From 18d105f6103501e0c708f5d026e722fe605a853f Mon Sep 17 00:00:00 2001
From: Oliver <okennedy@buffalo.edu>
Date: Sat, 18 Sep 2021 00:58:03 -0400
Subject: [PATCH] cleaning up a few terms

---
 appendix.tex | 22 +++++++++++-----------
 1 file changed, 11 insertions(+), 11 deletions(-)

diff --git a/appendix.tex b/appendix.tex
index 7895315..70000f9 100644
--- a/appendix.tex
+++ b/appendix.tex
@@ -71,7 +71,7 @@ We define the circuit for a select-union-project-join $Q$ recursively by cases a
   \begin{algorithmic}[1]
   \Require $\query$: query
   \Require $\dbbase$: a \dbbaseName
-  \Ensure $\circuit = \tuple{E, V, \ell, \phi}$: a circuit encoding the lineage of each tuple in $\query(\dbbase)$
+  \Ensure $\circuit = \tuple{E, V, \phi, \ell}$: a circuit encoding the lineage of each tuple in $\query(\dbbase)$
 
   \If{$\query$ is $R$}
     \State $V = \comprehension{v_t}{t \in \dbbase.R}$
@@ -82,7 +82,7 @@ We define the circuit for a select-union-project-join $Q$ recursively by cases a
     \EndFor
   \ElsIf{$\query$ is $\sigma_\theta(\query')$}
     \State $\tuple{V, E, \phi', \ell} = \abbrStepOne(\query', \dbbase)$
-    \For{$t \in \dbbase.R$}
+    \For{$t \in \domain(\phi')$}
       \If{$\theta(t)$}
         \State $\phi(t) = \phi'(t)$
       \Else
@@ -91,21 +91,21 @@ We define the circuit for a select-union-project-join $Q$ recursively by cases a
     \EndFor
   \ElsIf{$\query$ is $\pi_{\vec{A}}(\query')$}
     \State $\tuple{V', E', \phi', \ell'} = \abbrStepOne(\query', \dbbase)$
-    \State $V = V' \cup \comprehension{v_t}{t \in \pi_{\vec{A}}(\query)}$
-    \State $E = E' \cup \comprehension{(\phi(t'), v_t)}{t \in \pi_{\vec{A}}t', t' \in \query', t \in \pi_{\vec{A}}(\query')}$
+    \State $V = V' \cup \comprehension{v_t}{t \in \pi_{\vec{A}}(\domain(\phi'))}$
+    \State $E = E' \cup \comprehension{(\phi(t'), v_t)}{t \in \pi_{\vec{A}}t', t' \in \domain(\phi'), t \in \pi_{\vec{A}}(\domain(\phi'))}$
       \Comment{Nodes with in-degrees above 2 are corrected (with logarithmic overhead) with an equivalent fan-in tree.}
-    \For{$t \in \pi_{\vec{A}}(\query')$}
+    \For{$t \in \pi_{\vec{A}}(\query'(\dbbase))$}
       \State $\phi(t) = v_t$ \Comment{$v_t$ as defined above}
       \State $\ell(v_t) = +$
     \EndFor
   \ElsIf{$\query$ is $\query_1 \cup \query_2$}
     \State $\tuple{V_1, E_1, \phi_1, \ell_1} = \abbrStepOne(\query_1, \dbbase)$
     \State $\tuple{V_2, E_2, \phi_2, \ell_2} = \abbrStepOne(\query_2, \dbbase)$
-    \State $V = V_1 \cup V_2 \cup \comprehension{v_t}{t \in \query_1 \cap \query_2}$
-    \State $E = E_1 \cup E_2 \cup \comprehension{(\phi_1(t), v_t), (\phi_2(t), v_t)}{t \in \query_1 \cap \query_2}$
+    \State $V = V_1 \cup V_2 \cup \comprehension{v_t}{t \in \domain(\phi_1) \cap \domain(\phi_2)}$
+    \State $E = E_1 \cup E_2 \cup \comprehension{(\phi_1(t), v_t), (\phi_2(t), v_t)}{t \in \domain(\phi_1) \cap \domain(\phi_2)}$
     \State $\phi = \phi_1 \cup \phi_2$
     \State $\ell = \ell_1 \cup \ell_2$
-    \For{$t \in \query_1 \cap \query_2$}
+    \For{$t \in \domain(\phi_1) \cap \domain(\phi_2)$}
       \State $\phi(t) = v_t$ \Comment{$v_t$ as defined above}
       \State $\ell(v_t) = +$
     \EndFor
@@ -113,12 +113,12 @@ We define the circuit for a select-union-project-join $Q$ recursively by cases a
     \For{$i \in [1, k]$}
       $\tuple{V_i, E_i, \phi_i, \ell_i} = \abbrStepOne(\query_i, \dbbase)$
     \EndFor
-    \State $V = V_1 \cup \ldots \cup V_k \cup \comprehension{v_t}{t \in \query_1 \bowtie \ldots \bowtie \query_k}$
+    \State $V = V_1 \cup \ldots \cup V_k \cup \comprehension{v_t}{t \in \domain(\phi_1) \bowtie \ldots \bowtie \domain(\phi_k)}$
     \State $E = E_1 \cup \ldots \cup E_k \cup \bigcup_{i \in [1,k]}
-      \comprehension{(\phi_i(\pi_{sch}(\query_i)(t))}{t \in \query_1 \bowtie \ldots \bowtie \query_k}$\Comment{Nodes with in-degrees above 2 are corrected (with $\log_2(k)$ overhead) with an equivalent fan-in tree.}
+      \comprehension{(\phi_i(\pi_{sch}(\query_i)(t))}{t \in \domain(\phi_1) \bowtie \ldots \bowtie \domain(\phi_k)}$\Comment{Nodes with in-degrees above 2 are corrected (with $\log_2(k)$ overhead) with an equivalent fan-in tree.}
     \State $\phi = \phi_1 \cup \ldots \cup \phi_k$
     \State $\ell = \ell_1 \cup \ldots \cup \phi_k$
-    \For{$t \in \query_1 \bowtie \ldots \bowtie \query_k$}
+    \For{$t \in \domain(\phi_1) \bowtie \ldots \bowtie \domain(\phi_k)$}
       \State $\phi(t) = v_t$
       \State $\ell(v_t) = \times$
     \EndFor