Index: doc/aaron_comp_II/comp_II.tex
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--- doc/aaron_comp_II/comp_II.tex	(revision be0a9d8a687f930ea1282a53a5089a2db229863e)
+++ doc/aaron_comp_II/comp_II.tex	(revision a2c5c1799ec77ea4f7f10bd094619668fa8b0bf8)
@@ -407,5 +407,7 @@
 If cross-argument resolution dependencies cannot be completely eliminated, effective caching strategies to reduce duplicated work between equivalent argument-parameter matches in different combinations may mitigate the asymptotic defecits of the whole-combination matching approach. 
 The final area of investigation is heuristics and algorithmic approaches to reduce the number of argument interpretations considered in the common case; if argument-parameter matches cannot be made independent, even small reductions in $i$ should yield significant reductions in the $i^{p+1}$ resolver runtime factor. 
+
 The discussion below presents a number of largely orthagonal axes for expression resolution algorithm design to be investigated, noting prior work where applicable. 
+Though some of the proposed improvements to the expression resolution algorithm are based on heuristics rather than asymptoticly superior algorithms, it should be noted that user programmers often employ idioms and other programming patterns to reduce the mental burden of producing correct code, and if these patterns can be identified and exploited by the compiler then the significant reduction in expression resolution time for common, idiomatic expressions should result in lower total compilation time even for code including difficult-to-resolve expressions that push the expression resolver to its theoretical worst case.
 
 \subsection{Argument-Parameter Matching}
@@ -446,6 +448,7 @@
 \subsubsection{Hybrid}
 This proposal includes the investigation of hybrid top-down/bottom-up argument-parameter matching.
-A reasonable hybrid approach might be to take a top-down approach when the expression to be matched is known to have a fixed type, and a bottom-up approach in untyped contexts.
-This may include switches from one type to another at different levels of the expression tree, for instance:
+A reasonable hybrid approach might take a top-down approach when the expression to be matched has a fixed type, and a bottom-up approach in untyped contexts.
+This approach may involve switching from one type to another at different levels of the expression tree. 
+For instance:
 \begin{lstlisting}
 forall(otype T)
@@ -456,7 +459,7 @@
 int x = f( f( '!' ) );
 \end{lstlisting}
-Here, the outer call to ©f© must have a return type that is (implicitly convertable to) ©int©, so a top-down approach could be used to select \textit{(1)} as the proper interpretation of ©f©. \textit{(1)}'s parameter ©x© here, however, is an unbound type variable, and can thus take a value of any complete type, providing no guidance for the choice of candidate for the inner ©f©. The leaf expression ©'!'©, however, gives us a zero-cost interpretation of the inner ©f© as \textit{(2)}, providing a minimal-cost expression resolution where ©T© is bound to ©void*©.
-
-Deciding when to switch between bottom-up and top-down resolution in a hybrid algorithm is a necessarily heuristic process, and though finding good heuristics for it is an open question, one reasonable approach might be to switch from top-down to bottom-up when the number of candidate functions exceeds some threshold.
+The outer call to ©f© must have a return type that is (implicitly convertable to) ©int©, so a top-down approach is used to select \textit{(1)} as the proper interpretation of ©f©. \textit{(1)}'s parameter ©x©, however, is an unbound type variable, and can thus take a value of any complete type, providing no guidance for the choice of candidate for the inner call to ©f©. The leaf expression ©'!'©, however, determines a zero-cost interpretation of the inner ©f© as \textit{(2)}, providing a minimal-cost expression resolution where ©T© is bound to ©void*©.
+
+Deciding when to switch between bottom-up and top-down resolution to minimize wasted work in a hybrid algorithm is a necessarily heuristic process, and though finding good heuristics for which subexpressions to swich matching strategies on is an open question, one reasonable approach might be to set a threshold $t$ for the number of candidate functions, and to use top-down resolution for any subexpression with fewer than $t$ candidate functions, to minimize the number of unmatchable argument interpretations computed, but to use bottom-up resolution for any subexpression with at least $t$ candidate functions, to reduce duplication in argument interpretation computation between the different candidate functions. 
 
 \subsubsection{Common Subexpression Caching}
@@ -464,5 +467,5 @@
 
 \subsection{Implicit Conversion Application}
-Baker's\cite{Baker82} and Cormack's\cite{Cormack81} algorithms do not account for implicit conversions\footnote{Baker does briefly comment on an approach for handling implicit conversions.}; both assume that there is at most one valid interpretation of a given expression for each distinct type.
+Baker's and Cormack's algorithms do not account for implicit conversions\footnote{Baker does briefly comment on an approach for handling implicit conversions.}; both assume that there is at most one valid interpretation of a given expression for each distinct type.
 Integrating implicit conversion handling into their algorithms provides some choice of implementation approach.