October 01, 2017 Sunday
Bedtime Story
Creating Simple Functions True and False in Lambda Calculus
In programming languages, logical values
are used to make a choice in between doing two things.
If True, then do so and if false then do
something else.
This idea can be used to encode the logical
values True and False in lambda calculus.
Let us see how.
We make certain expressions in the syntax
of lambda calculus for true and false.
True = λx.λy.x
So what does this expression mean?
It means that the black box takes in two
inputs x and y and chooses x as the output.
Now with False it will be the opposite.
It will take in two inputs x and y and
choose y as output.
False = λx.λy.y
So now we have generated two lambda
functions, the first one, the True takes in two inputs x and y and gives output
as x.
The second False function takes in two
inputs x and y and gives output as y.
So what can be done with these two
functions?
Can we work out some way to use these two
functions to make a logical operator or what is popularly called a logic gate
in electronics?
Perhaps the simplest logical operator is
the operator NOT.
Electronically, it is called an inverter as
it simply converts an input A to not A or a 0 to 1 or a 1 to 0.
No operation can be simpler than that.
So can we create this operator from the two
lambda functions that we have created above?
In our case, if we can create an operator
that can convert True to False or False to True, then we have achieved creating
this operator of logical negation.
Let us take a logical value b or a Boolean,
a term that is commonly used in formal logic.
Not let us apply it (b) to both the
functions False and True.
Now we shall proclaim it to be a NOT
function.
So if we claim it to be a NOT function,
then we can write it in the syntax of lambda calculus as follows:
NOT = λb.b
Then this proclaimed NOT function is
applied to first False and then to True.
This will be written down in lambda
calculus as follows:
NOT = λb.b False True
Please stay with me on this idea and we
shall continue it in the nights to come.
Stay tuned to the voice of an average story storytelling
chimpanzee or login at http://panarrans.blogspot.com
Good night mon ami and my fellow cousin ape.
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Another great educator and a teacher that I am aware of is
Professor Subhashish Chattopadhyay in Bangalore, India.
While I narrate stories, Professor Subhashish an electronic
engineer and a former professor at BARC, does and teaches real mathematics and
physics.
He started the participation of Indian students at the
International Physics Olympiad.
Do visit him here:
All his books can be downloaded for free through this link:
For edutainment and English education of your children, I
recommend this large collection of Halloween Songs for Kids:
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