1 FOL 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 FOL Procedure for Refutation Proofs 1. Convert Predicate form to clause form 2. Add the negation of what is to be inferred or proved in the clause form 3. Resolve the clauses together 4. If a contradiction exists in the clause form, steps will generate an empty clause 5. Note all the substitutions used to generate the contradiction

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

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

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

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

7 FOL Resolution: brief summary Full first-order version: l 1  ···  l k, m 1  ···  m n ( l 1  ···  l i-1  l i+1  ···  l k  m 1  ···  m j-1  m j+1  ···  m n )  where Unify ( l i,  m j ) = . 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 FOL 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)]  [  Y loves(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 FOL Conversion to CNF contd.  Standardize variables: each quantifier should use a different one:  X [  Y animal(Y)   loves(X,Y)]  [  Z loves(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 FOL 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,m 1 ) missile(m 1 ) american(west) enemy(nono,america) Add the negation of the goal to the database  criminal(X)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

25 FOL Resolution proof: definite clauses  Be careful about the conventions