sys

sections/system.tex

@@ -70,7 +70,7 @@ If a row with dependent cells is deleted, the dependent cells need to be updated
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \subsection{Update Index}
 \label{sec:system-index}
-The update index stores sequence of updates ($\overlay = \overlay_k \circ \ldots \circ \overlay_1$) and provide efficient access to the cells of an overlay spreadsheet (denoted $\spreadsheet_\overlay$) where undefined cells have the value $\errorval$.
+The update index stores a sequence of updates ($\overlay = \overlay_k \circ \ldots \circ \overlay_1$) and provides efficient access to the cells of an overlay spreadsheet (denoted $\spreadsheet_\overlay$) where undefined cells have the value $\errorval$.
 This entails:
 (i) cell expressions $\spreadsheet_\overlay[\column, \row]$ (for cell evaluation);
 (ii) upstream dependencies of a range (for topological sort and computing the active set), and
@@ -161,10 +161,10 @@ Consider a running sum, such as the one in \Cref{ex:recursive-running-sum}.
 The $k$th element will have $O(k)$ upstream dependencies, and so naively following \Cref{alg:upstream} is in $O(k)$.
 However, observe that a single pattern is responsible for all of these dependencies, suggesting that a more efficient option may be available.
 
-This dependency chain arises from recursion over single pattern; most cells depend on other cells defined by the same pattern.
+This dependency chain arises from recursion over a single pattern; most cells depend on other cells defined by the same pattern.
 We refer to such a pattern as \emph{recursive}, even if it does not create dependency cycle over individual cells.
 
-As with cell execution, the transitive closure of the dependencies of a recursive pattern has a closed-form representation.
+As with cell execution, the transitive closure of the dependencies of a recursive pattern may permit a closed-form representation.
 In our running example, the upstream of any $\cellRef{D}{k}$ is exactly $\cellRef{D}{1-(k-1)}$ and $\cellRef{C}{1-k}$.
 %
 The \texttt{lineage} field of \Cref{alg:upstream} is used to track the set of patterns visited, and the offset(s) at which they were visited.