diff options
Diffstat (limited to 'tex/thesis/contribution')
| -rw-r--r-- | tex/thesis/contribution/contribution.tex | 30 |
1 files changed, 15 insertions, 15 deletions
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} |