CMPF144 FUNDAMENTALS OF COMPUTING THEORY Module 7: Beyond Classical Logic.

CMPF144FUNDAMENTALS OF COMPUTING THEORY

Module 7: Beyond Classical Logic

Module 7 CMPF144 Fundamentals of Computing Theory

Slide 2

Objectives

To present and discuss some example of classical paradoxes. To discuss the importance of recursion and its benefits and weaknesses


Slide 3

Paradox

Logical Paradox:

A paradox is generally a puzzling conclusion we seem to be driven towards by our reasoning, but which is highly counterintuitive, nevertheless.


Slide 4

Paradox

A paradox is an apparently true statement that seems to lead to a logical self-contradiction, or to a situation that contradicts common intuition.

The identification of a paradox based on seemingly simple and reasonable concepts has often led to significant advances in science, philosophy and mathematics.


Slide 5

The Barber Shop

In a certain village there is a man who is a barber. The barber shave all and only those men in the village who do not shave themselvesQuestion:

Does the barber shave himself?


Slide 6

Think!!!

Any man in this village is shaved by the barber if and only if he is not shaved by himself. Therefore in particular the barber shaves himself if and only if he does not. We are in trouble if we say the barber shaves himself and we are in trouble if we say he does not.


Slide 7

Others…

There are many other paradoxes that can be discussed.

Epimenides paradox:

A Cretan says "All Cretans are liars".

Basically, the reason for this paradoxes is what we called a self-referral or circular-referral.


Slide 8

Recursion

Recursion is a theme that is found and explored:

in language and literature

in music

in mathematics

in art

and in computing


Slide 9

Recursion

Recursion means self-referential. Something may be defined in terms of itself. Care must be taken not to create a circular reference which might lead to an endless regression, a paradox.

Recursion is a simple but very powerful concept.

Recursion means an instance (or instances) is derived from an exactly identical instance.

Recursion can go on and on infinitely.


Slide 10

Recursion (Paradox)

Paradox can be an example of recursionExample:

The Epimenides paradoxEpimenides was a Cretan. He made the statement:"All Cretans are liars.“

If we believe this to be true, then Epimenides was lying. If he was lying, then Cretans tell the truth. But if Cretans tell the truth then…We are stuck in a strange loop, a paradox.


Slide 11

Recursion (Paradox)


Slide 12

Recursion (Paradox)

Dutch artist, M.C. Escher, constructed optically challenging drawings.

Waterfall (see Figure 1), a lithograph, shows water flowing over the edge of an aqueduct turning a waterwheel in the same aqueduct.


Slide 13

Recursion (Paradox)

Did the right hand draw the left hand first? Or did the left hand draw the right hand that draws the left hand?


Slide 14

Recursion (Paradox)

Mathematician Kurt Godel, challenged our understanding of number systems by stating his incompleteness theorem:

"All consistent axiomatic formulations of number theory include undecidable propositions."


Slide 15

Recursion (Paradox)

These are the example of axioms:a) 1 + 1 = 2b) Distance between 1 and 2 to be

exactly the one between 2 and 3


Slide 16

Recursion

From the definition of recursion, it can often lead to paradox. Such a definition is circular, or endlessly regressive.

In the perspective of the computer programmer, an endless loop.


Slide 17

Recursion

A more precise definition says that recursion never defines something in terms of itself, but always in terms of simpler versions of itself. We try to break a problem into two parts:

1. The simplest case, sometimes called the base case2. And a simpler version of the problem that might approach the base case.


Slide 18

Factorial

"n factorial" n! is the product: n * (n – 1) * (n – 2) … * 1, with 1! factorial defined to be 1, and 0! defined to be 1.

i.e.: 0! = 1 1! = 1

for example: 5! = 5 * 4 * 3 * 2 * 1Or, since 4! = 4 * 3 * 2 * 1 we can say that 5! is equal to 5 * 4!

The simplest version of the problem occurs when n is zero or one.


Slide 19

Factorial

The slightly simpler version of the problem is generalized this way.

n! = n * (n – 1)!

(n – 1) * (n – 2)!

(n – 2) * (n – 3)!

n * (n – 1) * (n – 2) * (n – 3) * ….


Slide 20

Recursive Function

Let define a function called factorial(n)The definition of recursive function factorial(n) is:

factorial(n){

if(n == 0) then

return 1;else

n * factorial (n-1);}


Slide 21

Tracing

Example:factorial (5)

5*factorial(4)5*4*factorial(3)5*4*3*factorial(2)5*4*3*2*factorial(1)5*4*3*2*1*factorial(0)5*4*3*2*1*1 = 120


Slide 22

Recursive

In C or C++, when we say that a function is recursive, we mean that the function calls itself. In order to prevent endlessly regressive function calls, we carefully construct a recursive function such that it terminates when the base case is reached. If the base case is not reached, the function calls itself and sends a slightly simpler version of the problem as the argument.


Slide 23

Quiz 2

Based on the example, write a recursive function called power(m,n) where m is the base number and n is the power number. I.e power(4, 3) mean 43

or power (5,5) means 55


Slide 24

The END

CMPF144 FUNDAMENTALS OF COMPUTING THEORY Module 7: Beyond Classical Logic.

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