Formal Methods - Lecture03

Lecture No. 3

Equality and Definite Description

Dr. Nazir A. Zafar Formal Methods

Now we extend our language by adding a theory of equality between expressions.

Predicate calculus with equality is more expressive, because of asserting identity of two objects, or distinguishing between them.

The expressions which are equal may be substituted one for the other, without affecting the

Introduction

substituted one for the other, without affecting the truth of a statement.

This forms basis of our theory of equality. Properties such as symmetry and transitivity can

be derived from them.


Addition of equality allows us to formulate a simple rule for reasoning with quantifications:

the one-point rule. We show how this rule may be used to

introduce existential quantifier and eliminate the existential quantifier.

We also show how equality may be used in

Introduction

We also show how equality may be used in statements expressing uniqueness and numerical quantity.

We will conclude lecture by introducing notations for identifying objects by using a description of their properties, rather than referring by name.


The notion of equality is a familiar one: We learn that 1+1 equals 2

In the Christian religion, 25th Dec. equals Christmas Day.

It shows that the two expressions concerned have same value, or denote the same object.

Equality

In a formal description, we identify expressions using the equality symbol: e.g. 1+1 = 2 and 25th Dec. = Christmas Day.

Consequently, e = f, means e is identical to f and we cannot distinguish between them


Example 4.1:

In an identity parade, a witness may state that the man on the right is the man who stole my idea, is equivalent to:

man on the right = man who stole my idea

Equality

Reflection:

Everything is identical to itself: thus, if t is any expression, then t is equal to t .

This principle is known as the law of reflection:

Note: This principle is an axiom of standard Z


Leibnizs law: Another axiom involving equality is Leibnizs law, or the substitution of equals i.e.if s = t , then whatever is true of s is true of t .

Example 4.3:

Equality

Example 4.3: If s = t then p [t / x] = p [s / x] (by substitution)


Example 4.3

If we know that

1. Christmas Day = 25th December, and

2. 25th December falls on a Sunday this year

Then we may apply the Leibniz's law rule and conclude that

Equality

conclude that

Christmas Day falls on a Sunday this year

Note:

If two expressions e and f are not identical, then we write e f .

This is simply an abbreviation for : (e = f)


If the identity of bound variable is revealed within the quantified expression, then we may replace all instances of that variable, and remove the existential quantifier.

x : a p x = t If t is in a, and p holds with t substituted for x, then

The One Point Rule

If t is in a, and p holds with t substituted for x, then t is a good candidate for this value.

This is the basis of the one-point rule

( x : a p x = t) t a p [t / x]


The predicate , n : N 4 + n = 6 n = 2is equivalent to 2 N 4 + 2 = 6 and n does not appear free in expression 2,

Hence the proposition 2 N 4 + 2 = 6 is true

Example: One Point Rule

The predicate , n : N 6 + n = 4 n = -2 is equivalent -2 N 6 + (-2) = 4and n does not appear free in expression -2,

Hence proposition -2 N 6 - 2 = 4 is false.


Example:

The predicate

n : N ( m : N n m ) n = n +1can not be simplified using one point rule.

This is because n is free in the expression n + 1

The One Point Rule

This is because n is free in the expression n + 1


Example 4.8

Symbolize only Romeo likes Juliet using a conjunction

Symbolization:

1. Let L(x, y) means x likes y, and

Uniqueness and Equality (use of Only)

1. Let L(x, y) means x likes y, and

2. let Person be the set of all people then

L(Romeo, Juliet) p : Person L(p, Juliet) p = Romeo

That is, any person likes Juliet must be Romeo


Example 4.9:

Formalize he statement:

there is at most one person to whom Romeo likes

Formalization:

Uniqueness and Equality (use of At most)

1. Let L(x, y) mean that x likes y, and

2. let Person be the set of all people then

p, q : Person L(Romeo, p) L(Romeo, q) p = q


Example 4.10:

Formalize the statement:

no more than two visitors are permitted

Formalization:

let Visitors be the set of all visitors then

Use of at most and at least

let Visitors be the set of all visitors then

p, q, r : Visitors p = q q = r r = pExample 4.11:

The statement at least two person has appliedcould be formalized as

p, q : Applicants p q


Example 4.12:Formalize the statement:there is exactly one book on my desk.

Formalization:1. let Book be the set of all books

Uniqueness and Equality (use of Exactness)

1. let Book be the set of all books2. Desk(x) means that x is on my desk, then

b : Book Desk(b) ( c : Book | Desk(c) c = b) OR

1 b : Book Desk(c)


We often use a descriptive phrase to denote anobject, rather than a name.

For example, when a crime is committed, and the police have not yet learned who committed, the question is referred to as

the driver of the white car or

Definite Description

the driver of the white car or

the cat in the hat etc.

Here, the word the is important which indicates existence and uniqueness.


Example 4.14 :

Following phrases indicate that there is a unique object with a certain property

The man who shot John Lennon

The woman who discovered radium


The oldest college in Oxford

Notation in Z: ( x : a | p)denotes unique object x from a such that p.


Example 4.15 :

The phrases in Example 4.14 can be formalized as above:

1. ( x : Person | Shot(x, John Lennon))


2. ( y : Person | Discovered(x, radium))3. ( z : Colleges | Oldest-College(x , Oxford))


Formal Methods - Lecture03

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