Lparse%20Programs%20Revisited:%20Semantics%20and%20Representation%20of%20Aggregates

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Lparse Programs Revisited Semantics and
Representation of Aggregates

• Guohua Liu and Jia-Huai You
• University of Alberta
• Canada

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Outline

• Stable models of weight-constraint programs
• - closely related to answer sets of
  Son-Pontelli-Tu
• for LPs with constraint atoms
• - what about other lparse-stable models?
  Some
• of them may be circular
• - a translation to avoid circularity
• Can we represent commonly used aggregates by
  weight constraints?

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Lparse programs

• Weight constraint W
• Weight constraint rule
• where each is a weight constraint.

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Semantics

• The reduct of a weight constraint W w.r.t.
  M is the constraint
• where

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Semantics

• The reduct of a weight constraint W w.r.t.
  M is the constraint
• where
• The reduct is

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Lparse-stable models may be circular

• Consider the one-rule program
• a ? not a 1 0
• Both M1 and M2 a are lparse-stable
  models.
• In M2, we need assume a in order to derive a.
• The weight constraint in the body is actually
  monotone.

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Equivalent Expressions

• This weight constraint is equivalent to each of
  the following (satisfaction-preserving)
• a ? count(x x D) 1 where D a
• a ? (a, a) (body is an abstract
  constraint atom)

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Is non-minimal the culprit ?

• Not always. Consider program P
• a ? not a 1 0
• f ? not f, not a
• a is now minimal, but still circular.

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Strongly satisfiable weight constraints

• Notation W is a weight constraint M is a set
  of atoms
• A weight constraint W is strongly satisfiable by
  M iff
• M W implies, for any ,
• W is strongly satisfiable by any M if
• -lit(W) contains no negative literal
• - upper-bound free

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Theorem

• Let P be an lparse program and M a set of
  atoms. Suppose all the weight constraints
  appearing in the body of any rule in P are
  strongly satisfiable by M.
• Then, M is an lparse-stable model of P iff M
  is an answer set (in the sense of Son et al.) for
  P.

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Weak notion of non-circularity(unfoundedness)

• Definition (essentially that of Calimeri et al.
  IJCAI-05)
• An lparse-stable model M of a program P is
  circular if
• there is a non-empty set s.t.
  , M \U does
• not satisfy the body of any rule r in P,
  where

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Weak notion of non-circularity(unfoundedness)

• Definition (essentially that of Calimeri et al.
  IJCAI-05)
• An lparse-stable model M of a program P is
  circular if
• there is a non-empty set s.t.
  , M \U does
• not satisfy the body of any rule r in P,
  where
• Example
• a ?
• b ? 2 a1, not b 1
• b ? a1, not b 1 1
• a,b is an lparse-stable model, but not an
  answer set, and it is not circular by the above
  definition.

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Transformation to strongly satisfiable programs

• Let l W u denote a weight constraint. Transform
  it to the conjunction of
• l W
• l W
• where
• Example not a 10
• transformed to
• not a 1 and 1 not a 1

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Representation of Aggregates

• Take the form
• where
• aggr is from Sum,Count,Avg,Min,Max
• op is from
• Result is a numeric constant

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Linear size encoding

• sum and account straightforward
• Encode by
• max and min more complex, but can be done
• What cannot be encoded?
• Aggregate expressions involving
• Product constraint

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Experiments
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Experiments (2)
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Conclusion

• Lparse semantics is closely related to answer
  sets by Son et al. The gap can be closed by a
  simple transformation.
• Lparse programs are already an effective
  representation language for aggregates. It only
  needs a simple frond end.
• More efficient implementation of weight
  constraints
• is needed.