1. Extended LPs In Normal LPs all the negative information is implicit. Though that’s desired in some cases (e.g. the database with flight connections), sometimes an explicit form of negation is needed for Knowledge Representation “Penguins don’t fly” could be: noFly(X)  penguin(X) This does not relate fly(X) and noFly(X) in: fly(X)  bird(X) noFly(X)  penguin(X) For establishing such relations, and representing negative information a new form of negation is needed in LP: Explicit negation - ¬

2. Extended LP: motivation ¬ is also needed in bodies: “Someone is guilty if is not innocent” –cannot be represented by: guilty(X)  not innocent(X) –This would imply guilty in the absence of information about innocent –Instead, guilty(X)  ¬innocent(X) only implies guilty(X) if X is proven not to be innocent The difference between not p and ¬p is essential whenever the information about p cannot be assumed to be complete

3. ELP motivation (cont) ¬ allows for greater expressivity: “If you’re not sure that someone is not innocent, then further investigation is needed” –Can be represented by: investigate(X)  not ¬innocent(X) ¬ extends the relation of LP to other NMR formalisms. E.g –it can represent default rules with negative conclusions and pre-requisites, and positive justifications –it can represent normal default rules

4. Explicit versus Classical ¬ Classical ¬ complies with the “excluded middle” principle (i.e. F v ¬F is tautological) –This makes sense in mathematics –What about in common sense knowledge? ¬A is the the opposite of A. The “excluded middle” leaves no room for undefinedness hire(X)  qualified(X) reject(X)  ¬ qualified(X) The “excluded middle” implies that every X is either hired or rejected It leaves no room for those about whom further information is need to determine if they are qualified

5. ELP Language An Extended Logic Program P is a set of rules: L 0   L 1, …, L m, not L m+1, … not L n (n,m  0) where the L i are objective literals An objective literal is an atoms A or its explicit negation ¬A Literals not L j are called default literals The Extended Herbrand base H P is the set of all instantiated objective literals from program P We will consider programs as possibly infinite sets of instantiated rules.

6. ELP Interpretations An interpretation I of P is a set I = T U not F where T and F are disjoint subsets of H P and ¬L  T  L  F (Coherence Principle) i.e. if L is explicitly false, it must be assumed false by default I is total iff H P = T U F I is consistent iff ¬  L: {L, ¬L}  T –In total consistent interpretations the Coherence Principle is trivially satisfied

7. Answer sets It was the 1st semantics for ELPs [Gelfond&Lifschitz90] Generalizes stable models to ELPs D Let M - be a stable models of the normal P - obtained by replacing in the ELP P every ¬ A by a new atom A ’. An answer-set M of P is obtained by replacing A ’ by ¬ A in M - A is true in an answer set M iff A  S A is false iff ¬A  S Otherwise, A is unknown Furthermore, if M includes a pair of objective literals A and ¬A then M should include all objective literals Some programs have no consistent answer sets: –e.g. P = {a  ¬ a  }

8. Answer sets and Defaults D Let  P be the default theory obtained by transforming: L 0  L 1,…,L m, not L m+1,…, not L n into: L 1,…,L m : ¬L m+1,…, ¬L n L 0 where ¬¬A is (always) replaced by A T There is a one-to-one correspondence between consistent answer-sets of P and consistent default extensions of  P

9. Answer-sets and AEL D Let  P be the AEL theory obtained by transforming: L 0  L 1,…,L m, not L m+1,…, not L n into: L 1  L L 1  …  L m  L L m   ¬ L L m+1  …  ¬ L L m  L 0  L L 0  T There is a one-to-one correspondence between the answer-sets of P and the expansions of  P

10. Answer Sets and Stable Models Substitute every negative objective literal ¬ A by A ’ in program P’ Add to P’ the following rules for every pair of atoms A, B: A  B ’. A’  B ’. The answer sets of P are in 1-1 correspondence with the Stable Models of P’. If we remove the extra rules we obtain Paraconsistent Answer Set Semantics.

11. 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

12. 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”

13. 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  )

14. 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)}

15. 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)

16. 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

17. 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  )

18. 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}

19. 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

20. 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 }

21. 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 }