knowledge representation-part 2

7/28/2019 Knowledge Representation-Part 2

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First-Order Logic

Study Material:

Section 2.2-2.4,Chapter 2 in Luger, Artificial Intelligence:

Structures and StrategiesExample 3.3.5, Chapter 3 in Luger, Artificial Intelligence:Structures and Strategies, pp 115-116

1CP468, Dr. Reem K. Al-Halimi


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Propositional Logic Simple logic in which all sentences are assertions with

truth values.

e.g. it is raining

We cannot access the components of an individualassertion

e.g. if P represents A bug is in my soup we cannot

access the bug or soup components No variables are allowed cannot create general

assertions about entities.



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Example 2

Blocks WorldGoal: to create a set of expressionsthat is to represent a static snapshot ofthe blocks world.

Assumption: predicates evaluatedtop-down, and left-right.

b

da

c



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Example 2

Blocks Worldblock X is on block Y

on(X, Y)

block X is on the tableon_table(X)

block X has nothing on top of it

clear(x)

the robots hand is emptyhand_empty

b

da

c



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Example 2

Blocks WorldWhat is the domain of the variables X and

Y in this world for the previousexpressions?

D={a,b,c,d}

What are the possible interpretations forX and Y for on(X,Y)?

{, ,, ,,,,,,, }

Which interpretations make on(X,Y) truein this world?

{, }

b

da

c



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Example 2

Blocks WorldA description of this blocks world:

on(b,a)ontable(a)

ontable(d)

on(c,d)

clear(b)clear(c)

hand_empty

b

da

c



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Models and Interpretations Models are mathematical abstractions of the possible world.

Models are defined by: a set of objects.

the relations between them and the functions that can be applied to them

Interpretations map the symbols of a language to the model: constant symbols are mapped to objects

function symbols are mapped to functions

predicate symbols are mapped relations in the model. Interpretations and models together determine the truth of a

sentence.

m is a model of sentence smeans sentence s is true in m.



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Interpretations If an interpretation Imakes S true, we sayIsatisfiesS.

If there is at least one interpretation Ithat makes Strue, we saySis satisfiable.

If there is no interpretation I that makes Strue, we saySis unsatisfiable:

X (p(X) p(X))



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Interpretations If a sentence S is T under all interpretations, S is said

to be valid.

Examples: X (p(X) p(X))

true

The deduction theorem

For any sentences s1 and S2, S1 entails sS2 if and only ifthe sentence (S1S2) is valid.

(proof?)



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Interpretations

valid: All interpretations make S

true

satisfiableThere is aninterpretation

that makes S true

unsatisfiable

Nointerpretationmakes S true



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Example 2

Blocks WorldGoal: to create a set of expressions that isto represent a static snapshot of the blocks

world.Assumption: predicates evaluated top-

down, and left-right.

on(b,a)ontable(a)ontable(d)

on(c,d)clear(b)clear(c)hand_empty

b

da

c

The blocks worldin this figure is amodelfor thisinterpretation



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Example 2

Blocks WorldWhen is a block clear?

A block in our world is clear if there are no other blocks on top of it

X ((Y on(Y,X)) clear(X))How can we clear a block clear?

X ((Y on(Y,X)) clear(X))

conditions action

How can we stack a block on top of another?

XY (hand_empty clear(X) clear(Y) pick_up(X) put_down(X,Y) stack(X,Y))

b

da

c



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Using FOL to Represent a Domain Domain: A part of the world of which we would like to express some

knowledge. Components:

assertions axioms (basic sentences that do not need to be proven)

facts about the domainon(a, b) properties of objects or predicates (possibly partial)

XY(on(Y,X) ontable(Y)) definitions

X (((Y on(Y,X)) clear(X)) (clear(X)(Y on(Y,X)))

theorems (entailed by axioms )X ((Y on(Y,X)) clear(X)) (assuming the definition is used in KB)

queries/goals clear(a) => repsonse should be F X clear(X) => response: {X/b, X/c} Substitution list



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Knowledge Endgineering

The general process of knowledge base construction

(Russell and Norvig, 2003)



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Knowledge Endgineering Process1. Identify the task

2. Assemble the relevant information

3. Decide on a vocabulary of predicates, functions, andconstants.

4. Encode general knowledge about the domain

5. Encode a description of the specific problem instance

6. Pose queries to the inference procedure and get answers.7. Debug the knowledge base


E l 3


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Example 3

A Logic-Based Financial Advisor

Goal: to help the user decide whether to invest in a savings account, thestock market only, or a combination of the two.

Recommendations will be based on the investors current savings andtheir income according to the criteria:

1. Individuals with an inadequate savings account should make

increasing their savings a first priority.2. Individuals with an adequate savings account and an adequate

income should consider investing in the stock market.

3. Individuals with an adequate savings account and a lower incomeshould consider splitting their surplus income between savings

and stocks. the adequacy of their savings account is determined by the number

of dependents: s/he has to have at least $5000/dependent.

An adequate income must be steady, at least $15,000/year plus anadditional $4,000 per dependent.


E l 3


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Example 3


1. Identify the task

Goal: to help the user decide whether to invest in a savingsaccount, the stock market only, or a combination of the

two.


E l 3


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Example 3


2. Assemble the relevant information

Recommendations will be based on the investors current savings and theirincome according to the criteria:

1. Individuals with an inadequate savings account should makeincreasing their savings a first priority.

2. Individuals with an adequate savings account and an adequate incomeshould consider investing in the stock market.

3. Individuals with an adequate savings account and a lower income

should consider splitting their surplus income between savings andstocks.

the adequacy of their savings account is determined by the number ofdependents: s/he has to have at least $5000/dependent.

An adequate income must be steady, at least $15,000/year plus an

additional $4,000 per dependent. 32CP468, Dr. Reem K. Al-Halimi

E l 3


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Example 3



Recommendations will be based on the investors current savings andtheir income according to the criteria:

1. Individuals with an inadequate savings account should make

increasing their savings a first priority.

2. Individuals with an adequate savings account and an adequateincome should consider investing in the stock market.

3. Individuals with an adequate savings account and a lower incomeshould consider splitting their surplus income between savingsand stocks.

the adequacy of their savings account is determined by the numberof dependents: s/he has to have at least $5000/dependent.

An adequate income must be steady, at least $15,000/year plus an

additional $4,000 per dependent. 33CP468, Dr. Reem K. Al-Halimi

E l 3


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Example 3



Concepts

the investors current savings

adequate/inadequate

the investors income

adequate/inadequate

steady/unsteady

investment

savings stock

combination (savings and stock)

the number of dependents

An adequate income must be steady


E l 3


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Example 3



Symbol Type Concept Name

Function Symbols the investors current savings amount_saved

the investors income earningsConstants adequate adequate

inadequate inadequate

savings savings

stock stock

combination combination

steady steady

unsteady unsteady

Predicate Symbols invest invest

dependents dependents 35CP468, Dr. Reem K. Al-Halimi


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Example 3

A Logic-Based Financial Advisor4. Encode general knowledge about the domain

Decisions are based on savings_account(adequate)

savings_account(inadequate)

income(adequate)

income(inadequate)

decisions: invest(savings)

invest(stocks)

invest(combination)



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Example 3


Decision criteria:1. savings_account(inadequate) invest(savings)

2. savings_account(adequate) income(adequate) invest(stocks)

3. savings_account(adequate) income(inadequate) invest(combination)



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Example 3


adequate savings? the adequacy of their savings account is determined by the

number of dependents: s/he has to have at least

$5000/dependent.

How many dependents? Y such that dependents(Y) is true

What is the minimum savings required?

minsavings(Y) = 5000*Y What are the actual savings?

amount_saved(X) Are the actual savings greater than the minimum?

greaterThan(X, minsavings(Y))



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Example 3:A Logic-Based Financial Advisor

3. Decide on a vocabulary of predicates, functions, and constants.

Symbol Type Concept NameFunction Symbols the investors current savings amount_saved

the investors income earnings

minimum adequate savings minsavings

Constants adequate adequate


savings savings

stock stock

combination combination

steady steadyunsteady unsteady


dependents dependents

test if one number is greater than another greaterThan 39CP468, Dr. Reem K. Al-Halimi


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Example 3


adequate savings? Are the savings adequate (check all amounts saved)?

X amount_saved(X) Y (dependents(Y) greaterThan(X,minsavings(Y)) savings_account(adequate)

X amount_saved(X) Y (dependents(Y) greaterThan(X,minsavings(Y)) savings_account(inadequate)



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Example 3


adequate income? An adequate income must be steady, at least $15,000/year

plus an additional $4,000 per dependent

What is the minimum savings required? minincome(Y) = 15000 + 4000*Y

What are the actual savings? earnings(X, Z)

Are the actual earnings greater than the minimum? greaterThan(X, minincome(Y))



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Example 3:A Logic-Based Financial Advisor

3. Decide on a vocabulary of predicates, functions, and constants.

Symbol Type Concept NameFunction Symbols the investors current savings amount_saved

the investors income earnings

minimum adequate savings minsavings

minimum adequate income minincome

Constants adequate adequate


savings savings

stock stock

combination combinationsteady steady

unsteady unsteady


dependents dependents

test if one number is greater than another greaterThan 42CP468, Dr. Reem K. Al-Halimi


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Example 3


adequate income? An adequate income must be steady, at least $15,000/year

plus an additional $4,000 per dependent

Are the savings adequate (check all amounts saved)? X earnings(X, steady) Y (dependents(Y) greaterThan(X,

minincome(Y)) income(adequate) X earnings(X, steady) Y (dependents(Y) greaterThan(X,

minincome(Y)) income(inadequate)

X earnings(X, unsteady) income(inadequate)



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Example 3

A Logic-Based Financial Advisor5. Encode a description of the specific problem

instance

Given a particular investor, we add his/her informationto the knowledge base:

e.g. an investor with three children, $22,000 in savings,and $25,000 in steady income:

earnings(25000, steady)

amount_saved(22000)

dependents(3)



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Example 3

A Logic-Based Financial Advisor6. Pose queries to the inference procedure and get

answers.

What type of investment should the customer make?

X invest(X)

How can the system find out the answer?



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Components of First-Order Logic

Proof System In propositional logic we could test an assumption

using a truth table.

We can use the same approach for sentences notcontaining variables



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Proof System Let us take an example:

if it is sunny, the weather will be warm

S W (1) if it is warm, the school will be open

W O (2)

It is sunny

S (3)

Will the school be open?

i.e. is true that in every world in which all of the abovesentences are true that O is true as well? Does Ologically follow from sentences 1, 2, and 3 above?



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Proof SystemS W O SW WO

T T F T F

T T T T T

T F T F T

T F F F T

F T T T T

F T F T F

F F T T T

F F F T T



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T T F T F

T T T T T

T F T F T

T F F F T

F T T T T

F T F T FF F T T T

F F F T T



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T T F T F

T T T T T

T F T F T

T F F F T

F T T T T

F T F T FF F T T T

F F F T T

Yes! The School will be open!



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Proof System In propositional logic we could test an assumption

using a truth table.

this cannot necessarily be done in first-order logic.Why not?

Solution? Use rules to create new conclusions fromgiven predicates



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Proof System Logical inference is the ability to infer new correct

expressions from a set of true assertions.

Aproof procedure is a combination of an inference ruleand an algorithm for applying that rule on a set oflogical expressions to generate new sentences



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Knowledge-Based System(from previous lectures) Consists of

world knowledge (knowledge base) and a way to reason about this knowledge.

KnowledgeBase

InferenceEngine



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Knowledge-Based SystemWorld knowledge is expressed in a language using the

languages syntax and semantics rules.

KnowledgeBase

InferenceEngine

Chirpy is a birdBirds have wings


World knowledge is expressed as a set of trueassertions about a problem domain.


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Knowledge-Based System Inference rules are used to reason about the world inorder to determine when a predicate expressionlogically follows from a knowledge base.

KnowledgeBase

InferenceEngine

Chirpy is a birdBirds have wingsChirpy has wings



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Using Inference RulesA sentence (predicate expression) P is said to logically

followfroma set of assertions S if and only if everyinterpretation and variable assignment that makes S

true, also makes P true.

S PS entails P

S is true

interpretations

S is true



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Using Inference Rules In our financial advisor problem our question is:

given the knowledge base KB, does it logically followfrom the facts in KB that

X invest(X)



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Using Inference RulesAn inference rule is soundif every predicate calculus

expression produced by the rule from a set S ofpredicate calculus expressions logically follows from S.

An inference rule is complete if, given a set S ofpredicate calculus expressions, the rule can infer every

expression that logically follows from S.



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Using Inference Rules

Examples Modus Ponens

PQ

P

Q

Modus ponens is a sound inference rule.

Example:

lawyer(john)

rich(john)lawyer(john)

We can conclude:

rich(john)



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Examples Modus Tollens

PQ

Q

P

Modus tollens is a sound inference rule.

Example:

lawyer(john)

rich(john)rich(john)

We can conclude:

lawyer(john)



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Examples Elimination

P Q P Q

Q P Elimination is a sound inference rule.

Example:

cat(snowball) white(snowball)

We can conclude:

cat(snowball )

white(snowball)



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ExamplesAnd Introduction

P

Q

P Q

And introduction is a sound inference rule.

Example:

cat(snowball )white(snowball)

We can conclude:

cat(snowball) white(snowball)



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Examples Universal Instantionation

X P(X)

P(a) Universal instantiation is a sound inference rule.

Example:

X mortal(X)

if russell is in the domain of X, we can conclude:

mortal(russell)



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ExamplesX (man(X)mortal(X))

man(socrates)

Prove that Socrates is mortal?

X (man(X)mortal(X))

man(socrates)mortal(socrates) using Universal Instantiation

mortal(socrates) by applying modus ponens



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Unification Unification is an algorithm for determining the

substitutions needed to make two or more predicateexpressions match.

To use the unification algorithm all variables in alogical database must be universally quantified.



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Unification unify:

1. mortal(X) and mortal(socrates)

{socrates/X}

2. knows(john, X) and knows(john, jane){jane/X}

3. knows(john, X) and knows(Y, bill)

{bill/X, john/Y}

4. knows(john, X) and knows(Y, mother(Y))

{john/Y, mother(john)/X}

5. knows(john, X) and knows(X, bill)fail

Substitution list



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Unification substitution list {X/Y} means: X replaces Y in the

original expression



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Unification Rules1. constants cannot be replaced2. a variable cannot be unified with a term containing

it.

e.g. X cannot be replaced by f(X)

Why?

3. Once a variable has been bound, future unifications

must take the value of this binding into account.



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Unification Algorithm Basic idea:

1. sentences are represented as lists

2. equal constants return an empty substitution list

3. individual variables are bound to whatever occurs inthe other sentence.

4. if either expression is empty, unification fails

5. sentences are bound by

1. recursively unifying the first elements in each,2. then applying the substitution list to the rest of each list

3. recursively unifying the remainder of the two lists.

4. creating a single substitution list bycomposition of the listsfrom steps 5.1 and 5.3 above.


Using In erence Ru es


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Using In erence Ru esUnification Algorithm

Luger: Artificial Intelligence, 6th edition. Pearson Education Limited, 2009 70CP468, Dr. Reem K. Al-Halimi


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Unification Algorithm: Composition Composition the process of creating a single

substitution list from two or more lists by applying oneto the other:

{X/Y, W/Z} and {V/X} and {a/V, f(b)/W}

1. composing {V/X} and {a/V, f(b)/W}:

{a/X, a/V, f(b)/W}

2. composing {X/Y, W/Z} and {a/X, a/V, f(b)/W}{a/Y, a/X, a/V, f(b)/Z, f(b)/W}



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Unification Algorithm: Composition Composition is associative

composition(sub1, composition(sub2, sub3)) =composition(composition(sub1, sub2), sub3)

but not commutative

composition(sub1, sub2) composition(sub1, sub2)



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Unification Algorithm: Most General Unifier Unify:

knows(john, X)knows(Y,Z)

Possible substitutions:G = {john/Y, Z/X}S = {john/Y, john/X, john/Z}

G is the substitution list with the least restrictions on thevariable values => G is called the most general unifier The unification algorithm described earlier calculates the

most general unifier for any given set of expressions .



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Unification Algorithm Unify:

knows(john, X)

knows(Y,Z)



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Unification Algorithmunify((knows john X), (knows Y Z))

unify(knows, knows )

return {}

unify(( john X), (Y Z))

unify(john, Y)

return {john/Y}

unify((X), (Z))

unify(X, Z)

return {Z/X}

unify((), ())

return {}

return {Z/X}

return {john/Y, Z/X}

1 2 3

4 5 6

7 89

11

12

10



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Financial Advisor Example continued How can we choose which investment is best for the

investor?

One approach: Goal-directed, depth-first search

Goal: X invest(X)




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Using In erence Ru esFinancial Advisor Example continued

Luger: Artificial Intelligence, 6th edition. Pearson Education Limited, 2009


s ng n erence u es


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s ng n erence u esFinancial Advisor Example continued

Luger: Artificial Intelligence, 6th edition. Pearson Education Limited, 2009From Fig 3.26 And/or graph searched by the financial advisor.

1. savings_account(inadequate) invest(savings)




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Using In erence Ru esFinancial Advisor Example continued

Luger: Artificial Intelligence, 6th edition. Pearson Education Limited, 2009From Fig 3.26 And/or graph searched by the financial advisor.

X amount_saved(X) Y (dependents(Y) greaterThan(X,

minsavings(Y)) savings_account(inadequate)

Unificationresult

{20000/X}

Unificationresult {2/Y}



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Summary Presented First Order Logic (FOL) as a powerful knowledge

representation language. Discussed the component of FOL: alphabet, syntax, semantics,

and proof systems.

Defined models and interpretations.

Looked at the components of a knowledge base and the

procedure involved in creating a knowledge-based system. Discussed inference rules and their properties: valid, satisfiable,

and unsatisfiable.

Looked at the use of sound inference rules in generating newfacts from a given set of assertions.

Presented important sound inference rules: modus ponens,modus tollens, and elimination, and universal instantiation.

Looked at the use of inference rules and search in a simple expertsystem through the financial advisor example.

knowledge representation-part 2

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