diff --git a/text/accumulation.tex b/text/accumulation.tex index d0a4ba7..16ba293 100644 --- a/text/accumulation.tex +++ b/text/accumulation.tex @@ -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. @@ -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}