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CS.462 Artificial Intelligence

CS.462 Artificial Intelligence. SOMCHAI THANGSATHITYANGKUL Lecture 06 : First Order Logic Resolution Prove. Proving Theorem in FOL. Proving theorem in FOL raises two additional problems: Need to remove quantifiers Need to be able to compare variable (which represent unknown values).

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CS.462 Artificial Intelligence

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  1. CS.462Artificial Intelligence SOMCHAI THANGSATHITYANGKUL Lecture 06 : First Order Logic Resolution Prove

  2. Proving Theorem in FOL • Proving theorem in FOL raises two additional problems: • Need to remove quantifiers • Need to be able to compare variable (which represent unknown values)

  3. Substitution and Unification • A substitution is a set of variable/ term pairs that one can apply to sentences to perform some needed unification or other manipulation SUBST(, ) denoted the result of applying substitution  to sentence  e.g., SUBST({x/A, y/ f(x)}, P (x,y)) = P(A, f(A)) • Unification is the process of taking 2 atomic statements and finding a substitution that make them equivalent UNIFY(p,q) =  where SUBST(, p) = SUBST(, q) e.g. UNIFY(P(x), P(A) = { x/A }

  4. Unification Example Knows(John, x)  Hates(John, x) Knows(John, Jane) Knows(y, Leonid) Knows(y, Mother(y)) Knows(x, Elizabeth) UNIFY(Knows(John, x), Knows(John, Jane)) = {x/John} UNIFY(Knows(John, x), Knows(y, Leonid)) = {x/John, y/Leonid} UNIFY(Knows(John, x), Knows(y, Mother(y))) = {y/John, x/Mother(John)} UNIFY(Knows(John, x), Knows(x, Elizabeth) = fail

  5. Unification Example • Before beginning unification, all variable names must be unique UNIFY(Knows(John, x1), Knows(x2, Elizabeth) = {x1/Elizabeth, x2/John} • Unification makes as few commitments about variable bindings as possible UNIFY(Knows(John, x), Knows(y, z) = {y/John, x/z} Or {y/John, x/z, w/Freda} Or {y/John, x/John, z/John} Or … {y/John, x/z} is the most general unifier (MGU)

  6. First Order Resolution

  7. Convert to Clausal Form

  8. Convert to Clausal Form

  9. Convert to Clausal Form

  10. Example : Convert to Clausal Form

  11. Resolution Using Unification Given: P(A)  (x) (P(x)  Q(f(x))), prove: (z) (Q(z)) • Negate the theorem: (z) (Q(z)) • Drop universal quantifiers and separate into disjunctive clauses. • P(A) • (P(x)  Q(f(x))) • Q(z) • Clause 1 and 2 are resolved by substituting A for x {x/A}. Resulting clause is Q(f(A)), which is added to original set. • Q(f(A)) is resolved against clause 3, using {z/f(A)}. Result is Q(f(A)) and Q(f(A)) • Contradiction proves original theorem.

  12. Resolution Proof • Question (informal): Is Marcus alive in the year 2001? • Axioms (informal) • Marcus was a man. • Marcus was a Pompeian. • Marcus was born in 40 A.D. • All man are mortal. • The volcano erupted in 79 A.D. • All Pompeians died when the volcano erupted. • No mortal lives longer than 150 years. • Additional axioms are required. 8. Alive means not dead 9. If someone dies, they remain dead.

  13. Resolution Proof Axioms in clause form • Man(Marcus) • Pompeian(Marcus) • Born(Marcus, 40) • Man(x1)  Mortal(x1) from... x Man(x)  Mortal(x) • Erupted(Volcano, 79) • Pomepeian(x2)  Died(x2, 79) from... x Pompeian(x)  Died(x, 79) • Mortal(x3)  Born(x3, t1)  Gt(t2-t1, 150)  Dead(x3, t2) from... x,t1,t2 (Mortal(x)Born(x,t)Gt(t2-t1, 150)  Dead(x, t2))

  14. Resolution Proof 8a. Alive(x4, t3)  Dead(x4, t3) 8b. Dead(x4, t3)  Alive(x4, t3) from... x,t Alive(x, t)  Dead(x, t) 9. Died(x6, t5)  Gt(t6,t5)  Dead(x6,t6) from... x,t1,t2 Died(x, t1) Gt(t2,t1)  Dead(x, t2) Prove:  alive(Marcus, 2001) Negate clause to be proved and add to list: alive(Marcus, 2001)

  15. Alive(x4, t3)Dead(x4, t3) Alive(Marcus, 2001) {x4/Marcus, t3/2001} Mortal(x3)  Born(x3, t1)  Gt(t2-t1, 150)  Dead(x3, t2) Dead(x4, 2001) {x3/Marcus, t2/2001} Mortal(Marcus)  Born(Marcus, t1)  Gt(2001-t1, 150) Man(x1)  Mortal(x1) {x1/Marcus} Man(Marcus)  Born(Marcus, t1)  Gt(2001-t1, 150) Man(Marcus) Born(Marcus, t1)  Gt(2001-t1, 150) Born(Marcus, 40) {t1/40} Gt(2001-40, 150) Gt(1961, 150)

  16. Try this • Use Resolution prove that West is a criminal

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