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For Wednesday

- Read chapter 10
- Prolog Handout 4

Exam 1

- Monday
- Take home due at the exam

Try It

- reverse(List,ReversedList)
- evenlength(List)
- oddlength(List)

The Anonymous Variable

- Some variables only appear once in a rule
- Have no relationship with anything else
- Can use _ for each such variable

Arithmetic in Prolog

- Basic arithmetic operators are provided for by built-in procedures:+, -, *, /, mod, //
- Note carefully:?- X = 1 + 2.X = 1 + 2?- X is 1 + 2.X = 3

Arithmetic Comparison

- Comparison operators:><>==< (note the order: NOT <=)=:=(equal values)=\=(not equal values)

Arithmetic Examples

- Retrieving people born 1950-1960:?- born(Name, Year), Year >= 1950, Year =< 1960.
- Difference between = and =:=?- 1 + 2 =:= 2 + 1.yes?- 1 + 2 = 2 + 1.no?- 1 + A = B + 2.A = 2B = 1

Length of a List

- Definition of length/2length([], 0).length([_ | Tail], N) :-length(Tail, N1),N is 1 + N1.
- Note: all loops must be implemented via recursion

Counting Loops

- Definition of sum/3sum(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).

The Cut (!)

- A way to prevent backtracking.
- Used to simplify and to improve efficiency.

Negation

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

Dynamic Predicates

- A way to write self-modifying code, in essence.
- Typically just storing data using Prolog’s built-in predicate database.
- Dynamic predicates must be declared as such.

Using Dynamic Predicates

- assert and variants
- retract
- Fails if there is no clause to retract

- retractall
- Doesn’t fail if no clauses

Resolution

- Propositional version.
{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
- p1Ú ... pj ... Ú pm
- q1Ú ... qk ... Ú qn
such that q=UNIFY(pj , ¬qk), derive

SUBST(q, p1Ú...pj1Úpj+1...ÚpmÚq1Ú...qk1 qk+1...Úqn)

- For two literals pj and qk in two clauses

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...ÚqnÛ

p1Ù... Ù pmÞ q1Ú ...Ú qn

Conjunctive Normal Form (CNF)

- For resolution to apply, all sentences must be in conjunctive normal form, a conjunction of disjunctions of literals
(a1Ú ...Ú am) Ù

(b1Ú ... Ú bn) Ù

..... Ù

(x1Ú ... Ú xv)

- Representable by a set of clauses (disjunctions of literals)
- Also representable as a set of implications (INF).

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)

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)

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)

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

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

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.

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

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))

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.

- 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.

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

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))))

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))))

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?

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)

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