Complex Preferences for Answer Set Optimization

1
Complex Preferences for Answer Set Optimization

• Author Gerhard Brewka
• Published 9th International Conference
  Principles of Knowledge Representation and
  Reasoning, Whistler, Canada, June 2004.
• Xiaomeng Wu
• Nov 30, 2004

2
Outline

• Introduction
• A motivating example
• Preference Description Language (PDL)
• Special case
• Discussion

3
Introduction

• Introduction
• Example
• PDL
• Special case
• Discussion

• Answer Set Programming
• 1. Generate potential solutions
• 2. Specify conditions, destroy non-solutions
• Answer set optimization
• 3. Pick one of the solutions with maximal quality

-gt generate-and-test
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Introduction

• Introduction
• Example
• PDL
• Special case
• Discussion

• Optimization
• Quality of an answer set
• Preference ordering, relative quality
• Qualitative preference
• Numerical information -gt hard to obtain
• If numerical information is available -gt easy to
  use

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Introduction

• Introduction
• Example
• PDL
• Special case
• Discussion

• Related work
• Smodels weight constraints
• Maximize, minimize
• DLV weak constraints
• lt- body. wl
• (w numerical penalty, l priority level)
• Fixed built-in preference handling
• ! Need flexible preference strategies -gtPDL

6
A motivating example

• Introduction
• Example
• PDL
• Special case
• Discussion

• Scheduling problem
• Assign lecturers, time slots and rooms to courses
• Use ASP with cardinality constraints
• l a1, , ar u
• l a(x) b(x) u
• Rules
• 1 teaches( L,C )lecturer( L ) 1 lt- course( C
  )
• 1 in( R,C ) room( R ) 1 lt- course( C )
• 1 at( S,C ) slot( S ) 1 lt- course( C )

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A motivating example

• Introduction
• Example
• PDL
• Special case
• Discussion

• Several hard constraints
• One course per lecturer
• lt- teaches( L,C ), teaches( L, C), C!C
• Confliction check
• lt- in( R,C ), in( R, C ), at( S,C ), at( S,C),
  C!C
• Answer set lt-gt solution in the form
• teaches(l,c), in(r,c) and at(s,c)

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A motivating example

• Introduction
• Example
• PDL
• Special case
• Discussion

• How to add lecturers personal preference?
• Preferred courses to teach
• Preferred time slots, e.g. morning is preferred
• Preferred rooms, e.g. close to office
• Sometimes, impossible to satisfy everyone

-gt Rank using penalty values, overall penalty is
small
-gt may be purely qualitative
-gt may be purely qualitative
-gt can be profs are important than assitants
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PDL

• Introduction
• Example
• PDL
• Special case
• Discussion

• PDL Preference Description Language
• Goal to select maximally preferred answer sets
• Compared with rule-based, PDL
• Can do qualitative numerical combination
  preference representation
• Can use different preference combination
  strategies

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PDL

• Introduction
• Example
• PDL
• Special case
• Discussion

• Building block preference rules
• Represent context dependent preference
• Represent a ranking of answer sets
• In the form
• a, b literals built from atoms
• C boolean combinations over atoms e.g.
• p integers satisfying piltpj whenever iltj

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PDL

• Introduction
• Example
• PDL
• Special case
• Discussion

• PDL experession prex
• Represents a preorder, a transitive and reflexive
  relation, on an answer set
• A similar role as objective functions in
  numerical optimization
• Definitions
• Preorder
• Optimal answer set S according to the preorder

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PDL

• Introduction
• Example
• PDL
• Special case
• Discussion

• Penalty of answer sets pen(S, prex)
• Several complex preorders
• pareto standard Pareto ordering
• lex lexicographic ordering
• inc and rinc Inclusion based strategy
• card and rcard of orderings satisfied
• psum add the penalties, smaller preferred

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PDL

• Introduction
• Example
• PDL
• Special case
• Discussion

• Example, revisited
• Course preference
• Each course is assigned a penalty
• Each lecturer can assign 10 points in total
• therefore

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PDL

• Introduction
• Example
• PDL
• Special case
• Discussion

• Example, revisited
• Time preference
• Room preference

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PDL

• Introduction
• Example
• PDL
• Special case
• Discussion

• Example, revisited
• Prof vs. assistant
• Cp union of Ci such as li is a profs
• Ca union of Ci such as li is an assistants
• Pp time and room preferences of profs
• Pa time and room preference of assistants
• Wanted Cp more important than Ca, but Ca more
  important than Pp
• So (lex (psum Cp)(psum Ca)(pareto Pp)(pareto Pa))

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PDL

• Introduction
• Example
• PDL
• Special case
• Discussion

• How to deal with modifications?
• X start from scratch, rerun program
• Generate coherent solutions given small changes
  -gt an optimization problem
• A description of the old solution
• A specification of closeness
• In a qualitative setting, described by preferences

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PDL

• Introduction
• Example
• PDL
• Special case
• Discussion

• Deal with changes (contd)
• Example revisited
• In general, no changes is preferred
• If a change is necessary, then it is more
  desirable to change the room rather than time
• If time is needed to be changed, it is better to
  reschedule a course with few students.

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Special case

• Introduction
• Example
• PDL
• Special case
• Discussion

• PDL is a generalization of methods in the
  literature
• Pprefr1, , rk
• Meta-preferences (preferences rules in levels)
• Weak constraints (lt-body. wl)
• Smodels ( minimize a1w1,,akwk )

-gt (pareto r1rk)
-gt (lex(pareto r1,1r1,k1)(pareto rn,1rn,kn))
-gt (lex(psum r1,1r1,k1)(psum rn,1rn,kn))
-gt ( psum a1 w1 ak wk )
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Discussion

• Introduction
• Example
• PDL
• Special case
• Discussion

• Implementation
• Contributions
• Definition of PDL for complex preferences
• PDL can be compiled to LP to be used as a tester
  program in a generate-and-improve method for
  finding optimal answer sets.
• Extendible, in other optimization context

Generating program P (smodels, DLV..)
Tester program
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References

• Brewka. G, Niemela. I, Truszczynski. M Answer
  set optimization. IJCAI-03, pp867-872
• Brewka. G, Niemela. I, Syrjanen. T Implementing
  ordered disjunction using answer set solvers for
  normal programs. JELIA 2002, pp444-455.
• Brewka. G Answer sets From constraints
  programming towards qualitative optimization.
  LPNMR-04, pp34-46
• Lifschitz. V Answer set programming and plan
  generation. Artificial Intelligence Journal 138
  pp39-54
• Son. R, Pontelli. E Planning with preferences
  using logic programming and nonmonotonic
  reasoning, LPNMR04, pp247-260