Small fixes for POV and better display in Github (#196)

design/slot-reservations.md

@@ -85,51 +85,51 @@ $h := F(0.5)$, where $0 \lt h \lt 1$ and $h \neq 0.5$
 Changing the value of $h$ will [affect the curve of the rate of
 expansion](https://www.desmos.com/calculator/pjas1m1472) (interactive graph).
 
-#### Eligible address calculations in-depth
+#### Expansion function, $F(t_i)$, in-depth
 
-The following is taken from https://hackmd.io/@bkomuves/BkDXRJ-fC, with only
-slight variations to constrain size, reproduced with written
-permission from the author.
+$F(t_i)$ defines the expansion factor of eligible addresses in the network over
+time.
 
 ##### Assumptions
 
-We assume network address are randomly and more-or-less uniformly selected from
-a space of $2^{256}$.
+It is assumed network addresses are randomly, and more-or-less uniformly,
+selected from a space of $2^{256}$.
 
-We also assume that the window can only change in discrete step, based on some
-underlying blockchain's cadence (for example this would be approx every 12
-seconds in case of Ethereum), and that we measure time based on timestamps
-encoded in this blockchain blocks.
+It is also assumed that the window can only change in discrete steps, based on
+some underlying blockchain's cadence (for example this would be approx every 12
+seconds in the case of Ethereum), and that we measure time based on timestamps
+encoded in blockchain blocks.
 
-However with this assumption given, we want to be as granular and tunable as
-possibly.
+However, with this assumption given, it is desired to be as granular and tunable
+as possibly.
 
-We have a time duration in which we want to go from a single network address to
-the whole address-space.
+There is a time duration in which it is desired to go from a single network
+address to the whole address-space.
 
-To be able to make this work nicely, first I propose to define a linear time
-function $t_i$ which goes from 0 to 1.
+To be able to make this work nicely, first a linear time function $t_i$ which
+goes from 0 to 1, is defined.
 
 ##### Implementation
 
-At any desired block with timestamp $timestamp_i$ simply compute:
+At any desired block with timestamp $timestamp_i$, simply compute:
 
 $$t_i := \frac{timestamp_i - start}{expiry - start}$$
 
-Then to get a network range, you can plug in any kind of expansion function
-$F(x)$ with $F(0)=0$ and $F(1)=1$; for example a parametric exponential:
+Then to get a network range, any kind of expansion function $F(x)$ with $F(0)=0$
+and $F(1)=1$ can be plugged in; for example, a parametric exponential:
 
  $$ F_s(x) = \frac{\exp(sx) - 1}{\exp(s) - 1} $$
 
-Remark: with this particular function, you probably want $s<0$ (resulting in
-fast expansion initially, slowing down later). Here is a Mathematica one-liner
-to play with it:
+Remark: with this particular function, is is likely desired to have $s<0$
+(resulting in fast expansion initially, slowing down later). Here is a
+Mathematica one-liner to play with this idea:
 ```
     Manipulate[
       Plot[ (Exp[s*x]-1)/(Exp[s]-1), {x,0,1}, PlotRange->Full ],
         {s,-10,-1} ]
 ```
-You can easily do the same with eg. the online [Desmos](https://www.desmos.com/calculator) tool.
+As an alternative, the same can easily be done with eg. the online
+[Desmos](https://www.desmos.com/calculator) tool.
 
 ##### Address window
 
@@ -139,26 +139,31 @@ from the "window center" $A_0$ is smaller than $2^{256}\times F(t_i)$:
  $$ XOR(A,A_0) < 2^{256}\cdot F(t_i) $$
 
 Note: since $t_i$ only becomes 1 exactly at expiry, to allow the whole network
-to particite near the end, there should be a small positive $\delta > 0$ such
+to participate near the end, there should be a small positive $\delta > 0$ such
 that $F(t)=1$ for $t>1-\delta$, leaving the last about $100\delta$ percentage of
 the total slot fill window when the whole network is eligible to participate.
 
-Alternatively, you could rescale $t_i$ to achieve the same effect: $$ t_i' :=
-\min(\; t_i/(1-\delta)\;,\;1\;) $$ The latter is probably simpler because still
-have complete freedom in selecting the expansion function $F(x)$.
+Alternatively, $t_i$ could be rescaled to achieve the same effect:
+
+$$ t_i' := \min(\; t_i/(1-\delta)\;,\;1\;) $$
+
+The latter is probably simpler because it allows complete freedom in selecting
+the expansion function $F(x)$.
 
 ##### Parametrizing the speed of expansion
 
-While we could in theory have arbitrary expansions functions, we probably don't
-want more than an one parameter family, that is, a single parameter to set the
-curve. However, even if we have a single parameter, there could be any number of
-different ways to map a number to the same family of curves.
+While, in theory, arbitrary expansions functions could be used, it is likely
+undesirable to have more than an one parameter family, that is, a single
+parameter to set the curve. However, even with a single parameter, there
+could be any number of different ways to map a number to the same family of
+curves.
 
 In the above example $F_s(t)$, while $s$ is quite natural from a mathematical
 perspective, it doesn't really have any meaning for the user. A possibly better
 parametrization would be the value $h:=F_s(0.5)$, meaning "how big percentage of
-network is allowed to participate at half-time". You can of course compute $s$
-from $h$: $$ s = 2\log\left(\frac{1-h}{h}\right) $$
+network is allowed to participate at half-time". $s$ can be computed from $h$:
+
+$$ s = 2\log\left(\frac{1-h}{h}\right) $$
 
 ### Abandoned ideas