Lecture 2

Predicate Logic

Dr. Nazir A. Zafar Formal Methods

Now we introduce another part of our logical language.

The language of propositions allows us to make statements about specific objects

But that does not allow us to describe such as

Chapter 3: Predicate Logic

Every cloud has a silver lining

These are known as universal statements because it describe properties that must be satisfied by every object in some universe of discourse.

Dr. Nazir A. Zafar Formal Methods

Example 3.1:

The following are examples of universal statements:

1. Any student interested doing research with Dr. Zafar must register FM course.

Predicate Logic

Dr. Zafar must register FM course.

2. Nobody knows the trouble I seen.

3. Jim doesn't know anybody who can sign his bail application.

Dr. Nazir A. Zafar Formal Methods

Sometimes we wish to state that at least one thing has a particular property, without necessarily knowing which thing it is.

This leads to an existential statement.

Example 3.2

Predicate Logic

Example 3.2 The following are examples of existential

statements:1. I heard it from one of your friends.2. A mad dog has bitten Andy.3. Some people prefer logic.

Dr. Nazir A. Zafar Formal Methods

To formalize such statements, we require a language that reveals the internal structure of propositional statements

A language that allows us to take and apply the objects without proper taking names of it.

The language we require is the language of

Predicate Logic

The language we require is the language of predicate calculus

Also called predicate logic or first order logic

Second order logic is extension of first order logic

Higher order logic is extension of the second order logic

Dr. Nazir A. Zafar Formal Methods

Predicate is a statement with a place for an object

When these places are filled, our predicates become statements about the objects that fill them

Example:

The statement _ > 5 is a predicate, no body

Predicate Logic

The statement _ > 5 is a predicate, no body knows it is true or false

But 0 > 5, a proposition which is false

However, we could also choose to put an object variable in the empty place above. And still it will remain predicate e.g. x > 5,

Dr. Nazir A. Zafar Formal Methods

Example: Construct a predicate and a proposition from _ > 5 .Predicate:

The statement _ > 5 , _ is filled by x and the resultant, x > 5 , is called predicate

Note:

Predicate and Propositions

Note:The statement was filled but still not a proposition. This is because we cannot say whether it is true or false without knowing the value of x.

Proposition: One of the ways to make a proposition out of x > 5 is as x x > 5 (this is proposition)

Dr. Nazir A. Zafar Formal Methods

Meanings of: x x > 5 (True)Explanation: Let = {0, 1, 2, 3, . . .}The above statement can be written as

0 > 5 1 > 5 2 > 5 3 > 5 4 > 5 .

Predicate Logic

Meanings of: x x > 5 (False)Explanation:

This statement can be written as

0 > 5 1 > 5 2 > 5 3 > 5 4 > 5 .

Dr. Nazir A. Zafar Formal Methods

Example 3.1 (1) :

Formalize the statement

I heard it from one of your friends.

Formalization:

Formalization using Existential Quantifier

Notations:

Friends stands for set of all your friends,

told (x, y) means x has told y.

Formal Statement:

f Friends told (f, me)

Dr. Nazir A. Zafar Formal Methods

Example 3.1 (2) :

Formalize the statement A mad dog has bitten Andy

Formalization:

Formalization using Existential Quantifier

Formalization:

Notations:

MadDog stands for set of all mad dogs,

bit (x, y) means x has bitten y.

Formal Statement:

d MadDog bit (x, Andy)Dr. Nazir A. Zafar Formal Methods

Example 3.1 (3) :

Formalize the statement

Some people prefer logic.

Formalization:

Formalization using Existential Quantifier

Formalization:

Notations:

Person stands for the set of all people, and PL(x) means x prefers logic.

Formal Statement:

p Person PL(p)Dr. Nazir A. Zafar Formal Methods

Example 3.1 (1):

Formalize following statements

1. Any student interested doing research with Dr. Zafar must register FM course.

Formalization:

Formalization using For all Quantifier

Notations:

Student: set of all students doing research with Dr. Zafar

Register (x) means x must register the FM Course

Formal Statement:

s Student Register (s)

Dr. Nazir A. Zafar Formal Methods

Example 3.1 (1): Modified

Formalize following statements

1. Any student interested doing research with Dr. Zafar must register FM course.

Formalization:

Formalization using For all Quantifier

Notations:

Student: set of all MS students

fm-group: set of students doing research with Dr. Zafar,

Register (x) means that x must register the FM Course

Formal Statement:

s Student | s fm-group Register (s)

Dr. Nazir A. Zafar Formal Methods

Example 3.1 (2):

Formalize following statements

1. Nobody knows the trouble I have seen.

Formalization:

Notations:

Person stands for set of all people,

Formalization using For all Quantifier

Person stands for set of all people,

K(x) means that x knows the trouble I have seen

Formal Statement:

p Person K(p)What is negation of above statement?

Dr. Nazir A. Zafar Formal Methods

Example 3.1 (3):

Jim doesnt know anybody who can sign his bail application.

Notations:

P stands for set of all people,

Formalization using For all Quantifier

P stands for set of all people,

K(x, y) means x knows y

S(x, y) means x can sign ys application for bail

Formalization:

p P K(Jim, p) S(p, Jim)

Dr. Nazir A. Zafar Formal Methods

Jim doesnt know anybody who can sign his bail appl.

Notations:

Set of all persons = P = {p1, p2, p3, p4}

Set of persons known to Jim = P1 = {p1, p2}

Set of persons not known to Jim = P2 = {p3, p4}

K(x, y) mean that x knows y.

Example 3.1 (3): Validity Checking

K(x, y) mean that x knows y.

S(x, y) mean that x can sign ys bail appl.

K(Jim, p1) = true, K(Jim, p2) = true, K(Jim, p3) = false, K(Jim, p4) = false,

S(p1, Jim) = false, K(p2, Jim) = false, S(p3, Jim) = false/true, K(p4, Jim) = false/true

Dr. Nazir A. Zafar Formal Methods

Predicate Logic

p P K(Jim, p) S(p, Jim)

Propositional Logic

[K(Jim, p1) S(p1, Jim)] [K(Jim, p2) S(p2, Jim)] [K(Jim, p3) S(p3, Jim)] [K(Jim, p4)

Validity Checking

Jim)] [K(Jim, p3) S(p3, Jim)] [K(Jim, p4) S(p4, Jim)]

[t t] [t t] [f t/f] [t t/f] t t t t t, a tautology

Dr. Nazir A. Zafar Formal Methods

Sign-Appl (P, P1, Jim)

if P = [ ] then

true

else

for i = 1 to Length (P)

if K(Jim, P[i]) then

Algorithm

if K(Jim, P[i]) then

S(P[i], Jim)endif

endfor

Dr. Nazir A. Zafar Formal Methods

P1 = known to Jim;

Jim: P;

known = {(Jim, p1), (Jim, p2)}

K : P x P BoolK (Jim, x) ==

Algorithm

K (Jim, x) ==

if (Jim, x) known thentrue

else false

Dr. Nazir A. Zafar Formal Methods

P1 = known to Jim;

Jim: P;

known = {(Jim, p1), (Jim, p2)}

S : P x P BoolS (x, Jim) ==

Algorithm

S (x, Jim) ==

if x P1 thenfalse

endif

pre x Jim

Dr. Nazir A. Zafar Formal Methods

In the Z notation, two kinds of quantified expressions have a similar syntax

Q x : a | p qwhere

Quantifiers and Declaration

Q is the quantifier;

x is the bound variable;

a is the range of x;

p is the constraint; and

q is the predicate.

Dr. Nazir A. Zafar Formal Methods

1. ( x : a | p q) ( x : a p q) 2. ( x : a | p q) ( x : a p q)

Pronunciation:

x : a | p q is pronounced as there exists an x

Useful Equivalences

x : a | p q is pronounced as there exists an xin a satisfying p, such that q.

x : a | p q is pronounced for all x in asatisfying p, q holds

Dr. Nazir A. Zafar Formal Methods

Example: Define scope of the variable x and y

( x : a | p q r) s t y : a | p q ( y : b | r s t) u v

Scopes )|:( rqpax

Scope of Variable

)|:(first xofscope rqpax

yfirst ofscope

ysecondofscope

)|:(|: vutsrbyqpay

Dr. Nazir A. Zafar Formal Methods

Example 3.7

Consider the following quantified expression, which states that every natural number x is greater than or equal to zero

Independent Variable has no Importance

Formal Description:

num : N num 0 The choice of num as variable is not important;

the following expression has the same meaning

nat : N nat 0

Dr. Nazir A. Zafar Formal Methods

Example 3.8:

Consider the following quantified expression, which states that there is some natural number max such that every natural number num must be less than or equal to max

max : N ( num : N num max)

Dependant Variable has Importance

max : N ( num : N num max) This statement is false.

By changing the name of the universal quantifier variable from num to max, results confusion

max : N ( max : N max max) This statement is true; the meaning has changed.

Dr. Nazir A. Zafar Formal Methods

Example 3.10:

In the expression below, the first bound variable is used as the range of the second:

a : b c : a p

Bound Variable as a Range Variable

a : b c : a p

In this case, it would make no sense to merge the two quantifications.

Dr. Nazir A. Zafar Formal Methods

Example 3.11: In the expression below, there is a single free occurrence of variable z

x : N z xExample 3.12: There are both free and bound

Free Occurrences of a Variable

occurrences of variable x in expression below

x = 3 x : N 0 x

Note: Occurrence of x adjacent to quantifier is neither free nor bound;

Free Occurrence Bound Occurrence

Dr. Nazir A. Zafar Formal Methods

If we wish to change the subject of the statement from x to y, we must replace each free occurrence of x in p with an occurrence of y. This process is called substitution.

We write p [y / x] to denote the predicate that results from substituting y for each free

Substitution

results from substituting y for each free occurrence of x in predicate p

The expression y need not be another type variable. it can be any expression whose possible values match those with x.

Dr. Nazir A. Zafar Formal Methods

Example 3.13

1. (x y + 2) [0 / x] 0 y + 2

2. ( x : N x y + 2) [0 / x]

Substitution

2. ( x : N x y + 2) [0 / x] ( x : N x y + 2)

3. ( x : N x y + 2) [5 / y] ( x : N x 5 + 2)

Dr. Nazir A. Zafar Formal Methods

Example 3.14

1. (x y + 2) [0 / x] [5 / y] 0 y + 2 [5 / y] 0 5 + 2

Substitution

0 5 + 2

2. (x y + 2) [y / x] [5 / y] (y y + 2) [5 / y] 5 5 + 2)

Dr. Nazir A. Zafar Formal Methods

Example 3.15

1. (x y + 2) [y, 5 / x, y] Parallel y 5 + 2

Substitution

2. (x y + 2) [y / x] [5 / y] Sequential

(y y + 2) 5 5 + 2)

Dr. Nazir A. Zafar Formal Methods