Lecture 5.pdf

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Engineering Engineering Engineering Engineering Systems Systems Systems Systems Design and Analysis Design and Analysis Design and Analysis Design and Analysis (ENG (ENG (ENG (ENG 504504504504))))

أحمد محمد حلميأحمد محمد حلمي/ / ددقسم هندسة الحاسبات و المنظومات قسم هندسة الحاسبات و المنظومات

جامعة الزقازيقجامعة الزقازيق––كلية الهندسة كلية الهندسة

Email: [email protected] / [email protected]

Webpage: http://www.staff.zu.edu.eg/amhm/

Lecture Lecture Lecture Lecture 5555

Dr. A. Helmi Eng. Sys. Design and Analysis (ENG 504) Zagazig Univ-Eng. Faculty 2013/2014Eng. Sys. Design and Analysis (ENG 504)

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�Branch and bound algorithm

� Examples:

� Travelling Salesman Problem

� Scheduling Problems

� Expert systems

� Propositional logic

In the previous lectureIn the previous lectureIn the previous lectureIn the previous lecture

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2AgendaAgendaAgendaAgenda

� Expert systems

� Predicate logic

Predicate LogicPredicate LogicPredicate LogicPredicate LogicPredicate LogicPredicate LogicPredicate LogicPredicate Logic

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3Limitations of Propositional Logic Limitations of Propositional Logic Limitations of Propositional Logic Limitations of Propositional Logic

� Statements about similar objects and relations need to be enumerated:

� Doesn’t deal with relations.

� Can’t combine evidences.

� Statements referring to groups of objects require exhaustive enumeration

of objects:

� Can’t express quantification.

� Can’t deal with uncertain information.


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� It is called also First-Order Logic (FOL).

� In English, the predicate is the part of the sentence that tells you something

about the subject.

� A predicate is a verb phrase template that describes a property of objects,

or a relationship among objects represented by the variables.

� Example 1: "The car Tom is driving is blue",

“The sky is blue“,

"The cover of this book is blue"

� Predicate: “x is blue“ which describes the property of being blue.

� We can give this predicate a name, i.e., B(x)

� Thus, “x” is an arbitrary object and B(x) reads as “x is blue”.

Predicate Logic Predicate Logic Predicate Logic Predicate Logic

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5Predicate Logic Predicate Logic Predicate Logic Predicate Logic

�Example 2: "John gives the book to Mary",

"Jim gives a toy to Tom",

"Jane gives a lecture to Mary",

� Predicate: “x gives y to z“ which describes a relationship bet. 3 objects.

� Name: G(x,y,z)

� x,y,z are variables, while “John”, “Mary”, ”Jane”, ”Jim” and “Tom” are

objects.

Notice that a predicate becomes a proposition when specific

values (objects) are assigned to its variables.



� As we did when studying propositional logic, we have to look on:

� Syntax.

� Semantics.

� Inference rules.

� Syntax:

� Symbols:

� Constants: objects.

� Functions: to check a property of its variables.

� Predicate: a true/false relationships bet. objects.

� Connectives : AND, OR, NOT,.. (as in propositional logic).

� Quantifiers: i.e., “∃∃∃∃” which means “there exists” and “∀∀∀∀” which means

“for all”.

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� Each predicate variable has a Domain or Universe which is the set of values that

can be assigned to the variable.

� Thus, If P(x) is a predicate and x has domain U, the truth set of P(x) is the set

of all elements in U such that P(t) is true.

� Example:

� U={1,2,3,4,5,6,7,8,9,10}

� P(x): “x is prime”

� The truth set is {1,2,3,5,7}



� Quantifiers:

� “∀∀∀∀x” which means “for all x in its domain” is called the universal quantifier.

� Example: “All Human are Mortals”, here is its FOL translation:

� predicate: “All x are y”

� H(x): x is a human

� M(y): y is mortal

� ∀∀∀∀x, H(x) ⇒⇒⇒⇒ M(x)M(x)M(x)M(x)

� “∃∃∃∃x” which means “at least, there exists one value of x from its domain” is

called the existential quantifier.

� Example: “Some numbers are prime”

� P(x): “x is prime”

�∃∃∃∃x, P(x)

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� Example 1: Translate the following quantified statements into English:

1. U(x) = ℝ,

P(x): x2 > 0

∀x, P(x)

Answer: For every rational number, the quadratic function gives positive value.

2. U(x) = ℝ,

M(x): x > 3

B(x): x3 > 27

∃x, M(x) ∧ B(x)

Answer: There exists one rational number such that it is greater than 3 and its

cubic is greater than 27.



� Example 2: Write the FOL representation:

1. No School bus is black

Answer: Predicate: “No x is y”

S(x): “x is a school bus”

B(y): “y is black”

ﾡ ∀x, S(x) ⇒⇒⇒⇒ B(x) or ∀x, S(x) ⇒⇒⇒⇒ ﾡ B(x)

2. Some uncles are younger than their nephews

Answer: Predicate:”x is younger than y” S(x,y): “x is younger than y”

P(x): “x is an uncle”

N(y): “y is a nephew”

∃x,y, S(P(x),N(y))

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� Notice that: If U={1,2,3}, then

∀∀∀∀x, P(x) is equivalent to P(1) ∧ P(2) ∧ P(3)

∃∃∃∃x,P(x) is equivalent to P(1) ∨ P(2) ∨ P(3)



� DeMorgan’s low for quantifiers:

ﾡ∀∀∀∀x, P(x) ⇔ ⇔ ⇔ ⇔ ∃∃∃∃x ,ﾡP(x)

ﾡ∃∃∃∃x,P(x) ⇔⇔⇔⇔ ∀∀∀∀x, ﾡP(x)

� Exercise 1: Write the FOL representation for:

� Some drivers do not obey the speed limit

� Not all dogs are dangerous

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� Quantifiers plus ∧ and ∨:

� ∀ distributes over ∧ :

∀∀∀∀x, (P(x) ∧ Q(x)) ⇔⇔⇔⇔∀∀∀∀x,P(x) ∧ ∀∀∀∀x,Q(x)

� ∃ distributes over ∨:

∃x,(P(x) ∨ Q(x)) ⇔⇔⇔⇔∃x,P(x) ∨ ∃x,Q(x)

� But notice that: ∃x,(P(x) ∧ Q(x)) ⇒⇒⇒⇒ ∃x,P(x) ∧ ∃x,Q(x) is True.



� Rules of inference:

1. Universal Modus ponens:

where “c” is some element in P’s domain.

∀∀∀∀x, (P(x) ⇒⇒⇒⇒ Q(x))

P(c)

∴∴∴∴ Q(c)

Given

Conclusion

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2. Universal Modus Tollens :

∀∀∀∀x, (P(x) ⇒⇒⇒⇒ Q(x))

ﾡQ(c)

∴∴∴∴ ﾡP(c)

Example:

All dogs bark

Roy does not bark

∴∴∴∴ Roy is not a dog

U = all living creatures,

P(x): x is a dog,

Q(x): x barks.




3. Universal Hypothetical Syllogism:

like the chain rule!

∀∀∀∀x, (P(x) ⇒⇒⇒⇒ Q(x))

∀∀∀∀x, (Q(x) ⇒⇒⇒⇒ R(x))

∴∴∴∴ ∀∀∀∀x, (P(x) ⇒⇒⇒⇒ R(x))

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� Problem formulation in general:

� Given: a story that describes properties and relations between objects.

� Questions:

� A query which requires an answer

(i.e., is there an object that has such and such?)

� Check if a proposition is true or false.

� Proof a proposition.



Example Example 33::

A horse that is registered for today’s race is not a thoroughbred.

Every horse registered for today’s race has won a race this year.

Therefore a horse that has won a race this year is not a thoroughbred. Proof?

Solution:

� Predicates: P(x): x is registered for today’s race

Q(x): x is a thoroughbred

R(x): x won a race this year

� Domain: all horses.

� FOL representation: ∃x, (P(x) ∧ ﾡQ(x))

∀∀∀∀x, (P(x) ⇒ ⇒ ⇒ ⇒R(x))

∃x, (R(x) ∧ ﾡQ(x))

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Example Example 33: : Solution:

� To proof that ∃x, (R(x) ∧ ﾡQ(x)) we use the inference rules:

ReasonStep

Given∃x, (P(x) ∧ ﾡQ(x))1

From 1P(a) ∧ ﾡQ(a), for some a 2

From 1P(a) 3

From 1ﾡQ(a)4

Given∀∀∀∀x, (P(x) ⇒ ⇒ ⇒ ⇒R(x))5

From 5R(a) ⇒ ⇒ ⇒ ⇒P(a)6

From 3,6R(a)7

From 4,7R(a) ∧ ﾡQ(a)8

From 8∃x, (R(x) ∧ ﾡQ(x)) 9



Example Example 44::

Tony, Shi-Kuo and Ellen belong to the Hoofers Club. Every member of the

Hoofers Club is either a skier or a mountain climber or both. No mountain

climber likes rain, and all skiers like snow. Ellen dislikes whatever Tony likes and

likes whatever Tony dislikes. Tony likes rain and snow.

Query: Is there a member of the Hoofers Club who is a mountain climber but

not a skier?

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21Example Example 44: : Solution:

� Predicates: S(x) : x is a skier , Domain: Hoffers club

M(x) : x is a mountain climber, Domain: Hoffers club

L(x,y) : x likes y, Domain of y: snow and rain.

� FOL representation: ∀∀∀∀x,S(x) v M(x)

M(x) ⇒⇒⇒⇒ ﾡL(x, Rain)

∀∀∀∀x,S(x) ⇒⇒⇒⇒ L(x, Snow)

ﾡL(Tony, x) ⇒⇒⇒⇒ L(Ellen, x)

L(Tony, x) ⇒⇒⇒⇒ ﾡL(Ellen, x)

L(Tony, Rain)

L(Tony, Snow)

M(x) ∧ ﾡS(x) ? If true, who is x?



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Step

∀x,S(x) v M(x) 1

M(x) v S(x)2

M(x) ⇒⇒⇒⇒ﾡL(x, Rain) 3

∀∀∀∀x,S(x) ⇒⇒⇒⇒ L(x, Snow) 4

ﾡL(Tony, x) ⇒⇒⇒⇒ L(Ellen, x) 5

L(Tony, Snow) 6

ﾡL(Ellen, Snow) 7

ﾡS(Ellen) 8

M(Ellen)9

M(Ellen) ∧ ﾡS(Ellen) 10


Example Example 44: : Solution:

Reason

given

From 1

given

given

given

given

From 5,6

From 7,4

From 9,2

From 10

Answer is Ellen

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23Applications Applications Applications Applications

� There are programming languages that are built upon the concepts of predicate

logic, i.e., PROLOG (programming in logic).

� There are numerous examples of Expert Systems that are based on rule-based

reasoning:

� DENDRAL : Infers molecular structure from the information provided by

a mass spectrometer

� MYCIN: Diagnosis of blood infections, 450 rules; performs as well as

experts

� PROSPECTOR: Correctly recommended exploratory drilling at geological

site


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� Given the following paragraph:

“Jack owns a dog. Every dog owner is an animal lover.

No animal lover kills an animal. Either Jack or Curiosity killed the

cat, who is named Tuna. Did Curiosity kill Tuna? ”

Answer the query using predicate logic rules, your answer should

include:

1. Predicate statements.

2. FOL representation.

3. Inference steps.

4. Answer to the query.

Assignment #Assignment #Assignment #Assignment #2 2 2 2

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