A rewritting method for WellFounded 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 GelfondLifschitz88
• Well-Founded Semantics (WFS) van Gelder et al.
  91

• Bottom-up computation for WFS Brass et al. 01
• More efficient than van Gelders alternated
  fixpoint
• Based on program transformations

3
Introduction

• Extended Logic Programmingdefault negation (not
  p) plus explicit negation ( )
• Answer Sets GelfondLifschitz91
• WFS with explicit negation (WFSX)
  PereiraAlferes92

• Our work extend Brass et als method to WFSX
• Adding two natural transformations
• Helps to understand relation WFS vs. WFSX

4
Outline

• Some LP definitions
• Brass et als method
• WFSX
• Coherence transformations
• Conclusions

5
Outline

• Some LP definitions
• Brass et als method
• WFSX
• Coherence transformations
• Conclusions

6
Some LP definitions

• Logic program P set of rules like

a ? b , not c c ? not b b

• Reduct PI we use I to interprete all not p.
  Example take Ia,b

7
Some LP definitions

• Logic program P set of rules like

a ? b , not c c ? not b b

• Reduct PI we use I to interprete all not p.
  Example take Ia,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 als method
• WFSX
• Coherence transformations
• Conclusions

9
Outline

• Some LP definitions
• Brass et als method
• WFSX
• Coherence transformations
• Conclusions

10
Brass et als 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

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Brass et als 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
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Brass et als 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
Brass et als 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
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Brass et als 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 Fittings model but for WFS we must get
rid of positive cycles (d,e)
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Brass et als 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 nots
happened to be true?
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Brass et als 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
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Brass et als 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
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Brass et als 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
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Brass et als 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 cant go on ge get the WFM!
20
Outline

• Some LP definitions
• Brass et als method
• WFSX
• Coherence transformations
• Conclusions

21
Outline

• Some LP definitions
• Brass et als method
• WFSX
• Coherence transformations
• Conclusions

22
WFSX

• Extended LP two negations
• not p p is not known to be true
• p is known to be false

23
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 als method
• WFSX
• Coherence transformations
• Conclusions

25
Outline

• Some LP definitions
• Brass et als method
• WFSX
• Coherence transformations
• Conclusions

26
Coherence transformations

• We begin redefining trivial interpretation ...
• I facts(P) p
• I- HB heads(P) ,

a ? not b b ? not a ? b p
27
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
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Coherence transformations
p ? not q q ? not p q ? p
I I- p
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Coherence transformations
p ? not q q ? not p q ? p
I I- p
p
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Coherence transformations
p ? not q q ?
I I-
p , q
p , q
Delete rules containing not q in the body
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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

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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 als method
• WFSX
• Coherence transformations
• Conclusions

34
Outline

• Some LP definitions
• Brass et als 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