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The coherence principle

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) }

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The coherence principle

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  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: ¬LÎ TÞ LÎ F, for objective L • According to the WFS definition, everything is false that doesn’t 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 ÍGPs(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 ÍGPs(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 Jobert’s 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 (¬anesthesiadoesn’t belong to WFM) relies on contradiction (both anesthesia and notanesthesia belong to WFM)

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