Transcript Artificial Intelligence CS 165A Thursday, November 8, 2007  Inference in FOL (Ch 9)  Brief midterm review.

Artificial Intelligence
CS 165A
Thursday, November 8, 2007
 Inference in FOL (Ch 9)
 Brief midterm review
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What we’ve been talking about
• Complete and sound inference procedures
• New inference rules:
– Universal Instantiation (gets rid of )
– Existential Instantiation (gets rid of )
– Existential Introduction (adds )
• Generalized Modus Ponens
• Generalized (First-Order) Resolution
– Complete but semidecidable
• Unification
– Finds the substitution(s) necessary to make two sentences match
• Conjunctive normal form (CNF)
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FOL example
• Domain elements: {Bob, Alice, Carol, Ted}
• x Sleepy(x)
– Sleepy(Bob)  Sleepy(Alice)  Sleepy(Carol)  Sleepy(Ted)
• x Hungry(x)
– Hungry(Bob)  Hungry(Alice)  Hungry(Carol)  Hungry(Ted)
• Universal Instantiation: x Sleepy(x)
– Sleepy(Alice)
• Existential Instantiation: x Hungry(x)
– Hungry(k) where k is a constant (not a variable!)
• Existential Introduction: AtHome(Ted)
– x AtHome(x)
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Two versions of Generalized Resolution
For literals pi and qi , where UNIFY(pj , qk) = 
p1   p j   pm
Disjunctions
q1   qk   qn
SUBST( , p1    pm  q1    qn ) except p j and qk
For atomic sentences pi, qi, ri, si , where UNIFY(pj , qk) = 
p1   p j   pn1  r1   rn2
s1   sn3  q1   qk   qn4
Implications
SUBST( , p1   pn1  s1   sn3  r1   rn2  q1   qn4 ) except p j and qk
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Thursday quiz
Note: Everyone gets a 100% for last Thursday!
1. If the domain is natural numbers (0, 1, 2, ...), what is a
general unifier (if one exists) for these two atomic
sentences:
GreaterThan (x, 7)
GreaterThan (Squared (y), y)
E.g.,  = { a/13, b/4 }
2. What new (to us) dimension does situational calculus
allow us to represent and reason about?
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Conjunctive normal form (CNF)
• First we must convert all sentences to Conjunctive Normal
Form (CNF)
– A CNF sentence is a disjunctions of literals
 Literals: Possibly negated propositions, variables, constants, or
predicates
– So each sentence in the KB is a disjunction of literals, e.g.:
 P(x)  Q(y)
 Dog(x)  Cat(y)
 Big(x)  Slow(x)  Young(x)
– The KB itself is a conjunction of disjunctions of literals, e.g.:

P(x)Q(y)  Dog(x)Cat(y)  Big(x)Slow(x)Young(x)
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Another “canonical” form: INF
• Implicative Normal Form (INF)
– Each sentence in the KB is an implication with a conjunction of
atoms on the left and a disjunction of atoms on the right, e.g.:
 P(x)  Q(x)
 P(x)  Q(x)  R(x)  S(x)
• INF and CNF are logically equivalent
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Conversion to Conjunctive Normal Form (CNF)
• Replace (P  Q) with (P  Q) and (Q  P)
• Eliminate implications: Replace (P  Q) with (P  Q)
• Move  inwards: , , , (P  Q), (P  Q)
• Standardize variables apart: x P(x)  x Q(x) becomes
x1 P(x1)  x2 Q(x2) [Give all variables different names]
• Move quantifiers left in order: x P(x)  y Q(y) becomes
x y P(x)  Q(y)
• Eliminate  by “Skolemization” (coming)
• Drop universal quantifiers
• Distribute  over , e.g.: (P  Q)  R becomes (P  R) 
(Q  R)
[What about (P  Q)  R ?]
• Flatten nesting: (P  Q)  R becomes P  Q  R
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Conversion to Implicative Normal Form (INF)
• First convert to CNF
• Convert disjunctions to implications: negative literals to
the left, positive literals to the right
P  Q  R  S becomes P  Q  R  S
P  Q becomes
P  Q  False
R  S becomes True  R  S
Remember that P(x) is logically equivalent to True  P(x)
and P(x) is logically equivalent to P(x)  False
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Skolemization
• To “Skolemize” is to remove existential quantifiers by
elimination
– Existential elimination when x is on the outside
x Sleepy(x) …becomes… Sleepy(RipVanWinkle) [If what?]
– More complicated when inside a universal quantifier
x Person(x)  y Heart(y)  Has(x, y)
“Everyone has a heart”
x Person(x)  Heart(H1)  Has(x, H1)
“Everyone has the heart H1”
– Rather, introduce a “Skolem function” H(x)
x Person(x)  Heart(H(x))  Has(x, H(x))
where H() does not appear elsewhere in the KB
– Arguments of the Skolem function: all enclosing universally
quantified variables
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Skolemization examples
By Existential Elimination,
y MajorsIn(y, Sociology)
MajorsIn(Somedude, Sociology)
But it’s not true that
x y MajorsIn(y, x)
x MajorsIn(Somedude, x)
Rather,
x y MajorsIn(y, x)
x MajorsIn(P(x), x)
where P(x) refers to a different constant for every x
x y Student(x)  TakesCourses(x)  KnowsAbout(x, y)
x Student(x)  TakesCourses(x)  KnowsAbout(x, S(x))
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Simple examples
• How to put these into CNF and INF?
CNF
INF
 P(x)  Q(x)
P(x)  Q(x)
P(x)  Q(x)
P(x)
P(x)
True  P(x)
P(x)  Q(x)
P(x)  Q(x)
True  P(x)  Q(x)
P(x)  Q(y)
P(x)  Q(y)
True  P(x)  Q(y)
P(x)   Q(y)
P(x)  Q(y)
Q(y)  P(x)
P(x)  Q(x)
P(x), Q(x)
True  P(x), True  Q(x)
In practice, we can leave out the “True ”
in these sentences
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Moving  inwards
• x S(x) 
– x S(x)

(is equivalent to…)
x Hungry(x)   x Hungry(x)
• x S(x) 
– x S(x)

 x Hungry(x)  x Hungry(x)
• (P  Q) 
– (P  Q)
• (P  Q) 
– (P  Q) [In CNF that’s two sentences!]
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Generalized (first-order) resolution
For literals pi and qi , where UNIFY(pj , qk) = 
p1   p j   pm
Disjunctions
q1   qk   qn
SUBST( , p1    pm  q1    qn ) except p j and qk
Note:
Examples:
• p and q do not unify
pj: Engineer(x)
pj: Loves(x, Broccoli)
qj: Engineer(Bill)
qj: Loves(Joe, y)
 = { x/Bill }
 = { x/Joe, y/Broccoli }
• Rather, p and q unify
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Generalized resolution example
• KB:
Rich(x)  Famous(x)
Rich(x)  Content(x)
Famous(x)  Happy(x)
Content(x)  Happy(x)
Rich(Bob)
• Is Bob happy?
– ASK(KB, Happy(Bob))
– GMP can’t answer this
– GR can
 First put KB in CNF
• KB in CNF:
Rich(x)  Famous(x)
Rich(x)  Content(x)
Famous(x)  Happy(x)
Content(x)  Happy(x)
Rich(Bob)
and the negated query:
Happy(Bob)
What’s this method called?
- Proof by contradiction
- Refutation
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Rich(x)  Famous(x)
Rich(x)  Content(x)
Famous(x)  Happy(x)
Content(x)  Happy(x)
Rich(Bob)
Happy(Bob)
Example (cont.)
   ,   
 
Rich(x)  Content(x)
Content(x)  Happy(x)
{}
Rich(x)  Happy(x)
Rich(Bob)
{x/Bob}
Happy(Bob)
Happy(Bob)
{}
False
QED
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Forward and Backward Chaining
• A reasoning program needs
– A language for representing knowledge – First-Order Logic
– Rules – just one, Generalized Resolution
– Control mechanism– ???
• Two general approaches to control
– Start with the KB and generate new conclusions (which can enable
more inferences to be made)
 E.g., when a new fact is added to the KB
– Given what we want to prove, find implication sentences that
would allow us to conclude it, and attempt to establish their
premises
 E.g., when a goal is to be proved
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Forward and Backward Chaining
Forward chaining

S
G
Backward chaining
S

G
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Forward and Backward Chaining
• Forward chaining
– Data driven or data directed
– New version of TELL(KB, p)
 Add the sentence p, then apply inference rules to the updated
KB until no more rules apply (“chaining” – “chain reaction”)
• Backward chaining
–
–
–
–
Goal oriented
ASK(KB, q)
If SUBST(, q) is in KB, return q' = SUBST(, q)
Else, find implication sentences p  q then set p as a subgoals
 Keep doing this, working “backwards”
 If p is not in KB, look for r  p, then set r as a subgoal
 Etc…..
– Backward chaining is the basis for logic programming (e.g., Prolog)
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Midterm Review
• Tuesday during class
– Please be on time!
• Closed book exam
– You may bring one sheet of paper with notes (8.5” x 11”, both
sides)
– Test paper will be supplied (don’t need to bring your own)
– Calculator not necessary
• What are you responsible for?
– Possibly anything that has been covered in the reading (textbook
through Ch. 8 and articles), lectures, discussion sessions, and
homework assignments
– But you should probably focus on the lectures….
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You will be given…
• Inference rules for Propositional logic
– Modus ponens, and-elimination, and-introduction, or-introduction,
double-negation elimination, unit resolution, resolution
• FOL
– Universal elimination, existential elimination, existential
introduction
– Generalized modus ponens
– Generalized (first-order) resolution
• Procedure to convert logic sentences to CNF
These are posted on the course web
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Examples of things you might be asked
• Which search algorithm(s) would be most appropriate in this particular
situation?
• What are the main difference(s) between these two search algorithms
• What does it mean for a search alg. to be optimal? To be complete?
• Are these heuristics admissible or not?
• Do {iterative deepening search, A*, minimax, expectimax, …} on this
problem.
• Show the truth table for this propositional logic sentence.
• Are these logic sentences satisfiable, unsatisfiable, or valid?
• What does it mean for an inference procedure to be optimal? To be
complete?
• Does the KB entail this sentence S?
• How are these terms unified?
• Apply certain inference rules to this KB.
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