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  1. 1. Resolution(Decision)V. SaranyaAP/CSESri Vidya College of Engineering &Technology, Virudhunagar
  2. 2. • Conjunctive normal form• Resolution inference rule• Dealing with Equality• Resolution strategies• Completeness of resolution• Theorem provers
  3. 3. 1. Conjunctive Normal Form for FOLx American(x) ˄ Weapon(y) ˄ sells(x,y,z) ˄Hostile(z) => criminal(x)CNF:¬ American (x) ˄ ¬ Weapon(y) ˄ ¬sells(x ,y, z) ˄¬ hostile(z) ˄ criminal(x)
  4. 4. CNF converting Steps1. Eliminate implication2. Move ¬ inwards3. Standardize variables (use different variable)4. Skolemize (remove existential quantifier)5. Drop universal quantifiers6. Distribute ˄ and˄.
  5. 5. Converting to CNF1. Replace implication (A  B) by A  B2. Move  “inwards”• x P(x) is equivalent to x P(x) & vice versa3. Standardize variables• x P(x)  x Q(x) becomes x P(x)  y Q(y)4. Skolemize• x P(x) becomes P(A)5. Drop universal quantifiers• Since all quantifiers are now , we don’t need them6. Distributive Law
  6. 6. 2. Resolution Inference rule• Binary resolution Rule: resolves exactly 2 literals• Factoring: removal of redundant literals.• Combination of binary rule & factoring is“Complete”• Ex:• [animal(F(x)) v loves(G(x),x)] and (¬ loves(u,v) v¬kills(u,v)]• loves(G(x),x) and ¬ loves(u,v)• Ɵ = { u/G(x), v/x}  Unifier
  7. 7. Example• Everyone who loves all animal is loved bysomeone. Anyone who kills an animal isloved by no one.• Nikki loves all animals• Either nikki or zooro killed the cat, who isnamed teena.• Did zooro kill the cat?
  8. 8. 3. Dealing with EqualityAxiomatize Equality:• Reflexive: x x=x• Symmetric: x,y x=y, y=x• Transitive: x,y,z x=y ˄ y=z => x=z• Predicate name and function names are same– x,y x=y => (P1(x) P1(y))• Functional equality.
  9. 9. Additional inference rule• Demodulation• Para modulation• Extended unification algorithm
  10. 10. 4. Resolution strategies• To tell proof in a efficient way1. Unit Preference: ( prefer single literal)• Evil v devil => ghost• Devil => ghost2. Set of support: (subset of sentences)• Add relevant sentences.3. Input resolution: combine one of the i/p sentenceswith some other sentence.4. Linear Resolution: completeEx: P and Q are 2 predicates. P should be in KB or P is anancestor of Q.5. Subsumption: (elimination)
  11. 11. Resolution(decision)• Convert everything to CNF• Resolve, with unification• If resolution is successful, proof succeeds• If there was a variable in the item to prove, returnvariable’s value from unification bindings
  12. 12. Resolution (Review)• Resolution allows a complete inference mechanism (search-based) using only one rule of inference• Resolution rule:– Given: P1  P2  P3 … Pn, and P1  Q1 … Qm– Conclude: P2  P3 … Pn  Q1 … QmComplementary literals P1 and P1 “cancel out”• To prove a proposition F by resolution,– Start with F– Resolve with a rule from the knowledge base (that contains F)– Repeat until all propositions have been eliminated– If this can be done, a contradiction has been derived and the originalproposition F must be true.
  13. 13. Propositional Resolution Example• Rules– Cold and precipitation -> snow¬cold  ¬precipitation  snow– January -> cold¬January  cold– Clouds -> precipitation¬clouds  precipitation• Facts– January, clouds• Prove– snow
  14. 14. Propositional Resolution Example¬snow ¬cold  ¬precipitation  snow¬cold  ¬precipitation¬January  cold¬January  ¬precipitation ¬clouds  precipitation¬January  ¬cloudsJanuary¬clouds clouds
  15. 15. Examples
  16. 16. Another Resolution Example