Inference in FOL

1
Inference in FOL

• Compared to predicate logic,
• more abstract reasoning and specific conclusions

2
FOL knowledge bases

• Facts about environment involve statements about
  specific objects
• E.g., Dentist(Bill), Likes(Mary, Candy)
• General knowledge is mainly statements about sets
  of objects involving quantifiers
• E.g., ?x Dentist(x) ? Likes(x, Candy)

3
Deductive reasoningfrom general to specific

• how do quantified sentences get applied to facts?
• universal quantifier
• existential quantifier
• instantiation substituting a reference to an
  object for a variable
• inference conclusions entailed in KB

4
Instantiating Universal quantifier (UI)

• ?x p(x)
• statement is always true
• any substitution makes a legitimate statement
• format
• ?x p(x)
• subst( x/k, p(x) )
• (K is any constant or function from KB)
• p(K)

5
Instantiating Existential quantifier (UI)

• ?x p(x)
• statement is true for some object
• name the object for which it is true
• format
• ?x p(x)
• subst( x/k, p(x) )
• (k is a new constant, never used before
• Skolem constant)
• p(k)

6
Brute force reasoning

• use instantiation to create a propositional
  logic KB
• complete
• BUT...
• presence of functions causes infinitely large set
  of sentences (Father(Al), Father(Father(Al))
• ? semi-decidable (disproofs never end)

7
Direct reasoning

• ?x man(x) ? mortal(x)
• man(Socrates)
• Substitute for instantiation
• subst( x/Socrates, man(x) ? mortal(x))
• man(Socrates) ? mortal(Socrates)
• modus ponens
• mortal(Socrates)

8
Substitutions for reasoning

• generalized modus ponens
• p1, p2, p3, (p1 p2 p3 )gt q
• subst( x1/k1, x2/k2.., q)
• Unification substitutions so that the sentences
  are consistently instantiated

9
Substitutions for reasoning

• generalized modus ponens example
• Parent(Art,Barb),
• Parent(Barb,Carl),
• (Parent(x,y) Parent(y,z ) ? Grandparent(x,z)
• subst( x/Art, y/Barb,z/Carl, q)
• (Parent(Art,Barb) Parent(Barb,Carl )
• ? Grandparent(Art,Carl)
• Grandparent(Art,Carl)

10
Consistent substitutions

• unification algorithm p.278
• or variant here
• example
• ?x likes(Bill, x) (Bill likes everyone)
• ?y likes(y, Mary) (everyone likes Mary)
• subst( Bill/y, Mary/x, likes(Bill, Mary))
• makes two predicates identical

11
Application

• example
• ?x likes(Bill, x)
• ?y likes(y, Mary) gt trusts(y,Father(Mary))
• subst( Bill/y, Mary/x, likes(Bill, Mary))
• makes two predicates identical
• likes(Bill, Mary),
• likes(Bill, Mary) gt trusts(Bill,Father(Mary))
• ? trusts(Bill,Father(Mary))

12
Examples

• unify
• Likes(x,Art), Likes(Father(y), y)
• Art/y
• Likes(x,Art), Likes(Father(Art), Art)
• Art/y, Father(Art)/x
• unify
• Likes(x,Art), Likes(Bart, x)
• ? fails, cant subst x for Art and Bart

13
Examples

• unify
• Likes(x,Art), Likes(Bart, x)
• fails, cant subst x for Art and Bart
• BUT where did x come from?
• Art likes everybody ?x Likes(x, Art)
• Everybody likes Bart ?x Likes(Bart, x)
• standardize apart ?z0 Likes(Bart, z0)
• then Likes(Bart, Art) is OK with
• subst ( Bart/x, Art/z0 )

14
Unification algorithm

• Unify(L1, L2) // L1, L2 are both predicates or
  both objects
• If (L1 or L2 is variable or constant)
• if (L1L2) return (no subst required)
• if (L1 is variable) if L1 in L2 return fail
  else return L2/L1
• if (L2 is variable) if L2 in L1 return fail
  else return L1/L2
• return fail // both constants or functions
• // L1,L2 are predicates if we get to here
• If predicate symbols of L1,L2 not identical,
  return fail
• If L1,L2 have different number of arguments,
  return fail
• Subst
• For (i 1 to number of arguments in L1,L2)
• S Unify(L1.argumenti,L2.argumenti)
• if (Sfail) return fail
• if (S!)
• apply S to remainder of L1,L2
• Subst Subst U S
• Return Subst

15
Unification algorithm - examples

• Unify(L1, L2) // L1, L2 are predicates or
  objects
• If (L1 or L2 is variable or constant)
• if (L1L2) Art, Art x,x
• if (L1 is variable) if L1 in L2 return fail
  else return L2/L1
• x, Father(x) x, Mother(y)
• if (L2 is variable) if L2 in L1 return fail
  else return L1/L2 ltsimilargt
• return fail Art, Bart
• // L1,L2 are predicates if we get to here
• If predicates of L1,L2 not identical Likes(x,y)
  Brother(z,w)
• If L1,L2 have different of arguments
  Band(x,y,z), Band(t,v)
• Subst
• For (i 1 to of args in L1,L2)
• S Unify(L1.argi,L2.argi) Likes(Bill,x)
  Likes(y,Father(y))
• if (Sfail) return fail
• if (S!)
• apply S to remainder of L1,L2 Likes(Bill,x)
  Likes(Bill,Father(Bill))
• Subst Subst U S
• Return Subst

16
Inference Reasoning methods

• Forward chaining
• Backward chaining
• Resolution

17
Resolution

1. convert sentences to equivalent conjunctive
   normal form (CNF)
2. apply resolution refutation