1. Well-founded Semantics with Disjunction João Alcântara, Carlos Damásio and Luís Moniz Pereira e-mail: jfla|cd|lmp@fct.unl.pt Centro de Inteligência Artificial (CENTRIA) Depto. Informática, Faculdade de Ciências e Tecnologia Universidade Nova de Lisboa 2825-114 Caparica, Portugal ICLP'05 Sitges, October 2005

2. 2Outline Introduction Positive-Disjunctive Logic Programs Well-Founded Semantics with Disjunction Examples Conclusion and Future Works

3. 3Introduction Disjunctive reasoning in Logic Programming –Pioneer work by Minker – Generalized Closed World Assumption Positive-disjunctive logic programs –Semantics is obtained via minimal Herbrand models (Normal) Disjunctive logic programs –Stable Models –Well-founded Models  Many proposals!!!

4. 4Introduction A good well-founded semantics for disjunctive logic program should at least –Coincide with WFS on normal logic programs –Be uniquely defined for every program –Be definable both model-theoretically and by a fixpoint operator –Comply with Brass and Dix’s program transformations: unfolding, elimination of tautologies and non-minimal rules, positive and negative reduction –Allow further extensions to assimilate explicit negation

5. 5Introduction Our proposal –Well-founded semantics with disjunction (WFS d ) –Generalisation of the immediate consequences operator used to define WFS –Definition of a new domain to deal with default negation –It is intended to satisfy the conditions just enrolled

6. 6 Minimal Models of PDLP Disjunctive Logic Program: set of rules a 1 ...  a l ← b 1 ...  b m  not c 1 ...  not c n PDLP:  r  P, n = 0 Herbrand base (H BP ): set of all ground atoms over the language of P. Herbrand interpretation: any set I  H BP. – - the set of all Herbrand interpretations Coin – collection of Herbrand interpretations

7. 7 Operations over coins

8. 8 Example:

9. 9 Operations over coins [Seipel et al, 1997] I satisfies a rule a 1 ...  a l ← b 1 ...  b m  P iff it holds that  b i | 1 ≤ i ≤ m   I implies that  1 ≤ j ≤ l s.t. a j  I I is a model of P iff for every r  P, I satisfies r.

10. 10 Operations over coins [Seipel et al, 1997]

11. 11 Ordinal Powers of T [Seipel et al, 1997] For, the ordinal powers are defined by

12. 12 Ordinal Powers of T [Seipel et al, 1997] Although is not continuous, it reaches its least fixed point in at most iterations [Seipel et al, 1997] The least fixed point of, denoted by, is given by, in which is the smallest ordinal such that

13. 13 A fixed point based definition of WFS d We will study only finite disjunctive logic programs In normal logic programs, well-founded semantics may be defined in terms of partial interpretations Our idea is to exploit this notion of partial interpretations, but considering pairs of coins Pair of coins evaluating disjunctive clauses:

14. 14 A fixed point based definition of WFS d operator defined over is suitable to express what is true in a program P does not preserve the notion of falsity by default Example

15. 15 A fixed point based definition of WFS d Looking for alternative lattices works as dual of

16. 16 A fixed point based definition of WFS d Example ( cont ) Partial coin is a pair where and

17. 17 A fixed point based definition of WFS d Ordering partial coins: Minimal models

18. 18 A fixed point based definition of WFS d A divided program is obtained as follows

19. 19 A fixed point based definition of WFS d Observation:

20. 20 A fixed point based definition of WFS d

21. 21 A fixed point based definition of WFS d A least fixed point is guaranteed to exist: WFS d operator is monotonic with respect to

22. 22 WFS d vs Partial disjunctive stable models Partial Disjunctive Stable Models do not assign any meaning for P 1.

23. 23 WFS d vs Partial disjunctive stable models ( for stratified programs) P 2 has the partial stable models : {a,c} and {b,d} not e is obtained according to partial disjuntive stable models, perfect models, static and stationary semantics Atom “e” remains undefined in WFS d

24. 24 WFS d vs SWFS, WFDS and GDWFS WFS d : a is false; b remains undefined SWFS: a and b are undefined GDWFS: b is false; a is false WFDS: a and b are undefined

25. 25 WFS d vs D-WFS c is true in WFS d,but undefined in D-WFS. Although WFS d and D-WFS do not generally present the same results, yet D-WFS is strictly weaker than WFS d.

26. 26 Important results Theorem. Let P 1 and P 2 be disjunctive logic programs such that P 2 results from P 1 by unfolding, elimination of tautologies and nonminimal rules, and positive and negative reduction. We have WFS d (P 1 )= WFS d (P 2 ). Theorem. For normal logic programs, WFS d reduces to WFS.

27. 27 Conclusions and Future Works We have defined a new well-founded semantics for disjunctive logic programs: Well-Founded Semantics with Disjunction (WFS d ) It is based on a generalisation to a set of interpretations of the fixed point operator used to define WFS WFS d does not coincide with D-WFS, Static, GDWFS, WFDS, SWFS and Partial Disjunctive Stable Models. It is strictly stronger than D-WFS. We will extend WFS d to deal with explicit negation, either in its explosive or paraconsistent version Model-theoretical characterisation of WFS d

28. 28Questions ???