Proof Using Resolution Outline - PowerPoint Presentation

Slide1

Proof Using Resolution

Outline

I. Rule of resolution

III. Resolution refutation

* Figures are from the

textbook site

unless the source is specifically cited.

IV. Horn clauses

II. Conjunctive normal forms

Slide2

I. Resolution

An inference algorithm

is

 

sound

if

whenever

 

complete

if

whenever

 

resolution

+ a complete search algorithm

a complete inference algorithm

 

single inference rule



Inference rules covered so far are sound.



The inference algorithms using them may not be complete.

Slide3

Wumpus World Revisited

Agent:

 

 

 

 

 

 

 

 

 

//

 

 

Rules

KB:

Added to KB via inferences

Slide4

(cont’d)

 

Add to KB:

stench but no breeze

 

 

Similarly, as in deriving

 

 

 

 

 

 

biconditional elimination

Slide5

Resolvent

 

 

resolving the two literals that are

negations of each other

(

resolvent

)

 

If there’s a pit in one of

and

and it’s not in

then it’s in

or

 

 

 

 

Slide6

Simple Resolution Rule

 

(

and

are complementary

literals, i.e.,

or

.)

 

 

Since

is true, then

must be false. But one of

must be true.

Therefore, we can exclude

and assert that one of the remaining

literals must be true.

 

Clause

: a disjunction of literals.

 

Unit clause

: a single literal.

 

 

 

 

 

Slide7

Full Resolution Rule

 

 

If

is true, then

is false. Hence

must be true.

If

is false, then

must be true.

 

and

are complementary literals:

 

 

 

 

Slide8

One Pair at a Time

Only one pair of complementary literals can be resolved at each step.

 

 

true

 

 

 

Incorrect conclusion!

Slide9

II. Conjunctive Normal Form

The resolution rule applies to clauses only.

Every sentence of propositional logic is equivalent to a

CNF.

Conjunctive normal form

(CNF): a conjunction of clauses

Slide10

Converting to CNF

1. Eliminate

.

 

 

 

 

replaced with

 

2. Eliminate

.

 

 

 

 

3. Move

inwards, repeatedly applying

 

 

 

 

 

4. Apply the distributivity law

 

Slide11

CNF Conversion Algorithm (Optional)

 



C

onstruction is similar to the algorithm employing

a stack

to convert an infix expression into a postfix

expression (Com S 228).



Instead of outputting an operator after its two operands in the postfix format,

now you just make the logical operator the parent of the two roots of the subtrees

that store the same operator’s subexpression operands.

(infix expression)

 

(postfix expression)

Operators (connectives):

 

Operands (atomic sentences):

 



Parse every propositional sentence in the KB as an arithmetic

expression to construct an

expression tree

(Com S 228).

Slide12

Postorder Traversal



Perform a

postorder

traversal of the expression tree.



Conversion is done in five cases depending on the logical operator

stored at

.

 



When visiting

an internal node

(representing a connective), its left

and right children (or its unique child in the case of a

node)

store the CNFs for the expressions represented by the left and right subtrees.  

 

 

Slide13

Case 1

The node

stores

.

 

 

CNF

Slide14

Case 2

The node

stores

.

 

 

 

 

Slide15

Case 3

The node

stores

.

 

 

 



If

is a negative literal, i.e.,

, then

reduces to

because the two occurrences of

cancel out.

 

Slide16

Case 4

The node

stores

.

 

 



Logically equivalent to

 



First, c

onvert

into conjunctive normal form

(see case 3).

 



Then,

convert the disjunction

into conjunctive normal

form (see case 2)

 

Slide17

Case 5

The node

stores

.

 

 



Logically equivalent to

 



C

onvert

and

separately into conjunctive

normal forms

and

(see case 4).

 



Return

 

Slide18

Example

has the following conjunctive normal form:

 

Slide19

III. Proof by Resolution – An Example

KB

:

 

 

 

 

Q

:

KB

?

 

Converting sentences to CNF

 

 

 

 

 

 

 

 

 

Spilt each conjunction into clauses.

 

KB

:

KB

:

 

 

 

 

 

Slide20

Proof by Resolution

KB

(updated):

(1)

 

(2)

 

(3)

 

(4

)

 

(5)

 

(6)

 

(7)

 

(8)

 

(1)

 

(2)

 

(3)

 

(4)

 

resolve

Resolution tree

Slide21

Resolution Refutation

(Proof by contradiction)

To show that

KB

, we show that

KB

is unsatisfiable. .

 

KB (about a summer day):

(1) If it is raining and you are outside then you will get wet.

(2) If it is warm and there is no rain then it is a pleasant day.

(3) You are not wet.

(4) You are outside.

(5) It is a warm day.

It is a pleasant day.

Prove

* Example taken from http://watson.latech.edu/book/intelligence/intelligenceApproaches2b2.html

Slide22

KB in Propositional Sentences

(1) ( rain ∧ outside ) ⇒ wet

(2) ( warm ∧

rain ) ⇒ pleasant

(3)

wet

(4) outside

(5) warm

 

KB (rewritten):

We add

pleasant

to KB and try to derive an

empty clause

.

 

(1)

rain ∨ outside ∨ wet(2)

warm ∨ rain ∨ pleasant

(3)

wet

(4) outside                 (5) warm

 

converted into clauses

Slide23

Resolution Refutation Tree

pleasant

 

(2)

warm ∨ rain ∨

pleasant

 

warm

∨ rain

 

(5)

warm

rain

(1)

rain

∨

outside ∨ wet

 

outside

∨ wet

 

(4)

outside

wet

(3)

wet

 

 

contradiction!

(empty clause)

Slide24

Proving

in the Wumpus World

 

 

 

Slide25

Resolution Algorithm

The process ends in one of two situations below:



No new clauses can be added, in which case

KB

does not entail

 



Two clauses resolve to yield the empty clause, in which case

KB

entails

.

 

// no new clauses can be added.

Slide26

Completeness of Resolution

Given a set of clauses

S

, its

resolution closure RC

includes all theclauses in

as well as all the resolvents from repeated applicationsof the resolution rule.

 

RC

is

finite

because only

distinct clauses can be constructed out of

propositional symbols appearing in

S

.

 Ground Resolution Theorem: If is unsatisfiable, then RCcontains the empty clause

.

 Constructive proof by explicitly generating an assignment for

if RC

 

Slide27

IV. Horn Clauses

A clause is called a

Horn clause

if it contains

positive literal.

 

 

 

rain ∨

outside ∨ wet

 

 

rain

outside

wet

 

 

 

 

true

 

 

 

 

 



Definite clause

(

positive literal and

negative literal)

 



Fact

(

positive literal and

negative literal)

 



Goal clause

(

positive literal and

negative literal)

 

 

false

 

: positive literal

: negative literal

 

Slide28

Why Horn Clauses?



Every definite clause can be written as an implication.

 

 

 

Horn clauses are the basis of logic programming, and play an

important role in automated theorem proving.

 

Q :- P1, P2, …, Pk.

(Prolog programming language)



Horn clauses are

closed

under resolution, i.e., the resolvent of a Horn

clause is still a Horn clause.

 

 

 

 

 

 

 

 

 

Slide29

(cont’d)



Inferences with Horn clauses are through

forward- and backward-

chaining algorithms.

Logic programming

(natural inference steps easy for humans to follow)



Low computational complexity

: deciding entailment with Horn clauses

takes

time.

 

size of the KB