book/consensus: add formalisms for block rank calcs

doc/latex-macros.txt

@@ -5,4 +5,6 @@
 \cN:{\mathcal{N}}
 \cX:{\mathcal{X}}
 \cY:{\mathcal{Y}}
+\ran:{\textrm{ran}}
+\max:{\textrm{max}}
 

doc/src/arch/consensus.md

@@ -228,7 +228,7 @@ current state is:
 
     [C]--...--[C]--|--[M7] <-- F2
 
-# Appendix
+# Appendix: Data Structures
 
 This section gives further details about the high level structures that will be
 used by the protocol.
@@ -274,3 +274,83 @@ used by the protocol.
 | `canonical` | `Blockchain`      | Canonical (finalized) blockchain       |
 | `forks`     | `Vec<Blockchain>` | Fork chains containing block proposals |
 
+# Appendix: Ranking Blocks
+
+## Sequences
+
+Denote blocks by the symbols $bᵢ ∈ B$, then a sequence of blocks (alternatively
+a fork) is an ordered series $𝐛 = (b₁, …, bₘ)$.
+
+Use $S$ for all sets of sequences for blocks in $B$.
+
+## Properties for Rank
+
+Each block is associated with a target $T : B → 𝕀$ where $𝕀 ⊂ ℕ$.
+
+1. Blocks with lower targets are harder to create and ranked higher in a sequence of blocks.
+2. Given two competing forks $𝐚 = (a₁, …, aₘ)$ and $b = (b₁, …, bₙ)$,
+   we wish to select a winner. Assume $𝐚$ is the winner, then $∑ T(aᵢ) ≤ ∑ T(bᵢ)$.
+3. There should only ever be a single winner.
+   When $∑ T(aᵢ) = ∑ T(bᵢ)$, then we have logic to break the tie.
+
+Property (2) can also be statistically true for $p > 0.5$.
+
+This is used to define a *fork-ranking* function $W : S → ℕ$.
+This function must *always* have unique values for distinct sequences.
+
+### Additivity
+
+We also would like the property $W$ is additive on subsequences
+$$ W((b₁, …, bₘ)) = W((b₁)) + ⋯ + W((bₘ)) $$
+which allows comparing forks from any point within the blockchain. For example
+let $𝐬 = (s₁, …, sₖ)$ be the blockchain together with forks $𝐚, 𝐛$ extending $𝐬$
+into $𝐬 ⊕  𝐚 = (s₁, …, sₖ, a₁, …, aₘ)$ and $𝐬 ⊕  𝐛 = (s₁, …, sₖ, b₁, …, bₙ)$.
+Then we have that
+$$ W(𝐬 ⊕  𝐚) < W(𝐬 ⊕  𝐛) ⟺  W(𝐚) < W(𝐛) $$
+which means it's sufficient to compare $𝐚$ and $𝐛$ directly.
+
+## Proposed Rank
+
+With a PoW mining system, we are guaranteed to always have that the block hash
+$h(b) ≤ T(b)$. Since the block hashes $( h(b₁), …, h(bₘ) )$ for a sequence
+$( b₁, …, bₘ )$ have the property that $∑ h(bᵢ) ≤ T(bᵢ)$, as well as being
+sufficiently random, we can use them to define our work function.
+
+Because $W$ is required to be additive, we define a block work function
+$w : B → ℕ$, and $W(𝐛) = ∑ w(bᵢ)$.
+
+The block work function should have a statistically higher score for
+blocks with a smaller target, and always be distinct for unique blocks.
+We define $w$ as
+$$ w(b) = \max(𝕀) - h(b) $$
+since $h(b) < T(b) < \max(𝕀)$ this function is well defined on the codomain.
+
+## Hash Function
+
+Let $𝕀$ be a fixed subset of $ℕ$ representing the output of a hash function
+$[0, \max(𝕀)]$.
+
+**Definition:** a *hash function* is a function $H : ℕ → 𝕀$ having the
+following properties:
+
+1. *Uniformity*, for any $y ∈ 𝕀$ and any $n ∈ ℕ$, there exists an $N > n$
+   such that $H(N) = y$.
+2. *One-way*, for any $y ∈ 𝕀$, we are unable to construct an $x ∈ ℕ$ such
+   that $H(x) = y$.
+
+Note: the above notions rely on purely algebraic properties of $H$ without
+requiring the machinery of probability. The second property of being one-way
+is a stronger notion than $\ran(H)$ being statistically random. Indeed if the
+probability is non-zero then we could find such an $(x, y)$ which breaks the
+one-way property.
+
+**Theorem:** *given a hash function $H : ℕ → 𝕀$ as defined above, it's impossible to
+construct two distinct sequences $𝐚 = (a₁, …, aₘ)$ and $𝐛 = (b₁, …, bₙ)$
+such that $H(a₁) + ⋯ + H(aₘ) = H(b₁) + ⋯ + H(bₙ)$.*
+
+By property (2), we cannot find a $H(x) = 0$.
+This means we cannot construct an $x$ such that $H(x) + H(a) = H(b)$ for
+any $a, b ∈ ℕ$.
+
+Let $y = (H(b₁) + ⋯ + H(bₙ)) - (H(a₂) + ⋯ + H(aₘ))$, then we cannot find
+any $H(x) = y$ by property (2) for $H$.