Inference Rules with Quantifiers

1
Inference Rules with Quantifiers

• Substitution
• SUBST(?, ?) ??
• ? ? x1/t1, x2/t2, , xn/tn
• SUBST(x/John, y/mother(x), love(x, y))
  love(John, mother(John))

2
Inference Rules with Quantifiers

• Universal instantiation
• ?x a
• ?????????????
• SUBST(x/ground-term, ?)
• ?x king(x) ? greedy(x) ? evil(x)
• ???????????????????
• king(John) ? greedy(John) ? evil(John)

3
Inference Rules with Quantifiers

• Existential instantiation
• ?x a
• ??????????????
• SUBST(x/new-constant, ?)
• ?x crown(x) ? on-head(x, John)
• ??????????????
• crown(c) ? on-head(c, John)

4
Inference Rules with Quantifiers

• Reduction to propositional inference
• ?x king(x) ? greedy(x) ? evil(x)
• king(John)
• greedy(John)
• king(John) ? greedy(John) ? evil(John)

5
Inference Rules with Quantifiers

• An offspring of a horse is a horse
• ?
• Bluebeard is a horse
• ?
• Charlie is Bluebeards offspring
• ?
• Inferences horse(Charlie) ?

6
Inference Rules with Quantifiers

• An offspring of a horse is a horse
• ?x, y horse(x) ? offspring(x, y) ? horse(y)
• Bluebeard is a horse
• horse(Bluebeard)
• Charlie is Bluebeards offspring
• offspring(Bluebeard, Charlie)
• Inference horse(Charlie)

7
Generalized Modus Ponens

• p1, p2, , pn, (p1, p2, , pn ? q)
• ????????????????
• SUBST(?, q)
• pi and pi are atomic
• SUBST(?, pi) SUBST(?, pi)

8
Generalized Modus Ponens

• Unification
• UNIFY(p, q) unifier ? where SUBST(?, p)
  SUBST(?, q)
• knows(John, x) ? hates(John, x)
• knows(John, Jane)
• knows(y, Leonid)
• knows(y, mother(Y))
• knows(y, Elizabeth)
• UNIFY(knows(John, x), knows(John, Jane))
  x/Jane
• UNIFY(knows(John, x), knows(y, Leonid))
  x/Leonid, y/John
• UNIFY(knows(John, x), knows(y, mother(y)))
  y/John, x/mother(John)
• UNIFY(knows(John, x), knows(x, Elizabeth)) fail

9
Generalized Modus Ponens

• Standardization
• UNIFY(knows(John, x1), knows(x2, Elizabeth))
  x1/John, x2/Elizabeth
• Most generalized unifier
• UNIFY(knows(John, x), knows(y, z))
• y/John, x/John, z/John
• y/John, x/Jane, z/Jane
• y/John, x/v, z/v
• y/John, x/z, v/Jane
• y/John, x/z

10
Forward Chaining

• An offspring of a horse is a horse
• ?
• Bluebeard is a horse
• ?
• Bluebeard is Charlies parent
• ?
• Offspring and parent are inverse relations
• ?

11
Forward Chaining

• An offspring of a horse is a horse
• ?x,y horse(x) ? offspring(x, y) ? horse(y)
• Bluebeard is a horse
• horse(Bluebeard)
• Bluebeard is Charlies parent
• parent(Charlie, Bluebeard)
• Offspring and parent are inverse relations
• ?x,y offspring(x, y) ? parent(y, x)

12
Backward Chaining
?x,y horse(x) ? offspring(x, y) ?
horse(y) horse(Bluebeard) parent(Charlie,
Bluebeard)
horse(z)
horse(z)
z/Bluebeard
horse(Bluebeard)
offspring(Bluebeard, z)
parent(z, Bluebeard)
z/Charlie
13
Soundness and Completeness

• Soundness
• if KB derives a then KB a

14
Soundness and Completeness

• Modus Ponens is sound
• ?x p(x) ? q(x)
• ?x ?p(x) ? r(x)
• ?x q(x) ? s(x)
• ?x r(x) ? s(x)
• q(c)
• KB derives q(c) and KB s(c)

15
Soundness and Completeness

• Completeness
• if KB a then KB derives a

16
Soundness and Completeness

• Modus Ponens is not complete
• ?x p(x) ? q(x)
• ?x ?p(x) ? r(x)
• ?x q(x) ? s(x)
• ?x r(x) ? s(x)
• q(c)
• KB derives s(a) but KB ? s(a)

17
Generalized Resolution

• p1 ? pj ? pm
• q1 ? qk ? qn
• ????????????????????????????
• resolvent
• SUBST(?, p1 ? pj-1 ? pj1 ? pm ? q1 ? qk-1
  ? qk1 ? qn)
• UNIFY(pj , ?qk) ? for literals pi and qi

18
Generalized Resolution

• Conjunctive normal form
• ?x p(x) ? q(x)
• ?x ?p(x) ? r(x)
• ?x q(x) ? s(x)
• ?x r(x) ? s(x)
• ?p(x) ? q(x)
• p(y) ? r(y)
• ?q(z) ? s(z)
• ?r(t) ? s(t)

19
Generalized Resolution

• Refutation proof procedure
• KB a if and only if KB ? ?a false

20
Generalized Resolution
?p(x) ? q(x)
?q(z) ? s(z)
?p(z) ? s(z)
p(y) ? r(y)
?p(x) ? q(x) p(y) ? r(y) ?q(z) ? s(z) ?r(t) ? s(t)
s(y) ? r(r)
?r(t) ? s(t)
s(t)
?s(a)
sound and complete
21
Generalized Resolution

• Conversion to CNF
• Eliminate implications
• p ? q ? ?p ? q
• Move ? inwards
• ?(p ? q) ? ?p ? ?q
• ?(p ? q) ? ?p ? ?q
• ??x p ? ?x ?p
• ??x p ? ?x ?p
• ?? p ? p
• Standardize variables
• (?x p(x)) ? (?x p(x)) ? (?x p(x)) ? (?y q(y))

22
Generalized Resolution

• Conversion to CNF (cont.)
• Move quantifiers left
• p ? q ? ?X p ? q
• Skolemize (to remove ?)
• ?x p(x) ? p(c) Skolem constant
• ?x ?y p(x, y) ? ?x p(x, f(x)) Skolem function
• Distribute ? over ?
• (p ? q) ? r ? (p ? r) ? (q ? r)
• Flatten nested conjunctions and disjunctions
• (p ? q) ? r ? (p ? q ? r)
• (p ? q) ? r ? (p ? q ? r)

23
Generalized Resolution

• Jack owns a dog
• ?
• Every dog owner is an animal lover
• ?
• No animal lover kills an animal
• ?
• Either Jack or Fred killed cat Kiko
• ?

24
Generalized Resolution

• Jack owns a dog
• ?x dog(x) ? own(Jack, x)
• Every dog owner is an animal lover
• ?x (?y dog(y) ? own(x, y)) ? animal-loverx)
• No animal lover kills an animal
• ?x animal-lover(x) ? ?y animal(y) ? ?kill(x, y)
• Either Jack or Fred killed cat Kiko
• kill(Jack, Tuna) ? kill(Fred, Kiko)

25
Generalized Resolution

• dog(c)
• owns(Jack, c)
• ?dog(y) ? ?own(x, y)) ? animal-lover(x)
• ?animal-lover(x) ? ?animal(y) ? ?kill(x, y)
• kill(Jack, Kiko) ? kill(Fred, Kiko)
• cat(Kiko)
• ?cat(x) ? animal(x)
• ? kill(Fred, Kiko)

26
Logic Programming

• Robinson, J.A. 1965. A machine-oriented logic
  based on the resolution principle. Journal of ACM
  12 (1) 23-41.
• Kowalski, R.A. Kuehner, D. 1971. Linear
  Resolution with Selection Function. Artificial
  Intelligence 2 (3/4) 227-260.
• Kowalski, R.A. 1979. Algorithm Logic Control.
  Communication of ACM 22 (7) 424-436.
• Lloyd, J.W. 1987. Foundations of Logic
  Programming (2nd ed.).

27
Logic Programming

• Definite logic program
• A ? B1, B2, , Bm ? A ? ?B1 ? ?B2 ? ? ?Bm
  (program clause)
• head body
• ? A1, A2, , An ? ?A1 ? ?A2 ? ? ?An (goal
  clause)
• A, Bi, Aj atoms

28
Logic Programming

• PROLOG (Alain Colmerauer 1972) only Horn clauses
  (definite clauses) allowed

29
Logic Programming

• The occur-check is omitted from the unification ?
  unsound
• test ? p(x, x)
• p(x, f(x))

30
Logic Programming

• Backward chaining with depth-first search
  ? incomplete
• p(x, y) ? q(x, y)
• q(x, y) ? q(y, x)
• p(x, x)

31
Logic Programming

• Unsafe cut ? incomplete
• A ? B, C ? A
• B ? D, !, E
• D ? ? B, C
• ? D, !, E, C
• ? !, E, C

32
Logic Programming

• Negation as failure ? P if fails to prove P
• (closed-world assumption)

33
Exercises

• In Russell Norvigs AIMA (2nd ed.) Exercises
  of Chapter 9.