\floatname{algorithm}{Listing} \newcommand\stable{\mathsf{stable}} \newcommand\eval{\mathsf{\textsc{eval}}} \newcommand\solve{\mathsf{\textsc{solve}}} \newcommand\system{\mathsf{system}} \algblockx[Globals]{Globals}{EndGlobals}{\textbf{Globals:\\}}{} \algblockx[Assume]{Assumptions}{EndAssumptions}{\textbf{Assume:\\}}{} \chapter{Contribution} \label{chap:contribution} The main theoretical contribution of this paper is an improvement on a $\max$-strategy improvement algorithm for solving fixpoint equations over the integers with monotonic operators\cite{Gawlitza:2007:PFC:1762174.1762203}. The original algorithm is presented in Section \ref{section:basic-algorithm}. We employ the ideas of Seidl, et al. to design an algorithm which runs in considerably less time than the existing solver. In this chapter we will begin by presenting the Work-list Depth First Search (W-DFS) fixpoint algorithm developed by Seidl, et al.\cite{DBLP:tr/trier/MI96-11}. We will then present a modification to the algorithm to allow it to perform $\max$-strategy iteration rather than fixpoint iteration. The chapter will then conclude with our Local Demand-driven Strategy Improvement (LDSI) algorithm. The existing algorithm as presented in Section \ref{section:basic-algorithm} consists of two iterative operations: fixpoint iteration and max-strategy iteration. Each of these operations consists of naively ``evaluating'' the system repeatedly until a further evaluation yields no change. It is shown by Gawlitza, et al. that these iterations must converge in a finite number of steps\cite{Gawlitza:2007:PFC:1762174.1762203}, but in practice this naive approach performs many more operations than are necessary, in many cases merely re-calculating results which are already known. By making use of some data-dependencies within the equation systems it is possible to reduce the amount of work that is to be done quite considerably. In order to aid our explanation of these algorithms we will now define a few terms and notations. All variables are taken from the set $X$ and all values from the set $\D$. \begin{definition} \textbf{Variable Assignments:} $X \to \D$. A function from a variable to a value in our domain. An underlined value (eg. $\underline{\infty}$) indicates a variable assignment mapping everything to that value. Variable assignments may be combined with $\oplus$ in the following way: \begin{align*} \rho \oplus \varrho = \left\{\begin{array}{lc} \varrho(x) & x \in \mathsf{domain}(\varrho) \\ \rho(x) & \mbox{otherwise} \end{array}\right. \end{align*} \end{definition} \begin{definition} \textbf{Expressions:} For the purposes of this discussion we will consider expressions, $e \in E$, as $e : (X \to \D) \to \D$, a mapping from a variable assignment to the expression's value in that assignment. \end{definition} \begin{definition} \textbf{Equation System:} $\{ x = e_x \mid x \in X, e_x \in E \}$. An equation system can also be considered as a function $\varepsilon : (X \to D) \to (X \to D)$; $\varepsilon[\rho](x) = e_x(\rho)$. \end{definition} \begin{definition} \textbf{Dependencies:} A variable $x$ is said to \emph{depend on} $y$ if a change to the value of $y$ induces a change in the value of $x$. \end{definition} \section{Fixpoint Iteration} \subsection{Kleene Iteration} A simple approach to fixpoint iteration over monotonic equations is to simply iterate over the system repeatedly until a reevaluation results in no change to the values of any variables. This approach will always reach the least/greatest solution if there is one to be found, but it will often perform many more evaluations than are necessary. This algorithm is presented in Listing \ref{algo:kleene}. \begin{algorithm} \begin{algorithmic} \Assumptions \begin{tabularx}{0.9\textwidth}{rX} $\rho $:&$ X \to \D$, a variable assignment \\ $\varepsilon $:&$ (X \to \D) \to (X \to \D)$, an equation system \end{tabularx} \EndAssumptions \State $n = 0$ \State $\rho_0 = \underline{\infty}$ \Repeat \State $\rho_{n+1} = \varepsilon[ \rho_{n} ]$ \State $n = n + 1$ \Until {$\rho_{n-1} = \rho_n$} \State \Return $\rho_n$ \end{algorithmic} \caption{The Kleene iteration algorithm for solving fixpoint equations for their greatest solutions.} \label{algo:kleene} \end{algorithm} For each iteration the entire system is evaluated, irrespective of whether it could possibly have changed value. This results in a considerable inefficiency in practice and thus an approach which can evaluate smaller portions of the system in each iteration would be a sensible improvement. \subsection{W-DFS algorithm} The W-DFS algorithm presented by Seidl, et al. takes into account some form of data-dependencies as it solves the system. This gives it the ability to leave portions of the system unevaluated when it is certain that those values have not changed. \begin{algorithm} \begin{algorithmic} \Globals \begin{tabularx}{0.9\textwidth}{rX} $D : X \to \D$ & a mapping from variables to their current values, starting at $\{ x \mapsto \infty | \forall x \in X \}$ \\ I & A mapping from a variable to the variables which depend on it in their evaluation \\ stable & The set of all variables whose values have stabilised \\ system & The equation system, a mapping from a variable to its associated function \\ \end{tabularx} \EndGlobals something something \end{algorithmic} \begin{algorithmic} \Function {eval} {$x$, $y$} \Comment{Evaluate $y$ for its value and note that when $y$ changes, $x$ must be re-evaluated} \State $\solve(y)$ \State $I[y] = I[y] \cup \{x\}$ \State \Return $D[y]$ \EndFunction \end{algorithmic} \begin{algorithmic} \Function {solve} {$x$} \Comment{Solve a specific variable and place its value in $D$} \If {$x \not \in \stable$} \State $f = \system[x]$ \State $\stable = \stable \cup \{x\}$ \State $v = f( \lambda y . \eval(x, y) )$ \If {$v \ne D[x]$} \State $D = \{ x \mapsto v, \alpha \mapsto D[\alpha] \} \forall \alpha \ne x$ \State $W = I[x]$ \State $I(x) = \emptyset$ \State $\stable = \stable \backslash W$ \For {$v \in W$} \State $\solve(v)$ \EndFor \EndIf \EndIf \EndFunction \end{algorithmic} \caption{The W-DFS alluded to in \cite{DBLP:tr/trier/MI96-11} and presented in \cite{fixpoint-slides}, modified to find greatest-fixpoints of monotonic fixpoint equations} \label{algo:w-dfs} \end{algorithm} \section{$\max$-strategy Iteration} \subsection{Naive approach} TODO: Explanation of the naive approach \subsection{Adapted W-DFS algorithm} The $\max$-strategy iteration can be viewed as a fixpoint problem. We are attempting to find a strategy, $\sigma: E_{\max} \to E$ that will result in the greatest value for each $e \in E_{\max}$. Therefore if we consider our ``variables'' to be $\max$-expressions, our ``values'' to be their subexpressions and our ``comparison'' to be carried out using the result of evaluating the system with that strategy. This, then, allows us to use the W-DFS algorithm to re-evaluate only those parts of the strategy which have changed. Listing \ref{algo:w-dfs-max} presents this variation on W-DFS. \begin{algorithm} \begin{algorithmic} \Globals \begin{tabularx}{0.9\textwidth}{rX} $\sigma$ & A mapping from $\max$-expressions to their current sub-expressions, starting by mapping to the first sub-expression \\ I & A mapping from a $\max$-expression to the sub-expressions which depend on it in their evaluation \\ stable & The set of all $\max$-expressions whose strategies have stabilised \\ system & The equation system, a mapping from a variable to its associated function \\ bestStrategy & A function $(E_{\max}, (X \to D)) \to E$ mapping from an expression and a variable \\& assignment to the greatest subexpression in that context \end{tabularx} \EndGlobals \Function {eval} {$x$, $y$} \Comment{Evaluate $y$ for its value and note that when $y$ changes, $x$ must be re-evaluated} \State $\solve(y)$ \State $I[y] = I[y] \cup \{x\}$ \State \Return $\sigma[y]$ \EndFunction \Function {solve} {$x$} \Comment{Solve a specific expression and place its value in $\sigma$} \If {$x \not \in \stable$} \State $f = \system[x]$ \State $\stable = \stable \cup \{x\}$ \State $v = bestStrategy(x, \lambda y . \eval(x, y))$ \If {$v \ne \sigma[x]$} \State $\sigma = \{ x \mapsto v, \alpha \mapsto \sigma[\alpha] \} \forall \alpha \ne x $ \State $W = I[x]$ \State $I(x) = \emptyset$ \State $\stable = \stable \backslash W$ \For {$v \in W$} \State $\solve(v)$ \EndFor \EndIf \EndIf \EndFunction \end{algorithmic} \caption{W-DFS, this time modified to find the best $\max$-strategy.} \label{algo:w-dfs-max} \end{algorithm} \section{Combined W-DFS} W-DFS can be used to speed up both the $\max$-strategy iteration and the fixpoint iteration as two independent processes, but each fixpoint-iteration step still requires at least one complete evaluation of the equation system per $\max$-strategy improvement step. Ideally we would be able to adapt the W-DFS algorithm so that the fixpoint-iteration and $\max$-strategy iteration steps could provide some information to each other about what values have changed so that at each step only a subset of the entire system would have to be evaluated. The new algorithm, \emph{Combined W-DFS} seeks to do this. By adding an ``invalidate'' method to both W-DFS instances we provide an interface for the two sides of the algorithm to indicate which values have changed. This gives the other side enough information to avoid evaluating irrelevant portions of the equation system. This algorithm is presented in two parts. Listing \ref{algo:combined-fixpoint} presents the fixpoint-iteration portion of the algorithm, while Listing \ref{algo:combined-max} presents the $\max$-strategy portion. The correctness of this new algorithm is argued in \ref{sec:combined-correctness}. \begin{algorithm} \begin{algorithmic} \Globals \begin{tabularx}{0.9\textwidth}{rX} D & A mapping from variables to their current values, starting at $\{ x \mapsto \infty \}$ \\ I & A mapping from a variable to the variables which depend on it in their evaluation \\ stable & The set of all variables whose values have stabilised \\ system & The equation system, a mapping from a variable to its associated function \\ \end{tabularx} \EndGlobals \Function {eval} {$x$, $y$} \Comment{Evaluate $y$ for its value and note that when $y$ changes, $x$ must be re-evaluated} \State $\solve(y)$ \State $I[y] = I[y] \cup \{x\}$ \State \Return $D[y]$ \EndFunction \Function {invalidate} {$x$} \If {$x \in \stable$} \State $\stable = \stable \backslash \{x\}$ \State $D[x] = \infty$ \State $W = I[x]$ \State $I[x] = \emptyset$ \For {$v \in W$} invalidate(v) \EndFor \EndIf \EndFunction \Function {solve} {$x$} \Comment{Solve a specific expression and place its value in $D$} \If {$x \not \in \stable$} \State $f = \system[x]$ \State $\stable = \stable \cup \{x\}$ \State $v = f( \lambda y . \eval(x, y) )$ \If {$v \ne D[x]$} \State $D = \{ x \mapsto v, \alpha \mapsto D[\alpha] \} \forall \alpha \ne x$ \State $W = I[x]$ \State $I(x) = \emptyset$ \State strategy::invalidate($x$) \State $\stable = \stable \backslash W$ \For {$v \in W$} \State $\solve(v)$ \EndFor \EndIf \EndIf \EndFunction \end{algorithmic} \caption{The fixpoint portion of the Combined W-DFS.} \label{algo:combined-fixpoint} \end{algorithm} \begin{algorithm} \begin{algorithmic} \Globals \begin{tabularx}{0.9\textwidth}{rX} $\sigma$ & A mapping from $\max$-expressions to their current sub-expressions, starting by \\& mapping to the first sub-expression \\ D & A \emph{dynamic} variable assignment which will stay updated as $\sigma$ changes \\ $I$ & A mapping from a $\max$-expression to the sub-expressions which depend on it \\& in their evaluation \\ $I_V$ & A mapping from a variable to the $\max-$ expressions which depend on it in their \\& evaluation \\ stable & The set of all $\max$-expressions whose strategies have stabilised \\ system & The equation system, a mapping from a variable to its associated function \\ bestStrategy & A function $(E_{\max}, (X \to D)) \to E$ mapping from an expression and a variable \\& assignment to the greatest subexpression in that context \end{tabularx} \EndGlobals \Function {eval} {$x$, $y$} \Comment{Evaluate $y$ for its value and note that when $y$ changes, $x$ must be re-evaluated} \State $\solve(y)$ \State $I[y] = I[y] \cup \{x\}$ \State \Return $\sigma[y]$ \EndFunction \Function {invalidate} {$x \in X$} \Comment{X is vars} \State $\stable = \stable \backslash I[x]$ \State $W = I_v[x]$ \State $I_V = \emptyset$ \For {$v \in W$} \State solve(v) \EndFor \EndFunction \Function {solve} {$x$} \Comment{Solve a specific variable and place its value in $\sigma$} \If {$x \not \in \stable$} \State $f = \system[x]$ \State $\stable = \stable \cup \{x\}$ \State $v = bestStrategy(x, \lambda y . \eval(x, y))$ \If {$v \ne \sigma[x]$} \State $\sigma = \{ x \mapsto v, \alpha \mapsto \sigma[\alpha] \} \forall \alpha \ne x $ \State $W = I[x]$ \State $I(x) = \emptyset$ \State fixpoint::invalidate$(\mathsf{lookupVarFor}(x))$ \State $\stable = \stable \backslash W$ \For {$v \in W$} \State $\solve(v)$ \EndFor \EndIf \EndIf \EndFunction \end{algorithmic} \caption{The $\max$-strategy portion of the Combined W-DFS.} \label{algo:combined-max} \end{algorithm} \subsection{Correctness} \label{sec:combined-correctness} This algorithm relies on the correctness of the underlying W-DFS algorithm. This algorithm was presented in \cite{DBLP:tr/trier/MI96-11}. If we assume that W-DFS is correct then we only have to prove that the combination algorithm is correct. For this it is sufficient to show that the invalidate calls in both directions preserve the correctness of the original algorithm. // TODO finish this. General idea: \begin{itemize} \item Invalidate calls from fixpoint $\to$ max strategy are correct if the calls the other way are, because it completely re-solves the equations \item Invalidate calls from max strategy $\to$ fixpoint are correct because they effectively ``reset'' that part of the system, creating it to be entirely re-calculated. \end{itemize}