Theory of Computation in Multiple Dimensions
Understanding the human brain deeply
Support my writing
Join Medium.com as a member through the link below.
https://thomascherickal.medium.com/membership
In Retrospect — Obvious
Our computer programs are unidimensional.
They execute 1 instruction opcode at a time from a 2-dimensional sheet of code (assembler maybe).
What if the 1-instruction became a multiple-instruction sheet of opcodes?
A sheet of n instructions in length.
Being read from a 3-d computer code-cuboid (my own nomenclature), of size n by m by l.
Where l is the length of the code-cuboid, m is the number of memery available, and n is the number of processors fully connected with shared memory RAM and SSD both individual and commonly shared per processor.
Hold it, Thomas. You’re talking about MIMD distributed architectures in Flynn’s taxonomy. Been there, seen that.
True. But suppose all instructions were executed in one clock cycle so that there is automatic support for multiprocessing?
You can stop now, Thomas. That’s the approach of a symmetric multiprocessor and MPP (Massively Parallel Processing) supercomputers.
And multithreading is tough.
Writing code for shared processors is difficult.
MIMD is full of pitfalls and traps.
Let’s stick to a single computer, even in that our code fails so often. Multiple processors is a debugger’s nightmare. Multiple data stream, even — worse!
But can we stick to completely parallelizable programs? Which can all be executed independently as far as control and data structures go? Say, for example, the MPI (Message Passing Interface model)?
OK, that will work, but that’s a very small class of problems.
And suppose we used machine learning to implement the code, using algorithms that have been mathematically checked (formal method verification) with 100% accuracy?
Difficult. ML models don’t have 100% accuracy, and formal verification is an absolute bugbear.
And suppose we can perform error correction? And use quantum-inspired neural network systems (QiNNs) like the one below?
And even more — suppose we make the software seem sequential, but run in parallel, something which every processor does today? Something that is already happening today?
Instead of talking about it, can you show me what you’re talking about?
Current Processing and processing paradigms. Inherently sequential — Paper.
2-d programs and 3-d execution of code-cuboids, a plane passing through a cuboid something like this— Video.
Similar to a video from a TV every processor (here modeled by a pixel)is operational in one clock cycle at the same time. So a 3-d system is viewable on a 2-d plane cross-section, N x N pixels/processors at a time, a square.
Of course, the best way to explain this is to demonstrate a simulation. We’ll show you how it’s already been done (you should have guessed by now). And find out why this is a performance improvement.
3-Dimensional version of Turing Machines
Our aim is to show that classical Turing machines that are universal can be extended to multiple dimensions in which they show remarkable improvements and differing properties.
According to the halting problem, there exists computation that cannot be done by even a universal Turing machine — they are uncomputable. A ton of work can be done here!
But strings can be checked by a program written in two dimensions, hence functions uncomputable with 1-d language structure Turing machines can be computed with 2-d programs.
Hence the computational ability increases as the number of dimensions increase.
The same argument used for 1-D and 2-D string representations can be trivially extended to 2-D and 3-D as well.
So what does that mean for computation?
- Building computers with multiple cores exponentially increases their computing capacity.
- We have already seen it in action — the use of GPGPUs for AI and ML is a direct application of the improvement of computational capacity with the number of processors (example!).
- Parallel Processing is a well-established field in its own right but more analysis needs to be done on parallel computational complexity. There needs to be a measure of parallel complexity that extends big-O notation. A ton of work can be done here as well!
- And you can work on it — I’ll be exploring the capacities of 1-D, 2-D, 3-D, and even 4 or greater dimensionality strings used in soft computing with genetic algorithms and/or evolutionary computation!
But Where Does Understanding The Brain Come In?
Current programs are 1-d strings from 2-d sheets of code. I believe we can construct a more efficient processing technique with 2-d-instructions stored in a 3-d block of code (code-cube / code-cuboid to be pedantic). Our brain is really a programmed computer. Just a system acting under rules of emergence under complexity theory. This means that — the brain, which is a 3-d code system, must have been programmed in a 4-d environment. Which opens up a lot of treasures — and skeletons — in a lot of closets — philosophical, mathematical, physical, and even theological.
Until next time.
Happy researching!
Support My Writing
And visit Thomas’ Amazon Store to check out the best resources I’ve chosen out of all that Amazon has to offer on learning Python, Microsoft.NET, Design Patterns, Artificial Intelligence, Machine Learning, Quantum Computing, Software Engineering, Blockchain, and Clean Code. Visit my store and see what book suits you. Or electric violins, if you’re inclined in that direction! Have fun and thank me by purchasing something you like.
https://www.sites.google.com/view/thomascherickal/intro
Support my writing. Join Medium to read the best content on the web through the link below and 50% of your membership fee supports me every month. A sincere thanks to all my wonderful subscribers who help me and support my writing every month.
Join Medium with my referral link below — Thomas Cherickal
https://thomascherickal.medium.com/membership
God bless.
Coffee
If you enjoyed this article, do buy me a coffee every month for 2 USD:
