P iff (P-I) U 2 ?.
• D ?U, I? is a minimal diagnosis if there exists
  no diagnosis
• D2 ?U2, I2? for P such that (U2 I2) ½ (U I).

12
Example prioritized composition

• Given two preference relations Â1 and Â2, the
  prioritized composition  of Â1 and Â2 is defined
  as
• x  y x Â1 y Ç ( x 1 y Æ x Â2 y )
• x 1 y x 1 y Æ y 1 x
• Suppose that  is required to be strict partial
  order.
• Let a Â1 b b Â2 c
• c Â2 a ? cond
• b Â2 a

13
Then, Â is not a strict partial order.

• a  b
• b  c
• c  a

Suppose we want to revise only the preference
relation Â2 . Three possible revisions
14
This situation can be formalized as
? 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) p2(c,a) cond p2(b,a)
Ps
Pc

• P is contradictory MP . . . , p(a,b),
  p(b,c), p(c,a), ?
• P admits three minimal diagnoses
• D1 ? p2(a,c) , p2(c,a)cond ?, D2 ?
  p2(c,b), p2(b,c) ?
• D3 ? , p2(b,c), p2(c,a)cond ?

15
Computing minimal diagnoses

• 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 ?

16
Example

• Consider the program P Pc ? Ps
• with revisables b, d, e, f .
• P is contradictory since MP a, a, ? .
• The revisions of P are e, d,f, e,f, and
  d,e,f, where the first two are minimal.

a not b, not c a not d c e
? a, a ? b ? d, not f
Ps
Pc
17
Transformation ?

• 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 Pc
• Add the rule p(x1, . . ., xn) uncovered(
  p(x1, . . ., xn) )
• for each predicate p with arity n in C, where
  x1, . . ., xn are variables.
• Property Let P be a program over L and L a
  literal. Then,
• P ² L iff ?( P ) ² L

18
Example prioritized composition (cont)
?( P )
? 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 ) admits three minimal revisions wrt. the
  revisables of the form incorrect(.) and
  uncovered(.)
• Z1 uncovered(p2(a,c)), incorrect(p2(c,a)
  cond)
• Z2 uncovered(p2(c,b)), incorrect(p2(b,c))
• Z3 incorrect(p2(b,c)), incorrect(p2(c,a)
  cond)

19
Property

• 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 .

20
Selecting minimal diagnosis

• 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

21
Conclusions

• 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.