fix bug

app_notation-background.tex

@@ -61,7 +61,7 @@ In $\idb_{\semNX}$ we assign each tuple $\tup$ the polynomial:
  \idb_{\semNX}(\tup) = \sum_{i=1}^{\abs{\idb}} D_i(\tup)\cdot X_{i}
   \]
 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, $\semNX$ 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 $\assign_{\vct{w}}: \vct{X} \to \{0,1\}$ used here, uniquely extends to a semiring homomorphism $\textsc{Eval}_{\assign_{\vct{w}}}: \semNX \to \semN$. For a polynomial $\textsc{Eval}_{\assign_{\vct{w}}}(\poly)$ substitutes variables based on $\assign_{\vct{w}}$ and then evaluate the resulting expression in $\semN$. For instance, consider the polynomial $\poly = X + Y$ and assignment $\assign \defas X = 0, Y=1$. We get $\textsc_{\assign}(\poly) = 0 + 1 = 1$. % 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 implies that any assignment $\vct{X} \to \semN$, which includes as a special case the assignments $\assign_{\vct{w}}: \vct{X} \to \{0,1\}$ used here, uniquely extends to a semiring homomorphism $\textsc{Eval}_{\assign_{\vct{w}}}: \semNX \to \semN$. For a polynomial $\textsc{Eval}_{\assign_{\vct{w}}}(\poly)$ substitutes variables based on $\assign_{\vct{w}}$ and then evaluate the resulting expression in $\semN$. For instance, consider the polynomial $\poly = X + Y$ and assignment $\assign := X = 0, Y=1$. We get $\textsc{Eval}_{\assign}(\poly) = 0 + 1 = 1$. % 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.