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... Ü Coherence must be imposed

Imposing Coherence Coherence is: ¬L  T  L  F, for objective L According to the WFS definition, everything is false that doesn’t belong to  (T) To impose coherence, when applying  (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”

WFSX definition D The semi-normal version of P, P s, is obtained by adding not ¬L to every rule of P with head L D An interpretation (T U not F) is a PSM of ELP P iff: T =  P  Ps (T) T   Ps (T) F = H P -  Ps (T) T The WFSX semantics is determined by the knowledge ordering least PSM (wrt  )

WFSX example P:pacifist(X)  not hawk(X) hawk(X)  not pacifist(X) ¬ pacifist(a) Ps:pacifist(X)  not hawk(X), not ¬pacifist(X) hawk(X)  not pacifist(X ), not ¬hawk(X) ¬pacifist(a)  not pacifist(a) T 0 = {}  s (T 0 ) = {¬p(a),p(a),h(a),p(b),h(b)} T 1 = {¬p(a)}  s (T 1 ) = {¬p(a),h(a),p(b),h(b)} T 2 = {¬p(a),h(a)} T 3 = T 2 The WFM is: {¬p(a),h(a), not p(a), not ¬h(a), not ¬p(b), not ¬h(b)}

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)

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

WFSXp definition The paraconsistent version of WFSx is obtained by dropping the requirement that T and F are disjoint, i.e. dropping T   Ps (T) D An interpretation, T U not F, is a PSMp P iff: T =  P  Ps (T) F = H P -  Ps (T) T The WFSXp semantics is determined by the knowledge ordering least PSM (wrt  )

WFSXp example P:c  not ba b  a ¬ a d  not e Ps:c  not b, not ¬c a  not ¬a b  a, not ¬b ¬a  not a d  not e, not ¬d T 0 = {}  s (T 0 ) = {¬a,a,b,c,d} T 1 = {¬a,a,b,d}  s (T 1 ) = {d} T 2 = {¬a,a,b,c,d} T 3 = T 2 The WFM is: {¬a,a,b,c,d, not a, not ¬a, not b, not ¬b not c, not ¬c, not ¬d, not e}

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 Jobert’s manifestation, then pancreatitis can be exonerated –In both situations, the pacient should not be nourished, but should take H 2 antagonists

LP representation perforation  pain, abd-tender, per-irrit, not high-amylase pancreat  pain, abd-tender, per-irrit, not jobert ¬nourish  perforationh2-ant  perforation ¬nourish  pancreat h2-ant  pancreat surgery  perforationanesthesia  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 }

Results interpretation The symptoms are derived and non-contradictory Both perforation and pancreatitis are concluded He should not be fed ( ¬nourish ), but take H 2 antagonists The information about surgery is contradictory Anesthesia though not explicitly contradictory ( ¬anesthesia doesn’t belong to WFM) relies on contradiction (both anesthesia and not anesthesia belong to WFM) 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 }