The coherence principle

1
The coherence principle

• Generalizing WFS in the same way yields
  unintuitive results

pacifist(X) not hawk(X) hawk(X) not
pacifist(X) pacifist(a)

• Using the same method the WFS is pacifist(a)
• Though it is explicitly stated that a is
  non-pacifist, not pacifist(a) is not assumed, and
  so hawk(a) cannot be concluded.
• Coherence is not satisfied...
• Coherence must be imposed

2
Imposing Coherence

• Coherence is L Î T Þ L Î F, for objective L
• According to the WFS definition, everything is
  false that doesnt belong to G(T)
• To impose coherence, when applying G(T) simply
  delete all rules for the objective complement of
  literals in T

If L is explicitly true then when computing
undefined literals forget all rules with head L
3
WFSX definition

• The semi-normal version of P, Ps, is obtained by
  adding not L to every rule of P with head L

• An interpretation (T U not F) is a PSM of ELP P
  iff
• T GPGPs(T)
• T Í GPs(T)
• F HP - GPs(T)

• The WFSX semantics is determined by the knowledge
  ordering least PSM (wrt Í)

4
WFSX example
Ps pacifist(X) not hawk(X), not
pacifist(X) hawk(X) not pacifist(X ), not
hawk(X) pacifist(a) not pacifist(a)
T0 Gs(T0) p(a),p(a),h(a),p(b),h(b) T1
p(a) Gs(T1) p(a),h(a),p(b),h(b) T2
p(a),h(a) T3 T2
P pacifist(X) not hawk(X) hawk(X) not
pacifist(X) pacifist(a)
The WFM is p(a),h(a), not p(a), not h(a), not
p(b), not h(b)
5
Properties of WFSX

• Complies with the coherence principle
• Coincides with WFS in normal programs
• If WFSX is total it coincides with the only
  answer-set
• It is sound wrt answer-sets
• It is supported, cumulative, and relevant
• Its computation is polynomial
• It has sound implementations (cf. below)

6
Inconsistent programs

• Some ELPs have no WFM. E.g. a , a
• What to do in these cases?
• Explosive approach everything follows from
  contradiction
• taken by answer-sets
• gives no information in the presence of
  contradiction
• Belief revision approach remove contradiction by
  revising P
• computationally expensive
• Paraconsistent approach isolate contradiction
• efficient
• allows to reason about the non-contradictory part

7
WFSXp definition

• The paraconsistent version of WFSx is obtained by
  dropping the requirement that T and F are
  disjoint, i.e. dropping T Í GPs(T)

• An interpretation, T U not F, is a PSMp P iff
• T GPGPs(T)
• F HP - GPs(T)

• The WFSXp semantics is determined by the
  knowledge ordering least PSM (wrt Í)

8
WFSXp example
Ps c not b, not c a not a b a, not b
a not a d not e , not d
T0 Gs(T0) a,a,b,c,d T1
a,a,b,d Gs(T1) d T2 a,a,b,c,d T3 T2
P c not b a b a a d not e
The WFM is a,a,b,c,d, not a, not a, not b,
not b not c, not c, not d, not e
9
Surgery situation

• A patient arrives with sudden epigastric pain
  abdominal tenderness signs of peritoneal
  irritation
• The rules for diagnosing are
• if he has sudden epigastric pain abdominal
  tenderness, and signs of peritoneal irritation,
  then he has perforation of a peptic ulcer or an
  acute pancreatitis
• the former requires surgery, the latter
  therapeutic treatment
• if he has high amylase levels, then a perforation
  of a peptic ulcer can be exonerated
• if he has Joberts manifestation, then
  pancreatitis can be exonerated
• In both situations, the pacient should not be
  nourished, but should take H2 antagonists

10
LP representation
perforation pain, abd-tender, per-irrit, not
high-amylase pancreat pain, abd-tender,
per-irrit, not jobert nourish
perforation h2-ant perforation nourish
pancreat h2-ant pancreat surgery
perforation anesthesia surgery surgery
pancreat
pain. per-irrit. high-amylase. abd-tender.
jobert.
The WFM is pain, abd-tender, per-irrit,
high-am, jobert , not pain, not abd-tender,
not per-irrit, not high-am, not jobert,
nourish, h2-ant, not nourish, not h2-ant,
surgery, surgery, not surgery, not
surgery, anesthesia, not anesthesia, not
anesthesia
11
Results interpretation
The WFM is pain, abd-tender, per-irrit,
high-am, jobert , , nourish, h2-ant, not
nourish, not h2-ant, surgery, surgery, not
surgery, not surgery,anesthesia, not anesthesia,
not anesthesia

• The symptoms are derived and non-contradictory
• Both perforation and pancreatitis are concluded
• He should not be fed (nourish), but take H2
  antagonists
• The information about surgery is contradictory
• Anesthesia though not explicitly contradictory
  (anesthesia doesnt belong to WFM) relies on
  contradiction (both anesthesia and not anesthesia
  belong to WFM)