some misspells

approx_alg.tex

@@ -5,7 +5,7 @@
 
 In \Cref{sec:hard}, we showed that the answer to \Cref{prob:intro-stmt} is no.
 With this result, we now design an approximation algorithm for our problem that runs in $\bigO{\abs{\circuit}}$ for a very broad class of circuits (see the discussion after \Cref{lem:val-ub} for more).
-The folowing approximation algorithm applies to \abbrBIDB lineage polynomials (over $\raPlus$ queries), though our bounds are more meaningful for a non-trivial subclass of queries over \bis that contains all queries on \tis, as well as the queries of the PDBench benchmark~\cite{pdbench}.  All proofs and pseudocode can be found in \Cref{sec:proofs-approx-alg}.
+The following approximation algorithm applies to \abbrBIDB lineage polynomials (over $\raPlus$ queries), though our bounds are more meaningful for a non-trivial subclass of queries over \bis that contains all queries on \tis, as well as the queries of the PDBench benchmark~\cite{pdbench}.  All proofs and pseudocode can be found in \Cref{sec:proofs-approx-alg}.
 %it is then desirable to have an algorithm to approximate the multiplicity in linear time, which is what we describe next.
 
 \subsection{Preliminaries and some more notation}
@@ -148,7 +148,7 @@ if $\circuit$ is a tree, then
 we have $\abs{\circuit}(1,\ldots, 1)\le  \size(\circuit)^{O(k)}.$
 \end{Lemma}
 
-Note that the above implies that with the assumption $\prob_0>0$ and $\gamma<1$ are absolute constants from \Cref{cor:approx-algo-const-p}, then the runtime there simplies to $O_k\left(\frac 1{\inparen{\error'}^2}\cdot\size(\circuit)^2\cdot \log{\frac{1}{\conf}}\right)$ for general circuits $\circuit$. If $\circuit$ is a tree, then the runtime simplifies to $O_k\left(\frac 1{\inparen{\error'}^2}\cdot\size(\circuit)\cdot \log{\frac{1}{\conf}}\right)$, which then answers \Cref{prob:intro-stmt} is yes for such circuits.
+Note that the above implies that with the assumption $\prob_0>0$ and $\gamma<1$ are absolute constants from \Cref{cor:approx-algo-const-p}, then the runtime there simplifies to $O_k\left(\frac 1{\inparen{\error'}^2}\cdot\size(\circuit)^2\cdot \log{\frac{1}{\conf}}\right)$ for general circuits $\circuit$. If $\circuit$ is a tree, then the runtime simplifies to $O_k\left(\frac 1{\inparen{\error'}^2}\cdot\size(\circuit)\cdot \log{\frac{1}{\conf}}\right)$, which then answers \Cref{prob:intro-stmt} is yes for such circuits.
 
 Finally, note that by \Cref{prop:circuit-depth} and \Cref{lem:circ-model-runtime} for any $\raPlus$ query $\query$, there exists a circuit $\circuit^*$ for $\apolyqdt$ such that $\depth(\circuit^*)\le O_{|Q|}(\log{n})$ and $\size(\circuit)\le O_k\inparen{\qruntime{\query, \dbbase}}$. Using this along with \Cref{lem:val-ub}, \Cref{cor:approx-algo-const-p} and the fact that $n\le \qruntime{\query, \dbbase}$, we answer \Cref{prob:big-o-joint-steps} in the affirmative as follows:
 \begin{Corollary}

ra-to-poly.tex

@@ -16,7 +16,7 @@ For a probabilistic  database $\pdb = (\idb, \pd)$,  the result of a query is th
 Recall \Cref{fig:nxDBSemantics} which defines the lineage polynomial $\apolyqdt$ for any $\raPlus$ query.  We now make a meaningful connection between possible world semantics and world assignments on the lineage polynomial.
 
 \begin{Proposition}[Expectation of polynomials]\label{prop:expection-of-polynom}
-Given a \abbrBPDB $\pdb = (\idb,\pd)$, $\raPlus$ query $\query$, and lineage polynomial $\apolyqdt$ for aribitrary result tuple $\tup$, %$\semNX$-\abbrPDB $\pxdb = (\idb_{\semNX}',\pd')$ where $\rmod(\pxdb) = \pdb$,
+Given a \abbrBPDB $\pdb = (\idb,\pd)$, $\raPlus$ query $\query$, and lineage polynomial $\apolyqdt$ for arbitrary result tuple $\tup$, %$\semNX$-\abbrPDB $\pxdb = (\idb_{\semNX}',\pd')$ where $\rmod(\pxdb) = \pdb$,
 we have (denoting $\randDB$ as the random variable over $\idb$):
   $ \expct_{\randDB \sim \pd}[\query(\randDB)(t)] = \expct_{\vct{\randWorld}\sim \pdassign}\pbox{\apolyqdt\inparen{\vct{\randWorld}}}. $
 \end{Proposition}