Minor tweaks

app_notation-background.tex

@@ -46,7 +46,7 @@ Importantly, as the following proposition shows, any finite $\semN$-PDB can be e
 $\semNX$-PDBs are a complete representation system for $\semN$-PDBs that is closed under $\raPlus$ queries.
 \end{Proposition}
 
-%\subsection{Proof of \Cref{prop:semnx-pdbs-are-a-}}
+%\subsection{Proof of~\Cref{prop:semnx-pdbs-are-a-}}
 \begin{proof}
 To prove that $\semNX$-PDBs are complete consider the following construction that for any $\semN$-PDB $\pdb = (\idb, \pd)$ produces an $\semNX$-PDB $\pxdb = (\idb_{\semNX}, \pd')$  such that $\rmod(\pxdb) = \pdb$. Let $\idb = \{D_1, \ldots, D_{\abs{\idb}}\}$ and let $max(D_i)$ denote $max_{\tup} D_i(\tup)$. For each world $D_i$ we create a corresponding variable $X_i$.
 %variables $X_{i1}$, \ldots, $X_{im}$ where $m = max(D_i)$.
@@ -91,13 +91,13 @@ Denote the vector $\vct{p}$ to be a vector whose elements are the individual pro
   = \sum\limits_{\substack{\vct{w} \in \{0, 1\}^\numvar\\ s.t. w_j,w_{j'} = 1 \rightarrow \not \exists b_i \supseteq \{t_{i,j}, t_{i',j}\}}} \poly(\vct{w})\prod_{\substack{j \in [\numvar]\\ s.t. \wElem_j = 1}}\prob_j \prod_{\substack{j \in [\numvar]\\s.t. w_j = 0}}\left(1 - \prob_i\right)
 \end{align}
 %
-Recall that tuple blocks in a TIDB always have size 1, so the outer summation of \Cref{eq:tidb-expectation} is over the full set of vectors.
+Recall that tuple blocks in a TIDB always have size 1, so the outer summation of \cref{eq:tidb-expectation} is over the full set of vectors.
 \BG{Oliver's conjecture: Bag-\tis + Q can express any finite bag-PDB:
 A well-known result for set semantics PDBs is that while not all finite PDBs can be encoded as \tis, any finite PDB can be encoded using a \ti and a query. An analog result holds in our case: any finite $\semN$-PDB can be encoded as a bag \ti and a query (WHAT CLASS? ADD PROOF)
 }
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\subsection{Proof of \Cref{prop:expection-of-polynom}}
+\subsection{Proof of~\Cref{prop:expection-of-polynom}}
 \label{subsec:expectation-of-polynom-proof}
 \begin{proof}
 We need to prove for $\semN$-PDB $\pdb = (\idb,\pd)$ and $\semNX$-PDB $\pxdb = (\db',\pd')$ where $\rmod(\pxdb) = \pdb$ that $\expct_{\db \sim \pd}[\query(\db)(t)] = \expct_{\vct{W} \sim \pd'}\pbox{\polyForTuple(\vct{W})}$
@@ -116,7 +116,7 @@ By expanding $\polyForTuple$ and the expectation we have:
 \end{proof}
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\subsection{ \Cref{lem:pre-poly-rpoly}}\label{app:subsec-pre-poly-rpoly}
+\subsection{~\Cref{lem:pre-poly-rpoly}}\label{app:subsec-pre-poly-rpoly}
 \begin{Lemma}\label{lem:pre-poly-rpoly}
 If
 $\poly(X_1,\ldots, X_\numvar) = \sum\limits_{\vct{d} \in \{0,\ldots, B\}^\numvar}q_{\vct{d}} \cdot \prod\limits_{\substack{i = 1\\s.t. d_i\geq 1}}^{\numvar}X_i^{d_i}$
@@ -125,8 +125,8 @@ $\rpoly(X_1,\ldots, X_\numvar) = \sum\limits_{\vct{d} \in \eta} q_{\vct{d}}\cdot
 \end{Lemma}
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\begin{proof}%[Proof for \Cref{lem:pre-poly-rpoly}]
-Follows by the construction of $\rpoly$ in \Cref{def:reduced-bi-poly}. 
+\begin{proof}%[Proof for~\Cref{lem:pre-poly-rpoly}]
+Follows by the construction of $\rpoly$ in \cref{def:reduced-bi-poly}. 
 \qed
 \end{proof}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@@ -141,7 +141,7 @@ $%  \[
 $%    \]
 \end{Proposition}
 
-\begin{proof}%[Proof for \Cref{proposition:q-qtilde}]
+\begin{proof}%[Proof for~\Cref{proposition:q-qtilde}]
 Note that any $\poly$ in factorized form is equivalent to its \abbrSMB expansion.  For each term in the expanded form, further note that for all $b \in \{0, 1\}$ and all $e \geq 1$, $b^e = b$. 
 \qed
 \end{proof}
@@ -160,7 +160,7 @@ Then, in expectation we have
 &= \sum_{\vct{d} \in \eta}q_{\vct{d}}\cdot \prod_{\substack{i = 1\\s.t. d_i \geq 1}}^\numvar \prob_i\label{p1-s4}\\
 &= \rpoly(\prob_1,\ldots, \prob_\numvar)\label{p1-s5}
 \end{align}
-In steps \Cref{p1-s1} and \Cref{p1-s2}, by linearity of expectation (recall the variables are independent, or the monomial expectation is 0), the expecation can be pushed all the way inside of the product.  In \Cref{p1-s3}, note that $w_i \in \{0, 1\}$ which further implies that for any exponent $e \geq 1$, $w_i^e = w_i$.  Next, in \Cref{p1-s4} the expectation of a tuple is indeed its probability.
+In steps \cref{p1-s1} and \cref{p1-s2}, by linearity of expectation (recall the variables are independent, or the monomial expectation is 0), the expecation can be pushed all the way inside of the product.  In \cref{p1-s3}, note that $w_i \in \{0, 1\}$ which further implies that for any exponent $e \geq 1$, $w_i^e = w_i$.  Next, in \cref{p1-s4} the expectation of a tuple is indeed its probability.
 
 Finally, observe \Cref{p1-s5} by construction in \Cref{lem:pre-poly-rpoly}, that $\rpoly(\prob_1,\ldots, \prob_\numvar)$ is exactly the product of probabilities of each variable in each monomial across the entire sum.
 \qed
@@ -169,6 +169,6 @@ Finally, observe \Cref{p1-s5} by construction in \Cref{lem:pre-poly-rpoly}, that
 
 \subsection{Proof For Corollary ~\ref{cor:expct-sop}}
 \begin{proof}
-Note that \Cref{lem:exp-poly-rpoly} shows that $\expct\pbox{\poly} =$ $\rpoly(\prob_1,\ldots, \prob_\numvar)$.  Therefore, if $\poly$ is already in \abbrSMB form, one only needs to compute $\poly(\prob_1,\ldots, \prob_\numvar)$ ignoring exponent terms (note that such a polynomial is $\rpoly(\prob_1,\ldots, \prob_\numvar)$), which indeed has $O(\smbOf{|\poly|})$ computations.
+Note that \cref{lem:exp-poly-rpoly} shows that $\expct\pbox{\poly} =$ $\rpoly(\prob_1,\ldots, \prob_\numvar)$.  Therefore, if $\poly$ is already in \abbrSMB form, one only needs to compute $\poly(\prob_1,\ldots, \prob_\numvar)$ ignoring exponent terms (note that such a polynomial is $\rpoly(\prob_1,\ldots, \prob_\numvar)$), which indeed has $O(\smbOf{|\poly|})$ computations.
 \qed
 \end{proof}

intro-new.tex

@@ -134,7 +134,9 @@ Thus, the marginal probability of tuple $\tup$ is equal to the probability that
 
 For bag semantics, the lineage of a tuple is a polynomial over variables $\vct{X}=(X_1,\dots,X_n)$ with % \in \mathbb{N}^\numvar$ with
 coefficients in the set of natural numbers $\mathbb{N}$ (an element of semiring $\mathbb{N}[\vct{X}]$).
-Analogously to sets, evaluating the lineage for $t$  over an assignment corresponding to a possible world yields the multiplicity of the result tuple $\tup$ in this world. Thus, instead of using \Cref{eq:intro-bag-expectation} to compute the expected result multiplicity of a tuple $\tup$, we can equivalently compute the expectation of the lineage polynomial of $\tup$ which for this example we denote\footnote{In later sections we will simply refer to $\linsett{\query}{\pdb}{\tup}$ as $Q$.} as $\linsett{\query}{\pdb}{\tup}$ or $\Phi$ if the parameters are clear from the context. In this work, we study the complexity of computing the expectation of such polynomials encoded as arithmetic circuits.
+Analogously to sets, evaluating the lineage for $t$  over an assignment corresponding to a possible world yields the multiplicity of the result tuple $\tup$ in this world. Thus, instead of using \Cref{eq:intro-bag-expectation} to compute the expected result multiplicity of a tuple $\tup$, we can equivalently compute the expectation of the lineage polynomial of $\tup$ which for this example we denote\footnote{
+In later sections, where we focus on a single lineage polynomial, we will simply refer to $\linsett{\query}{\pdb}{\tup}$ as $Q$.
+} as $\linsett{\query}{\pdb}{\tup}$ or $\Phi$ if the parameters are clear from the context. In this work, we study the complexity of computing the expectation of such polynomials encoded as arithmetic circuits.
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \begin{Example}\label{ex:intro-lineage}