Well-founded Semantics with Disjunction

1. 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 Well-founded Semantics with Disjunction

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

3. Introduction • 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. Introduction • 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. Introduction • Our proposal • Well-founded semantics with disjunction (WFSd) • 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. Minimal Models of PDLP • Disjunctive Logic Program: set of rules a1 ...  al←b1  ... bmnotc1 ... notcn • PDLP: rP, n = 0 • Herbrand base (HBP): set of all ground atoms over the language of P. • Herbrand interpretation: any set I HBP . • - the set of all Herbrand interpretations • Coin – collection of Herbrand interpretations

7. Operations over coins

8. Operations over coins • Example:

9. Operations over coins [Seipel et al, 1997] • Isatisfies a rule a1 ...  al←b1  ... bm P iff it holds that bi | 1 ≤ i≤ mI implies that 1 ≤ j≤ l s.t. aj I • I is a model of P iff for every rP, I satisfies r.

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

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

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

13. A fixed point based definition of WFSd • 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. A fixed point based definition of WFSd • 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. A fixed point based definition of WFSd • Looking for alternative lattices works as dual of

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

17. A fixed point based definition of WFSd • Ordering partial coins: • Minimal models

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

19. A fixed point based definition of WFSd Observation:

20. A fixed point based definition of WFSd

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

22. WFSd vs Partial disjunctive stable models Partial Disjunctive Stable Models do not assign any meaning for P1.

23. WFSd vs Partial disjunctive stable models( for stratified programs) P2has 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 WFSd

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

25. WFSd vs D-WFS c is true in WFSd,butundefined in D-WFS. Although WFSdand D-WFS do not generally present the same results, yet D-WFS is strictly weaker than WFSd.

26. Important results • Theorem. Let P1 and P2 be disjunctive logic programs such that P2 results from P1 by unfolding, elimination of tautologies and nonminimal rules, and positive and negative reduction. We have WFSd(P1)= WFSd(P2). • Theorem. For normal logic programs, WFSd reduces to WFS.

27. Conclusions and Future Works • We have defined a new well-founded semantics for disjunctive logic programs: Well-Founded Semantics with Disjunction (WFSd) • It is based on a generalisation to a set of interpretations of the fixed point operator used to define WFS • WFSd 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 WFSdto deal with explicit negation, either in its explosive or paraconsistent version • Model-theoretical characterisation of WFSd

28. Questions ???