Small tweaks to S.2.

main.bbl

@@ -2,7 +2,7 @@
 %%% Do NOT edit. File created by BibTeX with style
 %%% ACM-Reference-Format-Journals [18-Jan-2012].
 
-\begin{thebibliography}{50}
+\begin{thebibliography}{52}
 
 %%% ====================================================================
 %%% NOTE TO THE USER: you can override these defaults by providing
@@ -372,6 +372,23 @@
 \newblock
 
 
+\bibitem[Impagliazzo et~al\mbox{.}(2001)]%
+        {eth}
+\bibfield{author}{\bibinfo{person}{Russell Impagliazzo},
+  \bibinfo{person}{Ramamohan Paturi}, {and} \bibinfo{person}{Francis Zane}.}
+  \bibinfo{year}{2001}\natexlab{}.
+\newblock \showarticletitle{Which Problems Have Strongly Exponential
+  Complexity?}
+\newblock \bibinfo{journal}{\emph{J. Comput. System Sci.}}
+  \bibinfo{volume}{63}, \bibinfo{number}{4} (\bibinfo{year}{2001}),
+  \bibinfo{pages}{512--530}.
+\newblock
+\showISSN{0022-0000}
+\urldef\tempurl%
+\url{https://doi.org/10.1006/jcss.2001.1774}
+\showDOI{\tempurl}
+
+
 \bibitem[Jha and Suciu(2012)]%
         {jha-12-pdwm}
 \bibfield{author}{\bibinfo{person}{Abhay~Kumar Jha} {and} \bibinfo{person}{Dan
@@ -382,6 +399,24 @@
 \newblock
 
 
+\bibitem[Joglekar et~al\mbox{.}(2016)]%
+        {ajar}
+\bibfield{author}{\bibinfo{person}{Manas~R. Joglekar}, \bibinfo{person}{Rohan
+  Puttagunta}, {and} \bibinfo{person}{Christopher R{\'{e}}}.}
+  \bibinfo{year}{2016}\natexlab{}.
+\newblock \showarticletitle{{AJAR:} Aggregations and Joins over Annotated
+  Relations}. In \bibinfo{booktitle}{\emph{Proceedings of the 35th {ACM}
+  {SIGMOD-SIGACT-SIGAI} Symposium on Principles of Database Systems, {PODS}
+  2016, San Francisco, CA, USA, June 26 - July 01, 2016}},
+  \bibfield{editor}{\bibinfo{person}{Tova Milo} {and}
+  \bibinfo{person}{Wang{-}Chiew Tan}} (Eds.). \bibinfo{publisher}{{ACM}},
+  \bibinfo{pages}{91--106}.
+\newblock
+\urldef\tempurl%
+\url{https://doi.org/10.1145/2902251.2902293}
+\showDOI{\tempurl}
+
+
 \bibitem[Karp et~al\mbox{.}(1989)]%
         {DBLP:journals/jal/KarpLM89}
 \bibfield{author}{\bibinfo{person}{Richard~M. Karp}, \bibinfo{person}{Michael

prob-def.tex

@@ -14,9 +14,10 @@ A circuit $\circuit$ is a Directed Acyclic Graph (DAG) with source gates (in deg
 Each gate has the following members: \type, \vari{input}, %\val, 
 \vpartial, \degval, \vari{Lweight}, and \vari{Rweight}, where \type is the value type $\{\circplus, \circmult, \var, \tnum\}$ and \vari{input} the list of inputs. Source gates have an extra member \val for the value.  $\circuit_\linput$ ($\circuit_\rinput$) denotes the left (right) input of \circuit.
 \end{Definition}
-We refer to the structure when the underlying DAG is a tree (with edges pointing towards the root) as an expression tree \etree.  Members not described in the definition are defined and used in the appendix proofs.  %In such a case, the root of \etree is analogous to the sink of \circuit.  The fields \vari{partial}, \degval, \vari{Lweight}, and \vari{Rweight} are used in the proofs of \Cref{sec:proofs-approx-alg}.
+We refer to the structure when the underlying DAG is a tree (with edges pointing towards the root) as an expression tree \etree.  The circuits $\inparen{1}$ and $\inparen{2}$ in column $\poly$ of \Cref{fig:two-step} are both expression trees.  %encode their respective polynomials in column $\poly$.
+Members not described in~\Cref{def:circuit} are defined and used in the appendix proofs.  %In such a case, the root of \etree is analogous to the sink of \circuit.  The fields \vari{partial}, \degval, \vari{Lweight}, and \vari{Rweight} are used in the proofs of \Cref{sec:proofs-approx-alg}.
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-The circuits $\inparen{1}$ and $\inparen{2}$ in column $\poly$ of \Cref{fig:two-step} are both expression trees.%encode their respective polynomials in column $\poly$.
+
 %Note that the ciricuit \circuit representing $AX$ and the circuit \circuit' representing $B\inparen{Y+Z}$ each encode a tree, with edges pointing towards the root.
 
 The function $\polyf\inparen{\cdot}$ (\Cref{def:poly-func}) maps a circuit to its corresponding polynomial.  We next define its inverse: