1. Lparse Programs Revisited: Semantics and Representation of Aggregates Guohua Liu and Jia-Huai You University of Alberta Canada

2. 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?

3. Lparse programs Weight constraint W : Weight constraint rule: where each is a weight constraint.

4. Semantics  The reduct of a weight constraint W w.r.t. M is the constraint where

5. Semantics  The reduct of a weight constraint W w.r.t. M is the constraint where The reduct is:

6. 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.

7. 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)

8. 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.

9. 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

10. 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.

11. 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

12. 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 [a=1, not b =1] b  [a=1, 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.

13. Transformation to strongly satisfiable programs Let l {W }u denote a weight constraint. Transform it to the conjunction of 1.l {W } 2.l’ {W} where Example: [not a =1]0 transformed to [not a =1] and 1 [not a =1]

14. Representation of Aggregates Take the form where aggr is from {Sum,Count,Avg,Min,Max} op is from Result is a numeric constant

15. 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:

16. Experiments

17. Experiments (2)

18. 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.