From a080384f843f51692703585170f7282b82a7d541 Mon Sep 17 00:00:00 2001
From: Carlo Zancanaro <carlo@pc-4w14-0.cs.usyd.edu.au>
Date: Mon, 12 Nov 2012 17:42:39 +1100
Subject: A few minor fixes to contribution.

---
 tex/thesis/contribution/contribution.tex | 30 +++++++++++++++---------------
 1 file changed, 15 insertions(+), 15 deletions(-)

(limited to 'tex/thesis/contribution/contribution.tex')

diff --git a/tex/thesis/contribution/contribution.tex b/tex/thesis/contribution/contribution.tex
index 18d956f..b5d47db 100644
--- a/tex/thesis/contribution/contribution.tex
+++ b/tex/thesis/contribution/contribution.tex
@@ -372,7 +372,7 @@ strategy. This means that each time we improve our strategy we must
 make it greater in at least one $\max$-expression, and no worse in the
 others.
 
-To this end a new function, $P_{\max}: ((E_{\max} \to E), (X \to D))
+To this end a new function, $P_{\max}: ((E_{\max} \to E) \times (X \to D))
 \to (E_{\max} \to E)$, is used below as a ``strategy improvement
 operator''. $P_{\max}$ takes a $\max$-strategy and a variable
 assignment and returns a new $\max$-strategy which constitutes an
@@ -398,14 +398,14 @@ the approach presented in Listing \ref{algo:naive-strategy}.
   \begin{algorithmic}
     \Assumptions
       \begin{tabularx}{0.9\textwidth}{rX}
-        $\sigma$ & $\in E_{\max} \to E$, a $\max$ strategy \\
+        $\sigma$ & $: E_{\max} \to E$, a $\max$ strategy \\
 
         $\system$ & $\in \Systems$, an equation
         system \\
 
-        $\rho$ & $\in X \to D$, a variable assignment \\
+        $\rho$ & $: X \to D$, a variable assignment \\
 
-        $P_{\max}$ & $ \in ((E_{\max} \to E), (X \to \CZ)) \to (E_{\max}
+        $P_{\max}$ & $: ((E_{\max} \to E) \times (X \to \CZ)) \to (E_{\max}
         \to E)$, a $\max$-strategy improvement operator \\
       \end{tabularx}
     \EndAssumptions
@@ -450,10 +450,10 @@ solve $\max$-strategy iteration problems.
   \begin{algorithmic}
     \Globals
     \begin{tabularx}{0.9\textwidth}{rX}
-      $\sigma$ & $\in (E_{\max} \to E)$, a mapping from
+      $\sigma$ & $: (E_{\max} \to E)$, a mapping from
       $\max$-expressions to a subexpression \\
 
-      $\inflSI$ & $\in (E_{\max} \to 2^{E_{\max}}$, a mapping from a
+      $\inflSI$ & $: (E_{\max} \to 2^{E_{\max}}$, a mapping from a
       $\max$-expression to the sub-expressions it influences \\
 
       $\stableSI$ & $\subseteq E_{\max}$, the set of all
@@ -461,7 +461,7 @@ solve $\max$-strategy iteration problems.
 
       $\system$ & $\in \Systems$, an equation system \\
       
-      $P_{\max}$ & $ \in ((E_{\max} \to E), (X \to \CZ)) \to (E_{\max}
+      $P_{\max}$ & $: ((E_{\max} \to E) \times (X \to \CZ)) \to (E_{\max}
       \to E)$, a $\max$-strategy improvement operator \\
     \end{tabularx}
     \EndGlobals
@@ -610,13 +610,13 @@ This algorithm is presented in three parts.
   \begin{algorithmic}
     \Globals
     \begin{tabularx}{0.9\textwidth}{rX}
-      $D$ & $\in X \to \CZ$, a mapping from variables to their current
+      $D$ & $: X \to \CZ$, a mapping from variables to their current
       value \\
 
-      $D_\old$ & $\in X \to \CZ$, a mapping from variables to their
+      $D_\old$ & $: X \to \CZ$, a mapping from variables to their
       last stable value \\
       
-      $\inflFP$ & $\in X \to 2^X$, a mapping from a variable to the
+      $\inflFP$ & $: X \to 2^X$, a mapping from a variable to the
       variables it \emph{may} influence \\
 
       $\stableFP$ & $\subseteq X$, a set of ``stable'' variables \\
@@ -627,17 +627,17 @@ This algorithm is presented in three parts.
 
       \\
 
-      $\sigma$ & $\in E_{\max} \to E$, a mapping from
+      $\sigma$ & $: E_{\max} \to E$, a mapping from
       $\max$-expressions to a subexpression \\
 
-      $\inflSI$ & $\in E_{\max} \to 2^{E_{\max}}$, a mapping from a
+      $\inflSI$ & $: E_{\max} \to 2^{E_{\max}}$, a mapping from a
       $\max$-expression to the sub-expressions it influences \\
 
       $\stableSI$ & $\subseteq E_{\max}$, the set of all
       $\max$-expressions whose strategies are stable \\
 
-      $P_{\max}$ & $ \in ((E_{\max} \to E), (X \to \CZ)) \to (E_{\max}
-      \to E)$, a $\max$-strategy improvement operator \\
+      $P_{\max}$ & $: ((E_{\max} \to E) \times (X \to \CZ)) \to
+      (E_{\max} \to E)$, a $\max$-strategy improvement operator \\
       
       \\
 
@@ -725,7 +725,7 @@ identify \emph{changed} values, rather than only \emph{unstable} ones.
 This procedure is similar to the equivalent method in the W-DFS
 algorithm, except for the fact that $\solve$ has been renamed to
 $\solve \fixpoint$. $\eval \fixpoint$ performs exactly the same
-function as the $\eval$ function in Figure \ref{algo:wdfs}.
+function as the $\eval$ function in Listing \ref{algo:wdfs}.
 
 \begin{algorithm}[H]
   \begin{algorithmic}
-- 
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