l6 logic programming

8/2/2019 l6 Logic Programming

1/44

Knowledge Engineering

Quang Nhat NGUYEN

(Academic year 2010-2011)

Hanoi University of Science and Technology

School of Information and Communication Technology


2/44

Content

Introduction

First-order logic

Knowledge representation

Semantic Web

og c programm ng

Resolution

Forward reasonin

Backward reasoning

Uncertain reasoning

Expert systems

Knowledge discovery by Machine learning

Knowledge discovery by Data mining

2Knowledge Engineering


3/44

Logical inference in First-order logic (1) Logical inference problem

, does the KB semantically entail ?KB ?

In all interpretations in which sentences in the KB are true, is

also true?

entailment for all possible input sentences in a finite

number of steps

For the propositional logic: Yes

However, the logical inference problem in the first-order

logic is undecidable



4/44

Logical inference in First-order logic (2) Is the Truth-table approach a viable approach for the

-

No!

It would require us to enumerate and list all possible

interpretations I

I= (assignments of symbols to objects, predicates to

relations and functions to relational mappings)

ere are oo many poss e n erpre a ons



5/44

Logical inference in First-order logic (3) Is the Inference-rule approach a viable approach for the

- -

Yes

The inference rules represent sound inference patterns

that one can apply to sentences in the KB

Reuse inference rules from the propositional logic, , - , - ,Or-introduction, Negation elimination

Additional inference rules are needed for sentences with

quan ers , We need to handle variables



6/44

Variable substitutions Variables in FOL sentences can be substituted with

Ground terms = {constants, variables, functions}

ground terms

{x1 /t1, x2/t2,}

Application of the substitution to FOL sentences

SUBST x/Tuan /Hai Likes x = Likes Tuan Hai

SUBST ({x/z, y/fatherOf(Vu)}, Likes(x, y)) = Likes (z,

fatherOf(Vu))



7/44

Inference rules for quantifiers (1) Universal instantiation (elimination)

o su s u e a var a e w a groun erm

x: (x) ; a is a ground term

a

Example

x: Likes(x, IceCream) Likes(Tuan, IceCream)

By the substitution of the variablexwith the constant Tuan

Denoted as: SUBST({x/a}, x: (x)) = (a)

Universal Instantiation can be applied many times to

produce many different consequences



8/44

Inference rules for quantifiers (2) Existential instantiation (elimination)

elsewhere in the KBx: (x) ; a does not appear in KB

(a)

Example

, ,

By the substitution of the variablexwith the constant Murderer

=,

Existential Instantiation can be applied once



9/44

Unification (1) Problem in inference: Universal instantiation gives many

x: (x) ; a is a ground term

(a)

Solution: Try substitutions of similar sentences in KB

Example Lets assume we have KB containing the 3 following sentences

x: King(x) Greedy(x) Evil(x)

King(John)

y: Greedy(y)

If we use a substitution = {x/John,y/John}, then we can infer

Evil John



10/44

Unification (2) A unification takes two similar sentences and computes

,

existsUNIFY(p,q) = such that SUBST(,p) = SUBST(,q)

Examples

UNIFY(Knows(John, x), Knows(John, Jane)) = {x/Jane}

UNIFY(Knows(John, x), Knows(y, Bill)) = {x/Bill, y/John}

UNIFY(Knows(John, x), Knows(y, MotherOf(y))) =

x o er o n , y o n UNIFY(Knows(John, x), Knows(x, Elizabeth )) = fail Because no

substitution can make the two sentence are the same (i.e.,xcannot take

on the values John and Elizabeth at the same time)



11/44

Most general unifier UNIFYshould return a substitution that makes the two

,

than one such unifier

For exam le, to unif Knows John,x and Knows ,z

1 = {y/John, x/z }

2 = {y/John, x/John, z/John} In the above example, the first unifier is more general

than the second

,most general unifier (MGU) that is unique up to

renaming of variables

In the above example: MGU = {y/John, x/z}



12/44

Resolution inference rule (1) Basic propositional version

,

Equivalent to

,



13/44

Resolution inference rule (2) FOL version:

1 j m, 1 k n

SUBST (, (p1 pj-1 pj+1 pm q1 qk-1 qk+1 qn))

k k j, k

is the most general unifier (MGU) ofpjand qk

The literalspjand qkare called complementary literals

Because each one unifies with the negation of the other

The resulting disjunction is called a resolvent



14/44

Resolution inference rule Examples (1) Example 1

,

Unhappy(Me)by the most general unifier = {x/Me}

Example 2

Lets assume that KB contains the following sentences PhD(x) HighlyQualified(x)

PhD(x) EarlyEarnings(x)

EarlyEarnings(x) Rich(x)

Tr resolution to infer Rich Me



15/44

Resolution inference rule Examples (2)

Convert KB to the equivalent sentences

PhD x Hi hl Qualified x

PhD(x) EarlyEarnings(x) HighlyQualified(x) Rich(x)

From PhD(x) HighlyQualified(x) and PhD(x) EarlyEarnings(x)

and using the MGU = {x/Me}, we infer HighlyQualified(Me)

EarlyEarnings(Me)

From HighlyQualified(x) Rich(x) and HighlyQualified(Me)

Earl Earnin s Me and usin the MGU = x/Me we infer

Rich(Me) EarlyEarnings(Me)

From Rich(Me) EarlyEarnings(Me) and EarlyEarnings(x)

,



16/44

Resolution-based proving

Proof by refutation: To prove KB, we need to prove thatKB is unsatisfiable

Main steps of the resolution-based proving procedure1. Convert {KB, } to CNF with ground terms and universal

variables only

2. Apply repeatedly the resolution rule while keeping track and

consistency of substitutions3. Stop when eitheran empty set (contradiction) is derivedorno

more new resolvents (conclusions) follow

,

clause (equivalent to false) is derived

p, p



17/44

Conjunctive normal form

To be able to do resolution, the given formulas have to

A literal is an atomic formula or the negation of anatomic formula

An atomic formula is also called a positive literal

The negation of an atomic formula is called a negative literal

A clause is a disjunction of literals

A special clause called the empty clause equivalent to false

is a conjunction of disjunctions of literals

Equivalently, if it is a set of clauses



18/44

Conversion to CNF (1)

1. Eliminate implications and equivalences using the laws

(p q) ((p q) (q p)) ((p q) (q p))

.

(p q) (p q)

(p q) (p q)(x: p(x)) (x: p(x))

(x: p(x)) (x: p(x))

3. Rename variables so that each quantifier has a unique variable



19/44


4. Eliminate existential quantifiers ()

quantifier (x), drop the existential quantifier and replace alloccurrences of the existential quantifier variablexby a new

E.g., (x: p(A) q(x)) (p(A) q(B))

If universal uantifiers ,, recede an existential

quantifier (x), drop the existential quantifier and replace all

occurrences of the existential quantifier variablexby the term

f(y1, , yn) where fis a new function symbol (called a Skolem

function) E.g., (x, y: p(x) q(y)) (x: p(x) q(f(x)) where f(x) is a

Skolem function



20/44


5.Drop all universal quantifiers ()

. .,

6.Convert to CNF by using the distributive law of over

, ,

E.g., (p (q r)) (p q) (p r)

7.Flatten nested con unctions or dis unctions. Then write

each disjunction on a separate line



21/44

Conversion to CNF Example (1)

Problem: Convert to CNF the following sentence:

, , ,

1. Eliminate implications

x: y: p x, y y: q x, y r x, y

2. Move inwardsx: ((y: p(x, y)) (y: (q(x, y) r(x, y))))

3. Rename variables

x: ((y: p(x, y)) (z: (q(x, z) r(x, z))))



22/44

Conversion to CNF Example (2)

4. Skolemize (i.e., eliminate existential quantifiers)

, 1 , 2 , 2

5. Drop universal quantifiers

p x, 1 x q x, 2 x r x, 2 x

6. Distribute over(p(x, f1(x)) q(x, f2(x))) (p(x, f1(x)) r(x, f2(x)))

7. Write each disjunction on a separate line

p(x, f1(x)) q(x, f2(x))p(x, f1(x)) r(x, f2(x))



23/44

Resolution-based proving Example

The crime example is described as follows:

weapons to hostile nations The countr Nono, an enem of America, has some

missiles

All of the country Nonos missiles were sold to it by

Use resolution to prove that West is a criminal!



24/44

Example: Formalization in FOL (1)

... it is a crime for an American to sell weapons to

x,y,z: American(x) Weapon(y) Sells(x, y, z) Hostile(z) Criminal(x)

Nono ... has some missiles

x: Owns(Nono, x) Missile(x)

All of the country Nonos missiles were sold to it by

Colonel West

x: Owns(Nono, x) Missile(x) Sells(West, x, Nono)



25/44

Example: Formalization in FOL (2)

Missiles are weapons

x: Missile x Wea on x

An enemy of America is a hostile nationx: Enemy(x, America) Hostile(x)

West, who is an American

American(West)

Nono, an enemy of America ...

Enemy(Nono, America)

To be proved that West is a criminalCriminal(West)



26/44

Example: Convert to CNF form

American(x) Weapon(y) Sells(x, y, z)

Owns(Nono, M1)

Missile(x) Owns(Nono, x) Sells(West, x, Nono)

Missile x Wea on x

Enemy(x, America) Hostile(x)

American West

Enemy(Nono, America)

Criminal West



27/44

Example: Proof



28/44

Horn normal form

A formula is in the Horn norm form if: The formula is a conjunction () of clauses

Each clause is a disjunction () of literals, in which at most 1 (possibly

no!) positive literal E.g., (p q) (p r s)

e now e ge ase can e represen e n e ornnorm form

Rules 1 2 n

Equivalent to the rule: (p1 p2 pn q)

Facts

,

Integrity constraints (p1 p2 pn)

Equivalent to the rule: (p1 p2 pn false)



29/44

Generalized Modus Ponens inference rule

(p1 p2 pn q), p1, p2, , pn

The application of the Generalized Modus Ponensn erence ru e requ res a a e sen ences n musbe in the Horn norm form

The Generalized Modus Ponens inference rule can beapplied in both reasoning strategies

Forward reasonin

Backward reasoning



30/44

Forward reasoning

Problem: Given a set of premises (i.e., rules and facts) stored in the

knowledge base KB, prove the conclusion Q

Idea: Iterate the two following steps until the conclusion is derived

Apply rules whose condition clause match KB

For each matched rule add its conclusion clause to KB



31/44

Forward reasoning Example (1)



32/44




33/44




34/44




35/44




36/44




37/44




38/44

Backward reasoning

Idea: the backward reasoning process starts from theconclusion Q

To prove Q by using the set of premises (i.e., rules and facts)in KB

If not yet, prove all the conditions of a rule (in KB) whoseconclusion clause is Q

Avoid loops

Check if the new derived clause has already been in the list of to-be-proved ones? If already, dont add it again!

Avoid to repeat the proof of a proved clause

The clause has already been proved true

e c ause as a rea y een prove unsa s a e a se n



39/44

Backward reasoning Example (1)



40/44




41/44




42/44




43/44




44/44

Forward vs. Backward reasoning

Forward reasoning is a data-driven inference process

. ., ,

Forward reasoning may do some redundant work

Backward reasoning is a goal-driven inference process,

-

E.g., How do I get into a PhD program?


l6 logic programming

Documents