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Resolution based Inferencing

Resolution based Inferencing. Resolution based rules studied earlier can lead to inferences, rules such as modus ponen, unit resolution etc… Other Approach include resolution through REFUTATTION Forward Chaining Backward Chaining Used in Automated Reasoning and PROLOG interpretators.

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Resolution based Inferencing

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  1. Resolution based Inferencing • Resolution based rules studied earlier can lead to inferences, rules such as modus ponen, unit resolution etc… • Other Approach include resolution through REFUTATTION • Forward Chaining • Backward Chaining • Used in Automated Reasoning and PROLOG interpretators

  2. Procedure for Refutation Proofs • Convert Predicate form to clause form • Add the negation of what is to be inferred or proved in the clause form • Resolve the clauses together • If a contradiction exists in the clause form, steps will generate an empty clause • Note all the substitutions used to generate the contradiction

  3. 1. Clause form English: All men are mortal, everyone who is mortal dies, Shahid is a man. Will shahid die? Clause form:  man(X)  mortal (X)  mortal(X)  die (X) man(shahid) Predicate form: All men are mortal X man(X)  mortal (X) Everyone who is mortal dies X mortal(X)  die (X) Shahid is a man man(shahid)

  4. 2. Negate Conclusion To be concluded: die(shahid) Add:  die (shahid)

  5. Final Clause form Complete Clause form: ( man(X)  mortal (X)) ^ ( mortal(X)  die (X)) ^ man(shahid) All men are mortal, everyone who is mortal dies, Shahid is a man. Will shahid die?

  6. ( man(X)  mortal (X)) ( mortal(X)  die (X))  man(X)  die (X) man(shahid) {shahid/X} die(shahid) die(shahid) [] Resolve All men are mortal, everyone who is mortal dies, Shahid is a man. Will shahid die? Under substitution shahid/X die(shahid) follows from the clause form

  7. Resolution: brief summary • Full first-order version: l1···lk, m1···mn (l1···li-1li+1 ···lkm1···mj-1mj+1···mn) where Unify(li, mj) = . • The two clauses are assumed to be standardized apart so that they share no variables. • For example,  healthy(X) unhappy(X) rich(imran) unhappy(imran) with  = {imran/X} • Apply resolution steps to CNF(KB ); complete for FOL

  8. Conversion to CNF • Everyone who loves all animals is loved by someone: X [Y animal(Y) loves(X,Y)]  [Y loves(Y,X)] • Eliminate biconditionals and implications X [Y animal(Y) loves(X,Y)]  [Yloves(Y,X)] • Move  inwards: X p X p,  X p X p X [Y (animal(Y) loves(X,Y))]  [Y loves(Y,X)] X [Y animal(Y) loves(X,Y)]  [Y loves(Y,X)] X [Y animal(Y) loves(X,Y)]  [Y loves(Y,X)]

  9. Conversion to CNF contd. • Standardize variables: each quantifier should use a different one: X [Y animal(Y) loves(X,Y)]  [Zloves(Z,X)] • Skolemize: a more general form of existential instantiation.Each existential variable is replaced by a Skolem function of the enclosing universally quantified variables: X [animal(f(X)) loves(X,f(X))] loves(g(X),X) • Drop universal quantifiers: [animal(f(X)) loves(X,f(X))] loves(g(X),X) • Distribute  over  : [animal(f(X)) loves(g(X),X)]  [loves(X,f(X)) loves(g(X),X)]

  10. Hostile Nation Example american(X)  weapon(Y)  sells(X,Y,Z)  hostile(Z)  criminal(X) • (american(X)  weapon(Y)  sells(X,Y,Z)  hostile(Z)) v criminal(X) • (american(X) v weapon(Y) v sells(X,Y,Z) v hostile(z)) v criminal(X) X owns(nono,X)  missile(X)  owns(nono,X) v missile(X) missile(X)  owns(nono,X)  sells(west,X,nono)  missile(X) v owns(nono,X) v sells(west,X,nono) missile(X)  weapon(X)  missile(X) v weapon(X) enemy(X,america)  hostile(X)  enemy(X,america) v hostile(X) owns(nono,m1) missile(m1) american(west) enemy(nono,america) Add the negation of the goal to the database  criminal(X)

  11. Resolution proof: definite clauses Be careful about the conventions

  12. Resolution proof: definite clauses Be careful about the conventions

  13. Resolution proof: definite clauses Be careful about the conventions

  14. Resolution proof: definite clauses Be careful about the conventions

  15. Resolution proof: definite clauses Be careful about the conventions

  16. Resolution proof: definite clauses Be careful about the conventions

  17. Resolution proof: definite clauses Be careful about the conventions

  18. Resolution proof: definite clauses Be careful about the conventions

  19. Resolution proof: definite clauses Be careful about the conventions

  20. Resolution proof: definite clauses Be careful about the conventions

  21. Resolution proof: definite clauses Be careful about the conventions

  22. Resolution proof: definite clauses Be careful about the conventions

  23. Resolution proof: definite clauses Be careful about the conventions

  24. Resolution proof: definite clauses Be careful about the conventions

  25. Resolution proof: definite clauses Be careful about the conventions

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