Resolution in FOPC. Deepak Kumar November Knowledge Engineering in FOPC

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1 Resolution in FOPC Deepak Kumar November 2017 Knowledge Engineering in FOPC Identify the task Assemble relevant knowledge Decide on a vocabulary of predicates, functions, and constants Encode general knowledge about the domain Encode a description of the specific problem instance Pose queries to the inference procedure and get answers Debug the knowledge base 2 1

2 FOPC Inference Rules Inference rules for Quantifiers (, ) Universal Instantiation Existential Instantiation Generalized Modus Ponens Unification Forward & Backward Chaining Definite Clauses Logic Programming in Prolog Resolution Reductio ad Absurdum 3 Resolution in Propositional Logic New rule of inference that combines Modus Ponens, as well as others, in one rule. Requires wffs to be in a special form: A wff is either a literal: P, P Or a clause a disjunction of literals: P Q R An empty set is equivalent to False: { } False Rule: Resolvent ω ω 1 ω k ω α 1 α m ω 1 ω k α 1 α m 4 2

3 Resolution Rule: ω ω 1 ω k ω α 1 α m ω 1 ω k α 1 α m Example: R P P Q R Q 5 Example: x y z [American(x) Weapon(y) Sells(x, y, z) Hostile(z) Criminal(x)] Becomes: American(x) Weapon(y) Sells(x, y, z) Hostile(z) Criminal(x) Procedure to convert FOPC wffs to CNF is similar to Propositional wffs except we need to - Eliminate - Have only quatified variables 3

4 2. Move inwards 3. Standardize Variables 4. Skolemize (i.e. Eliminate ) 5. Drop 6. Distribute over to get CNF 7. Write as clauses Example: Everyone who loves all animals is loved by someone. 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)] ] 4

5 x [ y [ Animal(y) Loves(x, y) ] [ y Loves(y, x)] ] ] x [ y Animal(y) Loves(x, y) ] [ y Loves(y, x)] ] 2. Move inwards 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)] ] 2. Move inwards x [ y Animal(y) Loves(x, y) ] [ y Loves(y, x)] ] 3. Standardize Variables x [ y Animal(y) Loves(x, y) ] [ z Loves(z, x)] ] 5

6 2. Move inwards x [ y Animal(y) Loves(x, y) ] [ y Loves(y, x)] ] 3. Standardize Variables x [ y Animal(y) Loves(x, y) ] [ z Loves(z, x)] ] 4. Skolemize (i.e. Eliminate ) Recall: x Owns(Nono, x) became Owns(Nono, M52) 4. Skolemize (i.e. Eliminate ) Recall: x Owns(Nono, x) became Owns(Nono, M52) But in: x [ y Animal(y) Loves(x, y) ] [ z Loves(z, x)] ] The existentially quantified variable is in the scope of a variable. We use a new function, F(x) such that F(x) = y, and G(x) = z. F and G are called Skolem Functions. x [Animal(F(x)) Loves(x, F(x)) ] [Loves(G(x), x)] ] 6

7 2. Move inwards 3. Standardize Variables 4. Skolemize (i.e. Eliminate ) x [Animal(F(x)) Loves(x, F(x)) ] [Loves(G(x), x)] ] 5. Drop [Animal(F(x)) Loves(x, F(x)) ] [Loves(G(x), x)] ] 2. Move inwards 3. Standardize Variables 4. Skolemize (i.e. Eliminate ) 5. Drop [Animal(F(x)) Loves(x, F(x)) ] [Loves(G(x), x)] ] 6. Distribute over [Animal(F(x)) Loves(G(x), x)] [ Loves(x, F(x)) Loves(G(x), x)] 7

8 2. Move inwards 3. Standardize Variables 4. Skolemize (i.e. Eliminate ) 5. Drop 6. Distribute over [Animal(F(x)) [Loves(G(x), x)] [ Loves(x, F(x)) Loves(G(x), x)] 7. Write as clauses Animal(F(x)) Loves(G(x), x) Loves(x, F(x)) Loves(G(x), x) Resolution in FOPC Rule: l 1 l k m 1 m n SUBST(θ, l 1 l i 1 l i+1... l k m 1 m j 1 m j+1... m n ) where θ = UNIFY(l i, m j ) Example: Human(Deepak) x Human(x) Mortal(x) Human(x) Mortal(x) Human(Deepak) Mortal(Deepak) θ = {x/deepak} 8

9 Resolution in FOPC Example: Animal(F(x)) Loves(G(x), x) Loves(u, v) Kills(u, v) Animal(F(x)) Kills(G(x), x) θ = {u/g(x), v/x} Using Resolution w/ Reductio ad Absurdum To prove KB resolution α show that KB α is unsatisfiable. Therefore, by refutation, it must be that KB resolution α 9

10 Example: Col. West Knowledge Base x y z [American(x) Weapon(y) Sells(x, y, z) Hostile(z) Criminal(x)] x [Owns(Nono, x) Missile(x)] x [Missile(x) Owns(Nono, x) Sells(CWest, x, Nono)] x [Missile(x) Weapon(x)] x [Enemy(x, America) Hostile(x)] American(CWest) Enemy(Nono, America) Is Colonel West a criminal? (Criminal(CWest)) Convert KB to Clause Form Knowledge Base x y z [American(x) Weapon(y) Sells(x, y, z) Hostile(z) Criminal(x)] x [Owns(Nono, x) Missile(x)] x [Missile(x) Owns(Nono, x) Sells(CWest, x, Nono)] x [Missile(x) Weapon(x)] x [Enemy(x, America) Hostile(x)] American(CWest) Enemy(Nono, America) Is Colonel West a criminal? (Criminal(CWest)) 10

11 Convert KB to Clause Form Knowledge Base x y z [American(x) Weapon(y) Sells(x, y, z) Hostile(z) Criminal(x)] American(x) Weapon(y) Sells(x, y, z) Hostile(z) Criminal(x) x [Owns(Nono, x) Missile(x)] Owns(Nono, m52) Missile(M52) x [Missile(x) Owns(Nono, x) Sells(CWest, x, Nono)] Missile(x) Owns(Nono, x) Sells(Cwest, x, Nono) x [Missile(x) Weapon(x)] x [Enemy(x, America) Hostile(x)] Missile(x) Weapon(x) Enemy(x, America) Hostile(x) American(CWest) Enemy(Nono, America) Is Colonel West a criminal? (Criminal(CWest)) Convert KB to Clause Form Knowledge Base American(x) Weapon(y) Sells(x, y, z) Hostile(z) Criminal(x) Owns(Nono, m52) Missile(M52) Missile(x) Owns(Nono, x) Sells(Cwest, x, Nono) Missile(x) Weapon(x) Enemy(x, America) Hostile(x) American(CWest) Enemy(Nono, America) Is Colonel West a criminal? (Criminal(CWest)) 11

12 Proof by Refutation 1. American(x) Weapon(y) Sells(x, y, z) Hostile(z) Criminal(x) 2. Owns(Nono, m52) 3. Missile(M52) 4. Missile(x) Owns(Nono, x) Sells(Cwest, x, Nono) 5. Missile(x) Weapon(x) 6. Enemy(x, America) Hostile(x) 7. American(CWest) 8. Enemy(Nono, America) 9. Criminal(CWest) Negation of Conclusion Proof by Refutation 1. American(x) Weapon(y) Sells(x, y, z) Hostile(z) Criminal(x) 2. Owns(Nono, m52) 3. Missile(M52) 4. Missile(x) Owns(Nono, x) Sells(Cwest, x, Nono) 5. Missile(x) Weapon(x) 6. Enemy(x, America) Hostile(x) 7. American(CWest) 8. Enemy(Nono, America) 9. Criminal(CWest) Negation of Conclusion 10. American(CWest) Weapon(y) Sells(CWest, y, z) Hostile(z) From (1) & (9) & {x/cwest} 11. Weapon(y) Sells(CWest, y, z) Hostile(z) From (10) & 7 & { } 12. Missile(y) Sells(CWest, y, z) Hostile(z) From (11) & (5) & {x/y} 13. Sells(CWest, M52, z) Hostile(z) From (12) & (3) & {y/m52} 14. Missile(M52) Owns(Nono, M52) Hostile(Nono) From (13) & (4) & {x/m52, z/nono} 15. Owns(Nono, M52) Hostile(Nono) From (14) & (3) & { } 16. Hostile(Nono) From (15) & (2) & { } 17. Enemy(Nono, America) From (16) & (6) & {x/nono} 18. Ø From (17) & (8) & { } Since KB Criminal(Cwest) leads to a contradiction, it must be that KB resolution Criminal(Cwest). 12

13 Proofs by Resolution & Refutation Do the did Curiosity Kill the Cat example from your text. Answer Extraction Green s Trick Knowledge Base American(x) Weapon(y) Sells(x, y, z) Hostile(z) Criminal(x) Owns(Nono, m52) Missile(M52) Missile(x) Owns(Nono, x) Sells(Cwest, x, Nono) Missile(x) Weapon(x) Enemy(x, America) Hostile(x) American(CWest) Enemy(Nono, America) Who is a criminal? Criminal(x) Answer(x) 13

14 Answer Extraction Green s Trick 1. American(x) Weapon(y) Sells(x, y, z) Hostile(z) Criminal(x) 2. Owns(Nono, m52) 3. Missile(M52) 4. Missile(x) Owns(Nono, x) Sells(Cwest, x, Nono) 5. Missile(x) Weapon(x) 6. Enemy(x, America) Hostile(x) 7. American(CWest) 8. Enemy(Nono, America) 9. Criminal(x) Answer(x) Negation of Conclusion 10. American(x) Weapon(y) Sells(x, y, z) Hostile(z) Answer(x) From (1) & (9) & {x/x} 11. Weapon(y) Sells(CWest, y, z) Hostile(z) Answer(CWest) From (10) & 7 & {x/cwest} 12. Missile(y) Sells(CWest, y, z) Hostile(z) Answer(CWest) From (11) & (5) & {x/y} 13. Sells(CWest, M52, z) Hostile(z) Answer(CWest) From (12) & (3) & {y/m52} 14. Missile(M52) Owns(Nono, M52) Hostile(Nono) Answer(CWest) From (13) & (4) & {x/m52, z/nono} 15. Owns(Nono, M52) Hostile(Nono) Answer(CWest) From (14) & (3) & { } 16. Hostile(Nono) Answer(CWest) From (15) & (2) & { } 17. Enemy(Nono, America) Answer(CWest) From (16) & (6) & {x/nono} 18. Answer(CWest) From (17) & (8) & { } CWest is the answer. Proofs by Resolution & Refutation Do the did Curiosity Kill the Cat example from your text. Try: Who killed the cat 14

Inf2D 12: Resolution-Based Inference

Inf2D 12: Resolution-Based Inference School of Informatics, University of Edinburgh 09/02/18 Slide Credits: Jacques Fleuriot, Michael Rovatsos, Michael Herrmann Last time Unification: Given α and β, find θ such that αθ = βθ Most general unifier