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 Proofs1. Convert Predicate form to clause form2. Add the negation of what is to be

inferred or proved in the clause form3. Resolve the clauses together4. 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 formEnglish: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 formAll 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:

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

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 implicationsX [Y animal(Y) loves(X,Y)] [Y loves(Y,X)]

Move inwards: X p X p, X p X pX [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,m1)missile(m1)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