[doc/architecture/blockchain] added inverse lead selection function

doc/src/architecture/blockchain.md

@@ -68,8 +68,10 @@ the probability that a party holding all the stake will be selected to
 be a leader. Stakeholder is selected as leader for slot j with
 probability $\phi_f(\alpha_i)$, $\alpha_i$ is $U_i$ stake.
 
+### absolute stake aggregation dependent leader selection functions
+
+#### linear functions
 
-### linear aggregation dependent leader selection
 in the previous leader selection function, it has the unique property of
 independent aggregation of the stakes, meaning the property of a leader winning leadership with stakes $\sigma$ is independent of whether the stakeholder would act as a pool of stakes, or distributed stakes on competing coins.
 "one minus the probability" of winning leadership with aggregated stakes is
@@ -80,13 +82,15 @@ $\prod_{i}^{n}(1-\phi(\sigma_i))=-(1-f)^{\sum_i(\sigma_i)}$
 thus:
 $$ 1-\phi(\sum_{i}\sigma_i) =\prod_{i}^{n}(1-\phi(\sigma_i)) $$
 
-#### linear leader selection
+##### linear leader selection
+
 $$y < T $$
 $$y = 2^lk \mid 0 \le k \le 1$$
 $$T = 2^l\phi(v)$$
 $$ \phi(v)=\frac{1}{v_{max}}v $$
 
-#### dependent aggregation
+##### dependent aggregation
+
 linear leader selection has the dependent aggregation property, meaning it's favorable to compete in pools with sum of the stakes over aggregated stakes of distributed stakes:
 
 $$\phi(\sum_{i}{\sigma_i})>\prod_{i}^{n}{\sigma_i}$$
@@ -95,17 +99,49 @@ let's assume the stakes are divided to stakes of value $\sigma_i=1$ for $\Sigma>
 $$V>(\frac{1}{v_{max}})^{n-1}$$
 note that $(\frac{1}{v_{max}})^{n-1} < 1, V>1$, thus competing with single coin of the sum of stakes held by the stakeholder is favourable.
 
-### scalar linear aggregation dependent leader selection
+##### scalar linear aggregation dependent leader selection
+
   a target function T with scalar coefficients can be formalized as
  $$T=2^lk\phi(\Sigma)=2^l(\frac{1}{v_{max}})\Sigma$$
  let's assume $v_{max}=2^v$, then:
  $$T=2^lk\phi(\Sigma)=2^{l-v}\Sigma$$
  then the lead statement is
- $$y<2^{l-v}\Sigma$$ for example for a group order or l=24 bits, and maximum value of $v_{max}=2^{10}$, then lead statement:
+ $$y<2^{l-v}\Sigma$$ for example for a group order or l=    24 bits, and maximum value of $v_{max}=2^{10}$, then lead statement:
  $$y<2^{14}\Sigma$$
 
-### pairing leader selection independent aggregation function
-TODO
+##### competing max value coins
+
+for a stakeholder with $nv_{max}$ absolute stake, $\mid n \in \mathbb{Z}$ it's advantageous for the stakeholder to
+distribute stakes on $n$ competing coins.
+
+
+#### inverse functions
+
+inverse lead selection functions doesn't require maximum stake, most suitable for absolute stake,
+it has the disadvantage that it's inflating with increasing rate as time goes on , but it can be function of the inverse of the slot
+to control the increasing frequency of winning leadership.
+
+##### leader selection without maximum stake upper limit
+
+the inverse leader selection without maximum stake value can be $\phi(v)=\frac{v}{v+c}$  where c is $ > 1$
+and inversely proportional with probability of winning leadership, let it be called leadership coefficient.
+
+
+##### decaying linear leader selection
+
+as the time goes one, and stakes increase, this means the combined stakes of all stakeholders increases the probability
+of winning leadership in next slots leading to more leaders at a single slot, to maintain, or to be more general to control this frequency of leaders per slot,
+c (the leadership coefficient) need to be function of the slot $sl$, i.e $c(sl) = \frac{sl}{R}$ where $R$ is epoch size (number of slots in epoch).
+
+##### pairing leader selection independent aggregation function
+
+the only family of functions that are isomorphic to summation on multiplication (having the independent aggregation property) is the exponential function,
+and since it's impossible to implement in plonk,
+
+###### TODO (proof)
+
+a re-formalization of the lead statement using pairing that is isomorphic to summation on multiplication is also an options.
+
 
 ## Leaky non-resettable beacon