1. A rewritting method for Well-Founded Semantics with Explicit Negation Pedro Cabalar University of Corunna, SPAIN.

2. 2 Introduction Logic programming (LP) semantics for default negation: –Stable models [Gelfond&Lifschitz88] –Well-Founded Semantics (WFS) [van Gelder et al. 91] Bottom-up computation for WFS [Brass et al. 01] –More efficient than van Gelder’s alternated fixpoint –Based on program transformations

3. 3 Introduction Extended Logic Programming: default negation (not p) plus explicit negation ( ) : –Answer Sets [Gelfond&Lifschitz91] –WFS with explicit negation (WFSX) [Pereira&Alferes92] p Our work: extend Brass et al’s method to WFSX –Adding two natural transformations –Helps to understand relation WFS vs. WFSX

4. Outline  Some LP definitions  Brass et al’s method  WFSX  Coherence transformations  Conclusions

5. Outline  Some LP definitions  Brass et al’s method  WFSX  Coherence transformations  Conclusions

6. 6 Some LP definitions Logic program P: set of rules like a  b, not c c  not b b Reduct P I : we use I to interprete all ‘not p’. Example: take I={a,b}

7. 7 Some LP definitions Logic program P: set of rules like a  b, not c c  not b b Reduct P I : we use I to interprete all ‘not p’. Example: take I={a,b}  (I) = least model of P I Stable model: any fixpoint I =  (I) Well-founded model (WFM): –Positive atoms I + = least fixpoint of  –Negative atoms I - = HB – greatest fixpoint of  l.f.p. g.f.p. + - HB

8. Outline  Some LP definitions  Brass et al’s method  WFSX  Coherence transformations  Conclusions

9. Outline  Some LP definitions  Brass et al’s method  WFSX  Coherence transformations  Conclusions

10. 10 Brass et al’s method Trivial interpretation: a 3-valued interpretation where –Positive atoms I + = facts(P) –Negative atoms I - = HB – heads(P) We exhaustively apply 5 program transformations PNSFL The trivial interpretation of the final program will be the WFM

11. 11 Brass et al’s method: an example a  not b, cd  not g, e b  not ae  not g, d c f  not d d  not cf  g, not e I + = facts(P) = {c} I - = HB – heads(P) = {g}

12. 12 Brass et al’s method: an example a  not b, cd  not g, e b  not ae  not g, d c f  not d d  not cf  g, not e I + = facts(P) = {c} I - = HB – heads(P) = {g} S Success:delete c from bodies Negative reduction:delete rules with not c in the body N

13. 13 Brass et al’s method: an example a  not b, cd  not g, e b  not ae  not g, d c f  not d d  not cf  g, not e I + = facts(P) = {c} I - = HB – heads(P) = {g} P Positive reduction:delete not g from bodies Failure:delete rules with g in the body F

14. 14 Brass et al’s method: an example a  not bd  e b  not ae  d c f  not d I + = facts(P) = {c} I - = HB – heads(P) = {g} Interesting property: exhausting {P,N,S,F} yields Fitting’s model … but for WFS we must get rid of positive cycles (d,e)

15. 15 Brass et al’s method: an example a  not bd  e b  not ae  d c f  not d I + = facts(P) = {c} I - = HB – heads(P) = {g} L Positive loop detection: delete rules with some p  (  ) optimistic viewing: “what if all not’s happened to be true?”

16. 16 Brass et al’s method: an example a  not bd  e b  not ae  d c f  not d I + = facts(P) = {c} I - = HB – heads(P) = {g} L Positive loop detection: delete rules with some p  (  )  (  ) = {a, b, c, f }

17. 17 Brass et al’s method: an example a  not bd  e b  not ae  d c f  not d I + = facts(P) = {c} I - = HB – heads(P) = {g} L Positive loop detection: delete rules with some p  (  )  (  ) = {a, b, c, f } i.e. delete rules with some {d, e, g}

18. 18 Brass et al’s method: an example a  not b b  not a c f  not d I + = facts(P) = {c} I - = HB – heads(P) = {g, e, d} P... we must go on until no new transformation is applicable. Positive reduction: delete not d from bodies

19. 19 Brass et al’s method: an example I + = facts(P) = {c, f } I - = HB – heads(P) = {g, e, d } We can’t go on: ge get the WFM! a  not b b  not a c f  not d

20. Outline  Some LP definitions  Brass et al’s method  WFSX  Coherence transformations  Conclusions

21. Outline  Some LP definitions  Brass et al’s method  WFSX  Coherence transformations  Conclusions

22. 22 WFSX Extended LP: two negations not p“p is not known to be true” “p is known to be false” p Objective literal L is any p or. We’ll denote L s.t. = ppp Answer sets: reject stable models containing both p andp WFS Coherence problem: should imply not pp p  not q q  not p p WFM + = { } WFM - = { } p q

23. 23 WFSX Given P we define its seminormal version P s p  not q q  not p p p  not q, not p q  not p, not q  not p p P P s The well-founded model is defined now as: –Positive atoms I + = least fixpoint of  s –Negative atoms I - =  s (I + ) In the example, we get I + = {, q } I - = { p, }pq

24. Outline  Some LP definitions  Brass et al’s method  WFSX  Coherence transformations  Conclusions

25. Outline  Some LP definitions  Brass et al’s method  WFSX  Coherence transformations  Conclusions

26. 26 Coherence transformations We begin redefining trivial interpretation... –I + = facts(P) = { p } –I - = HB – heads(P) = {, } ab a  not b b  not a  b p p

27. 27 Coherence transformations We begin redefining trivial interpretation... –I + = facts(P) = { p } –I - = HB – heads(P)  { L | L  facts(P) } = {,, } ab a  not b b  not a  b p p p

28. 28 Coherence transformations p  not q q  not p q  p p I + = { } I - = { p } p

29. 29 Coherence transformations p  not q q  not p q  p p I + = { } I - = { p } p R Coherence reduction:delete not p from bodies Coherence Failure:delete rules with p in the body C

30. 30 Coherence transformations p  not q q  p I + = { } I - = { } p, q N Delete rules containing not q in the body p, q

31. 31 Coherence transformations Theorem 2: transformations {P,S,N,F,L,C,R} are sound w.r.t. WFSX Theorem 3: Let W be the WFM under WFS: (i) if W contradictory (p, p  W + ) then P contradictory in WFSX (ii) the WFM under WFSX contains more or equal info than W The converse of (i) does not hold... Corollary: when WFS leads to complete (and not contradictory) WFM it coincides with WFSX a  not a a

32. 32 Coherence transformations Theorem 4 (main result) Given P... P' where x  { P, S, N, F, L, C, R } P' is the final program (free of contradictory facts) The trivial interpretation of P' is the WFM of P under WFSX. xx

33. Outline  Some LP definitions  Brass et al’s method  WFSX  Coherence transformations  Conclusions

34. Outline  Some LP definitions  Brass et al’s method  WFSX  Coherence transformations  Conclusions

35. 35 Conclusions We added two natural transformations w.r.t. coherence: "whenever L founded, L unfounded" Used and implemented for applying WFSX to causal theories of actions [Cabalar01] Can be used as slight efficiency improvement for answer sets? Explore a new semantics: Fitting's + coherence transformations