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Simplify definition of Q #111

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Oct 24, 2024
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12 changes: 6 additions & 6 deletions text/accumulation.tex
Original file line number Diff line number Diff line change
Expand Up @@ -77,13 +77,13 @@ \subsection{History and Queuing}
\end{aligned}\right.
\end{equation}

We further define the accumulation priority queue function $Q$, which provides the sequence of work-reports which are accumulatable given the work-reports which have hitherto been accumulated and a set of not-yet-accumulated work-reports and their dependencies.
We further define the accumulation priority queue function $Q$, which provides the sequence of work-reports which are accumulatable given a set of not-yet-accumulated work-reports and their dependencies.
\begin{equation}
Q\colon\left\{\begin{aligned}
&(\seq{(\mathbb{W}, \{\H\})}, \dict{\H}{\H}) \to \seq{\mathbb{W}} \\
&(\mathbf{r}, \mathbf{a}) \mapsto \begin{cases}
[] &\when \mathbf{g} = \{\} \\
\mathbf{g} \frown Q(E(\mathbf{r}, \srmap(\mathbf{g})), \mathbf{a} \cup \srmap(\mathbf{g}))\!\!\!\! &\otherwise \\
&\seq{(\mathbb{W}, \{\H\})} \to \seq{\mathbb{W}} \\
&\mathbf{r} \mapsto \begin{cases}
[] &\when \mathbf{g} = [] \\
\mathbf{g} \frown Q(E(\mathbf{r}, \srmap(\mathbf{g})))\!\!\!\! &\otherwise \\
\multicolumn{2}{l}{\,\where \mathbf{g} = [w \mid (w, \{\}) \orderedin \mathbf{r}]}
\end{cases}
\end{aligned}\right.
Expand All @@ -100,7 +100,7 @@ \subsection{History and Queuing}
We may now define the sequence of accumulatable work-reports in this block as $\mathbf{W}^*$:
\begin{align}
\using m &= \mathbf{H}_t \bmod \mathsf{E}\\
\mathbf{W}^* &\equiv \mathbf{W}^! \frown Q(\mathbf{q}, \accumulatedcup \cup\;\srmap(\mathbf{W}^!)) \\
\mathbf{W}^* &\equiv \mathbf{W}^! \frown Q(\mathbf{q}) \\
\quad\where \mathbf{q} &= E(\wideparen{\ready_{m\dots}} \concat \wideparen{\ready_{\dots m}} \concat \mathbf{W}^Q, \srmap(\mathbf{W}^!))
\end{align}

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