A rewritting method for Well-Founded Semantics with Explicit Negation

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

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. p Introduction • Extended Logic Programming:default negation (not p) plus explicit negation ( ) : • Answer Sets [Gelfond&Lifschitz91] • WFS with explicit negation (WFSX) [Pereira&Alferes92] • 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. Some LP definitions a  b , not c c not b b • Logic program P: set of rules like • ReductPI: we use I to interprete all ‘not p’. Example: take I={a,b}

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

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. We exhaustively apply 5 program transformationsP N S F L Brass et al’s method • Trivial interpretation: a 3-valued interpretation where • Positive atoms I+ = facts(P) • Negative atoms I- = HB – heads(P) • The trivial interpretation of the final program will bethe WFM

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

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

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

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

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

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

18. P 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} ... we must go on until no new transformation is applicable. Positive reduction: delete not d from bodies

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

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

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

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

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

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

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

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

31. a  not a a 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

32. x x 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.

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. 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