EFFORT
\(CB(n)\) or \(\delta(n)\) So, the chess board is a hypothetical Turing machine thing.
Definition[]
The maximum sum of n-1’s that can be written (in the finished plane), with n state and a n colour halting Turing machine. Note that we do not use a tape but a Plane. As well as that, we allow for movement in 8 directions. Up, down, left, right, up right, up left, down right, down left.
Simpler Explanation[]
Assume we have a robot, I.e. BusyBever 9000. It enters a infinite hotel, infinity*infinity in size. This hotel has infinitely many rows, and infinite rooms in each row. The robot has n states. Each state gives a different rule of instructions. So, it enters a room, and follows the instructions for whatever it should have. These rooms have different coloured lights. So, say it has an instruction telling it to turn on a red light when it enters. Now, it moves forward a row and continues, so on and so forth, until all the rooms are the n-1th colour. So, assigning each colour to a number, in a ascending number what is the total sum of all largest numbers? Say there are 3, and it is a 3 colour machine. Then, the total sum is 2+2+2, or 6
Better Explanation[]
So, instead of coloured lights and robots, we have a read and write head moving on the infinite paper. Instead of turning it into a 0 or 1, it turns it into 0,1,2, etc, with n-1 as the maximum number. It reads and writes, until it eventually halts. The final question, is what is the sum of all n-1s?
Augmented CB(n)[]
Also called \(\Delta(n)\) It should be more powerful.
Instead of asking what is the sum of all n-1s, we ask what is the sum of all numbers that are not 0.
Better Definition[]
The maximum sum of all nonzero numbers that can be written (in the finished plane), with a n state and a n colour halting Turing machine. Note that we do not use a tape but a Plane. As well as that, we allow for movement in 8 directions. Up, down, left, right, up right, up left, down right, down left.