ua citation

app_notation-background.tex

@@ -22,7 +22,7 @@ To justify the use of $\semNX$-databases, we need to show that we can encode any
 As mentioned above we will use $\semNX$-databases paired with a probability distribution as a representation system, referring to such databases as \abbrNXPDB\xplural.
 Given \abbrNXPDB $\pxdb$, one can think of the of $\pd$ as the probability distribution across all worlds $\inset{0, 1}^\numvar$.  Denote a particular world to be $\vct{w}$.  For convenience let $\assign_\vct{w}: \pxdb\rightarrow\pndb$ be a function that computes the corresponding $\semN$-\abbrPDB upon assigning all values $w_i \in \vct{w}$ to $X_i \in \vct{X}$ of $\db_{\semNX}$.  Note the one-to-one correspondence between elements $\vct{w}\in\inset{0, 1}^\numvar$ to the worlds encoded by $\db_{\semNX}$ when $\vct{w}$ is assigned to $\vct{X}$ (assuming a domain of $\inset{0, 1}$ for each $X_i$).   %and a probability distribution $\pd$ over assignments $\assign$ of the variables $\vct{X} = \{X_1, \ldots, X_\numvar\}$  occurring in annotations of $\idb_{\semNX}$ to $\{0,1\}$.
 \AH{There was a big ICDT reviewer complaint in this section, but I don't know that I think it confuses things to think of them both an assignment and/or a vector of variables.}
-%Note that an assignment $\assign: \vct{X} \to \{0,1\}^\numvar$ can be represented as a vector $\vct{w} \in \{0,1\}^n$ where $\vct{w}[i]$ records the value assigned to variable $X_i$. Thus, from now on we will solely use such vectors which we refer to as \emph{world vectors} and implicitly understand them to represent assignments. 
+%Note that an assignment $\assign: \vct{X} \to \{0,1\}^\numvar$ can be represented as a vector $\vct{w} \in \{0,1\}^n$ where $\vct{w}[i]$ records the value assigned to variable $X_i$. Thus, from now on we will solely use such vectors which we refer to as \emph{world vectors} and implicitly understand them to represent assignments.
 We can think of $\assign_\vct{w}(\pxdb)\inparen{\tup}$ as the semiring homomorphism $\semNX \to \semN$ that applies the assignment $\vct{w}$ to all variables $\vct{X}$ of a polynomial and evaluates the resulting expression in $\semN$.
 \BG{explain connection to homomorphism lifting in K-relations}
 
@@ -56,7 +56,7 @@ Importantly, as the following proposition shows, any finite $\semN$-PDB can be e
 \begin{proof}
 To prove that \abbrNXPDB\xplural are complete consider the following construction that for any $\semN$-PDB $\pdb = (\idb, \pd)$ produces an \abbrNXPDB $\pxdb = (\db_{\semNX}, \pd')$  such that $\rmod(\pxdb) = \pdb$. Let $\idb = \{D_1, \ldots, D_{\abs{\idb}}\}.$ %and let $max(D_i)$
 \AH{What are we using $max(D_i)$ for?}
- %denote $max_{\tup} D_i(\tup)$. 
+ %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 $m = max(D_i)$.
 In $\db_{\semNX}$ we assign each tuple $\tup$ the polynomial:
@@ -66,7 +66,7 @@ In $\db_{\semNX}$ we assign each tuple $\tup$ the polynomial:
   \]
 The probability distribution $\pd'$ assigns all world vectors zero probability except for $\abs{\idb}$ world vectors (representing the possible worlds) $\vct{w}_i$. All elements of $\vct{w}_i$ are zero except for the position corresponding to variables $X_{i}$ which is set to $1$. Unfolding definitions it is trivial to show that $\rmod(\pxdb) = \pdb$. Thus, \abbrNXPDB\xplural are a complete representation system.
 
-Since $\semNX$ is the free object in the variety of semirings, Birkhoff's HSP theorem implies that any assignment $\vct{X} \to \semN$, which includes as a special case the assignments $\assign_{\vct{w}}$ used here, uniquely extends to the semiring homomorphism alluded to above, $\assign_\vct{w}\inparen{\pxdb}\inparen{\tup}: \semNX \to \semN$. For a polynomial $\assign_\vct{w}\inparen{\pxdb}\inparen{\tup}$ substitutes variables based on $\vct{w}$ and then evaluates the resulting expression in $\semN$. For instance, consider the polynomial $\pxdb\inparen{\tup} = \poly = X + Y$ and assignment $\vct{w} := X = 0, Y=1$. We get $\assign_\vct{w}\inparen{\pxdb}\inparen{\tup} = 0 + 1 = 1$. % It is trivial to show that an assignment  is a semiring homomorphism.
+Since $\semNX$ is the free object in the variety of semirings, Birkhoff's HSP theorem~\cite{graetzer-08-un} implies that any assignment $\vct{X} \to \semN$, which includes as a special case the assignments $\assign_{\vct{w}}$ used here, uniquely extends to the semiring homomorphism alluded to above, $\assign_\vct{w}\inparen{\pxdb}\inparen{\tup}: \semNX \to \semN$. For a polynomial $\assign_\vct{w}\inparen{\pxdb}\inparen{\tup}$ substitutes variables based on $\vct{w}$ and then evaluates the resulting expression in $\semN$. For instance, consider the polynomial $\pxdb\inparen{\tup} = \poly = X + Y$ and assignment $\vct{w} := X = 0, Y=1$. We get $\assign_\vct{w}\inparen{\pxdb}\inparen{\tup} = 0 + 1 = 1$. % It is trivial to show that an assignment  is a semiring homomorphism.
 Closure under $\raPlus$ queries follows from this and from \cite{DBLP:conf/pods/GreenKT07}'s Proposition 3.5, which states that semiring homomorphisms commute with queries over $\semK$-relations.