Virtual dimensions reduced to actual dimensions — Dr. Işık Barış Fidaner

Dr. Işık Barış Fidaner

So how is a potentiality linked to an event of tossing? Through a reduction of dimensions.

A potentiality reduces the dimensions of a virtual ground is to the dimensions of an actual existence.

In tossing a regular two-sided coin, two virtual dimensions are reduced to a single actual dimension.

In tossing an n-sided coin, n virtual dimensions are reduced to a single actual dimension.

An n-sided coin can be simulated by an ensemble of n one-sided coins each with p = 1/n, which makes their total information log(n).

Consider a more general case: an ensemble of n one-sided coins each with p = a/n, which makes their total information a*log(n/a) = log((n/a)^a).

In tossing/actualizing such an ensemble, n virtual dimensions are reduced into a actual dimensions.

The fraction (n/a) indicates that the virtual dimensions are equally distributed among the actual dimensions.

The term (n/a)^a indicates that the distributed element occurs as many as the actual dimensions.

For example:

4 virtual dimensions reduced to 1 actual dimension give two bits (it’s A/B/C/D, it’s 2 bits simply because it’s one of the four). 4 virtual dimensions reduced to 2 actual dimensions also give two bits (the first one is A/B and the second is C/D, which also makes up 2 bits, one bit for each).

9 virtual dimensions reduced to 1 actual dimension give two trits (it’s A/B/C/D/E/F/G/H/I, it’s 2 trits simply because it’s one of the nine). 6 virtual dimensions reduced to 2 actual dimensions also give two trits (the first one is A/B/C and the second is D/E/F, which also makes up 2 trits, one trit for each).

Information per one-sided coin, a*log(n/a)/n gives the information per virtual dimension. This is the entropy formula from [1] when the number of actual dimensions is equal to the block size: a = |B|.

[1] Fidaner, I. B. & Cemgil, A. T. (2013) Summary Statistics for Partitionings and Feature Allocations. In Advances in Neural Information Processing Systems, 26.

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