Compression-Examples

Definition

$t=$ Time

$c(t)=$ The best achievable compression rate (space saving) function of the foundation model that changes with time $t$

$\Delta t = const$ (recommended setting: 1 day to 1 week)

$t_i=t_{i-1} + \Delta t$

$\Delta c_i = c(t_i) - c(t_{i-1})$

PoWoC Token Incentive Model

Every time interval $\Delta t$ generates 1 fixed coin, meaning at each time point $t_i$ 1 fixed coin is generated, and this 1 coin is distributed as follows:

If $\Delta c_i > 0$, meaning the foundation model has made substantial progress
- Let $\Delta c_i =a + b + c + ...$, driven by $n$ contributors, then the coin amounts received by the $n$ contributors are respectively $\frac{a}{\Delta c_i} , \frac{b}{\Delta c_i} , \frac{c}{\Delta c_i},...$
  
  $The diagram illustrates the process in which one coin is created and distributed over a unit of time, $\Delta t$ . Between $t_2$ and $t_3$, a change of $\Delta c_3> 0$ is generated (the height of the yellow area >0), and this $\Delta c_3$ is driven by three contributors, {a, b, c}, in three stages (with a represented by red, b by green, and c by blue). Consequently, contributors a, b, and c will share the coin equally based on the height of the red, green, and blue bars, respectively (a gets the most, and c the least).$
  
  The diagram illustrates the process in which one coin is created and distributed over a unit of time, $\Delta t$ . Between $t_2$ and $t_3$, a change of $\Delta c_3> 0$ is generated (the height of the yellow area >0), and this $\Delta c_3$ is driven by three contributors, {a, b, c}, in three stages (with a represented by red, b by green, and c by blue). Consequently, contributors a, b, and c will share the coin equally based on the height of the red, green, and blue bars, respectively (a gets the most, and c the least).
If $\Delta c_i = 0$, meaning the foundation model has not made substantial progress
- Then, the coin distribution plan at time point $t_i$ $=$ the coin distribution plan at time point $t_{i-1}$

c[t_] := 1 - (1/Ceiling[(t + 0.5)*3] + 0.85);
d = 0.1;
e = 4.5
Show[
 Plot[c[t], {t, 1, e}, PlotRange -> {0, c[e]}, Filling -> None, 
  AxesLabel -> {t, c}, Ticks -> {Range[2, 10], Range[0, c[10], 0.02]},
   PlotLabel -> "PoWoC"],
 Plot[c[t], {t, 2, 3}, PlotRange -> {0, c[e]}, Filling -> 0],
 Plot[c[t], {t, 2, 3}, PlotRange -> {0, c[e]}, Filling -> c[2], 
  FillingStyle -> Directive[Yellow, Opacity[0.2]]],
 Plot[c[t], {t, 2.1 + 0/3, 2.1 + 0/3 + d}, PlotRange -> {0, c[e]}, 
  Filling -> c[2], FillingStyle -> Red],
 Plot[c[t], {t, 2.1 + 1/3, 2.1 + 1/3 + d}, PlotRange -> {0, c[e]}, 
  Filling -> c[2.5], FillingStyle -> Green],
 Plot[c[t], {t, 2.1 + 2/3, 2.1 + 2/3 + d}, PlotRange -> {0, c[e]}, 
  Filling -> c[2.8], FillingStyle -> Blue]
 ]

(Lossy) Compression Rate $c$

We denote $c$ as space saving [1]:

$$ c:=1-\frac{compressed}{original}= 1- \frac{p+(1-\alpha)d}{d}=\alpha-\frac{p}{d} $$

$p$ is the

Free Parameters