10-09-2020, 02:11 PM
Inspired by You are not allowed to view links. Register or Login to view. by Koen G I was playing with entropy having my key and two questions in mind:
1) the simpliest way to increase h2/h1.
The starting values where h1=3.86 and h2=2.13 (calculated using Takahashi transcription).
It appears that the following adjustments do some improvement:
cGh -> Gr (G is any gallow)
ch -> er
sh -> prl
t -> ll
a, i, n removed
Resulting h1=3.12, h2=2.26 (so h2/h1=0.72)
2) how merging letters changes entropy?
The "input" of letter X in entropy formula is p * log2(p). In the ideal case if X represents two letters with equal probability (so p/2 for each) then p * log2(p) turns into 2 * (p/2) * log2(p/2) = p * (log2(p) - 1), so the entropy increases. In the idealest case when each letter represents two letters with equal probabilities entropy increases by exactly 1. In the most idealest case when each letter represents two letters with equal probabilities independently of context both h1 and h2 change by the same value (1).
All this is quite logical since twice bigger alphabet requires 1 more bit per character. However interesting property of merging/splitting is that (in ideal case) it doesn't affect (h2-h1) difference, so can be used to manipulate h2/h1 without any side effects.
1) the simpliest way to increase h2/h1.
The starting values where h1=3.86 and h2=2.13 (calculated using Takahashi transcription).
It appears that the following adjustments do some improvement:
cGh -> Gr (G is any gallow)
ch -> er
sh -> prl
t -> ll
a, i, n removed
Resulting h1=3.12, h2=2.26 (so h2/h1=0.72)
2) how merging letters changes entropy?
The "input" of letter X in entropy formula is p * log2(p). In the ideal case if X represents two letters with equal probability (so p/2 for each) then p * log2(p) turns into 2 * (p/2) * log2(p/2) = p * (log2(p) - 1), so the entropy increases. In the idealest case when each letter represents two letters with equal probabilities entropy increases by exactly 1. In the most idealest case when each letter represents two letters with equal probabilities independently of context both h1 and h2 change by the same value (1).
All this is quite logical since twice bigger alphabet requires 1 more bit per character. However interesting property of merging/splitting is that (in ideal case) it doesn't affect (h2-h1) difference, so can be used to manipulate h2/h1 without any side effects.