I would like to share a bit of M381 Mathematical Logic Unit-2 with you.

Every computer program can be emulated on a URM machine which has only four basic instructions: Z(n), S(n), C(m,n) and J(m,n,q). We can give these instructions a unique code as follows:

$Z(n) = 6n-3$

$S(n) = 6n$

$C(m,n) = 2^m3^n+1$

$J(m,n,q) = 2^m3^n5^q+2$

By using this scheme each URM instruction is assigned a unique code. By using these codes and the set of prime numbers it is possible to assign a unique number to every (different) URM program. I.e. to establish a bijection between the set of possible URM programs and $\mathbf{N}$

Conclusion: we can reduce any program to a single integer, and each integer represents a program! ( This conclusion is essential in understanding Goedel's Theorems )

( Do I have readers who still haven't seen Pi, the movie, directed by Darren "Black Swan" Aronofsky? In Pi Maximillian Cohen said, "numbers, everything can be represented by numbers". If every song, every movie, every picture can be mapped to a single integer so can a sequence of pictures. One of the premises in Scientology is that we store at every time unit a dump of sensory images to our memory banks ( vision, smell, sound, touch, taste and a large sequence of emotions ) which form collectively a detailed recording of our life. All our actions, visions and feelings can be represented by a unique integer as well. You know where I am getting at. )

We are numbers.

1-2017 More on the randomness of randomness.

4 months ago

I'm afraid I still haven't seen pi I'll put it on my DVD rental list.

ReplyDeleteNo need, check your yahoo mail.

ReplyDeletewow, I'm looking forward to this course...! This is an entirely believable result. If you consider any computer program is just a sequence of bytes, then you can combine these linearly to give one huge binary number, then this shows that every program has unique binary (and hence integer) representation!!

ReplyDeleteYes Colin, I do recommend this course!

ReplyDelete