1. For Monday Read 10.1-10.2 Prolog handout 4 Homework:
Chapter 9, exercises 9-11

2. Exam 1 Wednesday Covers chapters 1-9 Focus on search and logic
No Lisp or Prolog
Take home handed out that day, due the following Monday.
Note: no class on March 3

3. Homework
Prolog homework
Logic homework

4. Counting Loops
Definition of sum/3 sum(Begin, End, Sum) :- sum(Begin, End, Begin, Sum). sum(X, X, Y, Y). sum(Begin, End, Sum1, Sum) :- Begin < End, Next is Begin + 1, Sum2 is Sum1 + Next, sum(Next, End, Sum2, Sum).

5. Negation Can’t say something is NOT true Use a closed world assumption
Not simply means “I can’t prove that it is true”

6. Resolution Propositional version. First­order form
{a Ú b, ¬b Ú c} |- a Ú c OR {¬aÞ b, b Þ c} |- ¬a Þ c
Reasoning by cases OR transitivity of implication
First­order form
For two literals pj and qk in two clauses
p1 Ú ... pj ... Ú pm
q1 Ú ... qk ... Ú qn
such that q=UNIFY(pj , ¬qk), derive
SUBST(q, p1Ú...pj­1Úpj+1...ÚpmÚq1Ú...qk­1 qk+1...Úqn)

7. Implication form
Can also be viewed in implicational form where all negated literals are in a conjunctive antecedent and all positive literals in a disjunctive conclusion.
¬p1Ú...Ú¬pmÚq1Ú...Úqn Û
p1Ù... Ù pm Þ q1Ú ...Ú qn

8. Conjunctive Normal Form (CNF)
For resolution to apply, all sentences must be in conjunctive normal form, a conjunction of disjunctions of literals
(a1 Ú ...Ú am) Ù
(b1 Ú ... Ú bn) Ù
..... Ù
(x1 Ú ... Ú xv)
Representable by a set of clauses (disjunctions of literals)
Also representable as a set of implications (INF).

9. Example Initial CNF INF P(x) Þ Q(x) ¬P(x) Ú Q(x) P(x) Þ Q(x)
¬P(x) Þ R(x) P(x) Ú R(x) True Þ P(x) Ú R(x)
Q(x) Þ S(x) ¬Q(x) Ú S(x) Q(x) Þ S(x)
R(x) Þ S(x) ¬R(x) Ú S(x) R(x) Þ S(x)

10. Resolution Proofs INF (CNF) is more expressive than Horn clauses.
Resolution is simply a generalization of modus ponens.
As with modus ponens, chains of resolution steps can be used to construct proofs.
Factoring removes redundant literals from clauses
S(A) Ú S(A) -> S(A)

11. Sample Proof P(w)  Q(w) Q(y)  S(y) {y/w}
P(w)  S(w) True  P(x)  R(x)
{w/x}
True  S(x)  R(x) R(z)  S(z) {x/A, z/A}
True  S(A)

12. Refutation Proofs
Unfortunately, resolution proofs in this form are still incomplete.
For example, it cannot prove any tautology (e.g. PÚ¬P) from the empty KB since there are no clauses to resolve.
Therefore, use proof by contradiction (refutation, reductio ad absurdum). Assume the negation of the theorem P and try to derive a contradiction (False, the empty clause).
(KB Ù ¬P Þ False) Û KB Þ P

13. Sample Proof P(w)  Q(w) Q(y)  S(y) {y/w}
P(w)  S(w) True  P(x)  R(x)
{w/x}
True  S(x)  R(x) R(z)  S(z) {z/x}
S(A)  False True  S(x)
{x/A}
False

14. Resolution Theorem Proving
Convert sentences in the KB to CNF (clausal form)
Take the negation of the proposed theorem (query), convert it to CNF, and add it to the KB.
Repeatedly apply the resolution rule to derive new clauses.
If the empty clause (False) is eventually derived, stop and conclude that the proposed theorem is true.

15. Conversion to Clausal Form
Eliminate implications and biconditionals by rewriting them.
p Þ q -> ¬p Ú q
p Û q ­> (¬p Ú q) Ù (p Ú ¬q)
Move ¬ inward to only be a part of literals by using deMorgan's laws and quantifier rules.
¬(p Ú q) -> ¬p Ù ¬q
¬(p Ù q) -> ¬p Ú¬q
¬"x p -> $x ¬p
¬$x p -> "x ¬p
¬¬p -> p

16. Conversion continued
Standardize variables to avoid use of the same variable name by two different quantifiers.
"x P(x) Ú $x P(x) -> "x1 P(x1) Ú $x2 P(x2)
Move quantifiers left while maintaining order. Renaming above guarantees this is a truth­preserving transformation.
"x1 P(x1) Ú $x2 P(x2) -> "x1 $x2 (P(x1) Ú P(x2))

17. Conversion continued
Skolemize: Remove existential quantifiers by replacing each existentially quantified variable with a Skolem constant or Skolem function as appropriate.
If an existential variable is not within the scope of any universally quantified variable, then replace every instance of the variable with the same unique constant that does not appear anywhere else.
$x (P(x) Ù Q(x)) -> P(C1) Ù Q(C1)
If it is within the scope of n universally quantified variables, then replace it with a unique n­ary function over these universally quantified variables.
"x1$x2(P(x1) Ú P(x2)) -> "x1 (P(x1) Ú P(f1(x1)))
"x(Person(x) Þ $y(Heart(y) Ù Has(x,y))) ->
"x(Person(x) Þ Heart(HeartOf(x)) Ù Has(x,HeartOf(x)))
Afterwards, all variables can be assumed to be universally quantified, so remove all quantifiers.

18. Conversion continued
Distribute Ù over Ú to convert to conjunctions of clauses
(aÙb) Ú c -> (aÚc) Ù (bÚc)
(aÙb) Ú (cÙd) -> (aÚc) Ù (bÚc) Ù (aÚd) Ù (bÚd)
Can exponentially expand size of sentence.
Flatten nested conjunctions and disjunctions to get final CNF
(a Ú b) Ú c -> (a Ú b Ú c)
(a Ù b) Ù c -> (a Ù b Ù c)
Convert clauses to implications if desired for readability
(¬a Ú ¬b Ú c Ú d) -> a Ù b Þ c Ú d

19. Sample Clause Conversion
"x((Prof(x) Ú Student(x)) Þ ($y(Class(y) Ù Has(x,y)) Ù $y(Book(y) Ù Has(x,y))))
"x(¬(Prof(x) Ú Student(x)) Ú ($y(Class(y) Ù Has(x,y)) Ù $y(Book(y) Ù Has(x,y))))
"x((¬Prof(x) Ù ¬Student(x)) Ú ($y(Class(y) Ù Has(x,y)) Ù $y(Book(y) Ù Has(x,y))))
"x((¬Prof(x) Ù ¬Student(x)) Ú ($y(Class(y) Ù Has(x,y)) Ù $z(Book(z) Ù Has(x,z))))
"x$y$z((¬Prof(x)Ù¬Student(x))Ú ((Class(y) Ù Has(x,y)) Ù (Book(z) Ù Has(x,z))))
(¬Prof(x)Ù¬Student(x))Ú (Class(f(x)) Ù Has(x,f(x)) Ù Book(g(x)) Ù Has(x,g(x))))
(¬Prof(x) Ú Class(f(x))) Ù
(¬Prof(x) Ú Has(x,f(x))) Ù
(¬Prof(x) Ú Book(g(x))) Ù
(¬Prof(x) Ú Has(x,g(x))) Ù
(¬Student(x) Ú Class(f(x))) Ù
(¬Student(x) Ú Has(x,f(x))) Ù
(¬Student(x) Ú Book(g(x))) Ù
(¬Student(x) Ú Has(x,g(x))))

20. Clause Conversion
(¬Prof(x)Ù¬Student(x))Ú (Class(f(x)) Ù Has(x,f(x)) Ù Book(g(x)) Ù Has(x,g(x))))
(¬Prof(x) Ú Class(f(x))) Ù
(¬Prof(x) Ú Has(x,f(x))) Ù
(¬Prof(x) Ú Book(g(x))) Ù
(¬Prof(x) Ú Has(x,g(x))) Ù
(¬Student(x) Ú Class(f(x))) Ù
(¬Student(x) Ú Has(x,f(x))) Ù
(¬Student(x) Ú Book(g(x))) Ù
(¬Student(x) Ú Has(x,g(x))))

21. Sample Resolution Problem
Jack owns a dog.
Every dog owner is an animal lover.
No animal lover kills an animal.
Either Jack or Curiosity killed Tuna the cat.
Did Curiosity kill the cat?

22. In Logic Form A) $x Dog(x) Ù Owns(Jack,x)
B) "x ($y Dog(y) Ù Owns(x,y)) Þ AnimalLover(x))
C) "x AnimalLover(x) Þ ("y Animal(y) Þ ¬Kills(x,y))
D) Kills(Jack,Tuna) Ú Kills(Cursiosity,Tuna)
E) Cat(Tuna)
F) "x(Cat(x) Þ Animal(x))
Query: Kills(Curiosity,Tuna)

23. In Normal Form A1) Dog(D) A2) Owns(Jack,D)
B) Dog(y) Ù Owns(x,y) Þ AnimalLover(x)
C) AnimalLover(x) Ù Animal(y) Ù Kills(x,y) Þ False
D) Kills(Jack,Tuna) Ú Kills(Curiosity,Tuna)
E) Cat(Tuna)
F) Cat(x) Þ Animal(x)
Query: Kills(Curiosity,Tuna) Þ False

24. Resolution Proof