Merge pull request gavofyork#6 from davxy/davxy-notation-review

Notation section review

text/ack.tex

@@ -2,7 +2,7 @@ \section{Acknowledgements}
 
 Much of this present work is based in large part on the work of others. The Web3 Foundation research team and in particular Alistair Stewart and Jeff Burdges are responsible for \textsc{Elves}, the security apparatus of Polkadot which enables the possibility of in-core computation for \Jam. The same team is responsible for Sassafras, \textsc{Grandpa} and \textsc{Beefy}.    
 
-Safrole is a mild simplification of Sassafras and was made under the careful review of Davide Gallosi and Alistair Stewart.
+Safrole is a mild simplification of Sassafras and was made under the careful review of Davide Galassi and Alistair Stewart.
 
 The original CoreJam \textsc{rfc} was refined under the review of Bastian Köcher and Robert Habermeier and most of the key elements of that proposal have made their way into the present work.
 

text/header.tex

@@ -1,6 +1,6 @@
 \section{The Header}\label{sec:header}
 
-We must first define the header in terms of its components. The header comprises a parent hash and prior state root ($\mathbf{H}_p$ and $\mathbf{H}_r$), an extrinsic hash $\mathbf{H}_x$, a time-slot index $\mathbf{H}_t$, the epoch, winning-tickets and judgements markers $\mathbf{H}_e$, $\mathbf{H}_w$ and $\mathbf{H}_j$, a Bandersnatch block author key $\mathbf{K}_k$ and two Bandersnatch signatures; the entropy-yielding \textsc{vrf} signature $\mathbf{H}_v$ and a block seal $\mathbf{H}_s$. Headers may be serialized to an octet sequence with and without the latter seal component using $\mathcal{E}$ and $\mathcal{E}_U$ respectively. Formally:
+We must first define the header in terms of its components. The header comprises a parent hash and prior state root ($\mathbf{H}_p$ and $\mathbf{H}_r$), an extrinsic hash $\mathbf{H}_x$, a time-slot index $\mathbf{H}_t$, the epoch, winning-tickets and judgements markers $\mathbf{H}_e$, $\mathbf{H}_w$ and $\mathbf{H}_j$, a Bandersnatch block author key $\mathbf{H}_k$ and two Bandersnatch signatures; the entropy-yielding \textsc{vrf} signature $\mathbf{H}_v$ and a block seal $\mathbf{H}_s$. Headers may be serialized to an octet sequence with and without the latter seal component using $\mathcal{E}$ and $\mathcal{E}_U$ respectively. Formally:
 \begin{equation}\label{eq:header}
   \qquad\mathbf{H} \equiv (\mathbf{H}_p, \mathbf{H}_r, \mathbf{H}_x, \mathbf{H}_t, \mathbf{H}_e, \mathbf{H}_w, \mathbf{H}_j, \mathbf{H}_k, \mathbf{H}_v, \mathbf{H}_s)
 \end{equation}

text/notation.tex

@@ -28,8 +28,7 @@ \subsection{Functions and Operators}\label{sec:functions}
 
 The substitute-if-nothing function $\mathcal{U}$ is equivalent to the first argument which is not $\none$, or $\none$ if no such argument exists:
 \begin{align}\label{eq:substituteifnothing}
-%  \mathcal{U}(a_0, \dots a_n ) \equiv a_x : (a_x \ne \none \vee x = n), \bigwedge_{i=0}^{x-1} a_i = \none
-  \mathcal{U}(a_0, \dots) \equiv a_x : (a_x \ne \none \vee x = n), \bigwedge_{i=0}^{x-1} a_i = \none
+  \mathcal{U}(a_0, \dots a_n ) \equiv a_x : (a_x \ne \none \vee x = n), \bigwedge_{i=0}^{x-1} a_i = \none
 \end{align}
 Thus, \eg $\mathcal{U}(\none, 1, \none, 2) = 1$ and $\mathcal{U}(\none, \none) = \none$.
 
@@ -56,7 +55,7 @@ \subsection{Numbers}\label{sec:numbers}
 
 \subsection{Dictionaries}\label{sec:dictionaries}
 
-A \emph{dictionary} is a possibly partial mapping from some domain into some co-domain in much the same manner as a regular function. Unlike functions however, with dictionaries the total set of pairings are necessarily enumerable, and we represent them in some data structure as the set of all $(key \mapsto value)$ pairs. (In such data-defined mappings, it is common to name the values within the range a \emph{key} and the values within the domain a \emph{value}, hence the naming.)
+A \emph{dictionary} is a possibly partial mapping from some domain into some co-domain in much the same manner as a regular function. Unlike functions however, with dictionaries the total set of pairings are necessarily enumerable, and we represent them in some data structure as the set of all $(key \mapsto value)$ pairs. (In such data-defined mappings, it is common to name the values within the domain a \emph{key} and the values within the co-domain a \emph{value}, hence the naming.)
 
 Thus, we define the formalism $\dict{\mathrm{K}}{\mathrm{V}}$ to denote a dictionary which maps from the domain $\mathrm{K}$ to the range $\mathrm{V}$. We define a dictionary as a member of the set of all dictionaries $\mathbb{D}$ and a set of pairs $p = (k \mapsto v)$:
 \begin{align}
@@ -74,7 +73,7 @@ \subsection{Dictionaries}\label{sec:dictionaries}
     v & \text{if}\ \exists k : (k \mapsto v) \in \mathbf{d} \\
     \none & \otherwise
   \end{cases}\\
-  \forall \mathbf{d} \in \mathbb{D}&, \mathbf{s}: a \setminus \mathbf{s} \equiv \{ (k \mapsto v): (k \mapsto v) \in \mathbf{d}, k \not \in \mathbf{s} \}
+  \forall \mathbf{d} \in \mathbb{D}&, \mathbf{s} \subseteq K: \mathbf{d} \setminus \mathbf{s} \equiv \{ (k \mapsto v): (k \mapsto v) \in \mathbf{d}, k \not \in \mathbf{s} \}
 \end{align}
 
 Note that when using a subscript, it is an implicit assertion that the key exists in the dictionary. Should the key not exist, the result is undefined and any block which relies on it must be considered invalid.
@@ -85,17 +84,17 @@ \subsection{Dictionaries}\label{sec:dictionaries}
   \dict{K}{V} &\equiv \big \{ \{ (k \mapsto v) \mid k \in K \wedge v \in V \} \big \}
 \end{align}
 
-To denote the active domain (\ie set of keys) of a dictionary $\mathbf{d} \in \dict{K}{V}$, we use $\keys{\mathbf{d}} \subset K$ and for the range (\ie set of values), $\mathcal{V}(\mathbf{d}) \subset V$. Formally:
+To denote the active domain (\ie set of keys) of a dictionary $\mathbf{d} \in \dict{K}{V}$, we use $\keys{\mathbf{d}} \subseteq K$ and for the range (\ie set of values), $\mathcal{V}(\mathbf{d}) \subseteq V$. Formally:
 \begin{align}
   \keys{\mathbf{d} \in \mathbb{D}} &\equiv \{\ k \mid \exists v : (k \mapsto v) \in \mathbf{d}\ \} \\
   \mathcal{V}(\mathbf{d} \in \mathbb{D}) &\equiv \{\ v \mid \exists k : (k \mapsto v) \in \mathbf{d}\ \}
 \end{align}
 
-Note that since the domain of $\mathcal{V}$ is a set, should different keys with equal values appear in the dictionary, the set will only contain one such value.
+Note that since the co-domain of $\mathcal{V}$ is a set, should different keys with equal values appear in the dictionary, the set will only contain one such value.
 
 \subsection{Tuples}\label{sec:tuples}
 
-Tuples are groups of values where each item typically belongs to a different set. They are denoted with parentheses, \eg the tuple $t$ of the integers $3$ and $5$ is denoted $t = (3, 5)$, and it exists in the set of integer pairs sometimes denoted $\N \times \N$, but denoted in the present work as $(\N, \N)$.
+Tuples are groups of values where each item may belong to a different set. They are denoted with parentheses, \eg the tuple $t$ of the integers $3$ and $5$ is denoted $t = (3, 5)$, and it exists in the set of integer pairs sometimes denoted $\N \times \N$, but denoted in the present work as $(\N, \N)$.
 
 We have frequent need to refer to a specific item within a tuple value and as such find it convenient to declare a name for each item. \Eg we may denote a tuple with two named integer components $a$ and $b$ as $T = \ltuple\isa{a}{\N}\ts\isa{b}{\N}\rtuple$. We would denote an item $t \in T$ through subscripting its name, thus for some $t = \ltup\is{a}{3}\ts\is{b}{5}\rtup$, $t_a = 3$ and $t_b = 5$.
 
@@ -157,7 +156,7 @@ \subsubsection{Signing Schemes}\label{sec:signing}
 
 We denote the set of valid Bandersnatch public keys as $\H_B$, defined in appendix \ref{sec:bandersnatch}. $\bandersig{k \in \H_B}{x \in \Y}{m \in \Y} \subset \Y_{96}$ is the set of valid singly-contextualized signatures of utilizing the secret counterpart to the public key $k$, some context $x$ and message $m$.
 
-$\bandersnatch{r \in \Y_R}{x \in \Y}{m \in \Y} \subset \Y_{388}$, meanwhile, is the set of valid Bandersnatch Ring\textsc{vrf} deterministic singly-contextualized proofs of knowledge of a secret within some set of secrets identified by some root in the set of valid \emph{roots} $\Y_R \in \Y_{196608}$. We denote $\mathcal{R}(\mathbf{s} \in \seq{\H_B}) \in \Y_R$ to be the root specific to the set of public key counterparts $\mathbf{s}$. A root implies a specific set of Bandersnatch key pairs, knowledge of one of the secrets would imply being capable of making a unique, valid---and anonymous---proof of knowledge of a unique secret within the set.
+$\bandersnatch{r \in \Y_R}{x \in \Y}{m \in \Y} \subset \Y_{784}$, meanwhile, is the set of valid Bandersnatch Ring\textsc{vrf} deterministic singly-contextualized proofs of knowledge of a secret within some set of secrets identified by some root in the set of valid \emph{roots} $\Y_R \in \Y_{196608}$. We denote $\mathcal{R}(\mathbf{s} \in \seq{\H_B}) \in \Y_R$ to be the root specific to the set of public key counterparts $\mathbf{s}$. A root implies a specific set of Bandersnatch key pairs, knowledge of one of the secrets would imply being capable of making a unique, valid---and anonymous---proof of knowledge of a unique secret within the set.
 
 Both the Bandersnatch signature and Ring\textsc{vrf} proof strictly imply that a member utilized their secret key in combination with both the context $x$ and the message $m$; the difference is that the member is identified in the former and is anonymous in the latter. Furthermore, both define a \textsc{vrf} \emph{output}, a high entropy hash influenced by $x$ but not by $m$, formally denoted $\banderout{\bandersnatch{r}{x}{m}} \subset \H$ and $\banderout{\bandersig{k}{x}{m}} \subset \H$.