\floatname{algorithm}{Listing} \newcommand\stable{\mathsf{stable}} \newcommand\eval{\mathsf{\textsc{eval}}} \newcommand\stabilise{\mathsf{\textsc{stabilise}}} \newcommand\solve{\mathsf{\textsc{solve}}} \newcommand\system{\mathsf{system}} \newcommand\invalidate{\mathsf{\textsc{invalidate}}} \newcommand\fixpoint{\mathsf{\textsc{fixpoint}}} \newcommand\strategy{\mathsf{\textsc{strategy}}} \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 (in the best case, and in most practical cases) 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. The subset of expressions of the form $\max(e_1, e_2, ... e_n)$, with $e_1, e_2, ..., e_n \in E$ are referred to as \emph{$\max$-expressions}, denoted by $E_{\max} \subset E$. \end{definition} \begin{definition} \textbf{Equation System:} $\{ x = e_x \mid x \in X, e_x \in E \}$. The values $x \in X$ are called ``variables'' while the values $e_x \in E$ are called ``right-hand-sides''. 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 or expression $x$ is said to \emph{depend on} $y$ if a change to the value of $y$ induces a change in the value of $x$. If $x$ depends on $y$ then $y$ is said to \emph{influence} $x$. \end{definition} \begin{definition} \textbf{Local:} A solver is said be local if, for some $e_x \in E$, the evaluation of $e_x$ only requires the evaluation of other variables which $e_x$ may depend on. \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}[H] \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, requiring the evaluation of many right-hand-sides repeatedly for the same value. Thus an approach which can evaluate smaller portions of the system in each iteration would be a significant improvement. An additional deficiency of Kleene iteration is that it is not guaranteed to terminate. In some cases Kleene iteration must iterate an infinite number of times in order to reach a fixpoint. An example system is presented as Figure \ref{fig:kleene-infinite}. In this case $x$ will take the value of $0$ in the first iteration, then $y$ will evaluate to $-1$. In the next iteration $x$ will also take the value $-1$, thereby requiring $y$ to take the value $-2$. This will continue without bound, resulting in the Kleene iteration never reaching the greatest fixpoint of $\{ x \mapsto -\infty, y \mapsto -\infty \}$. \begin{figure}[H] \begin{align*} x &= \min(0, x) \\ y &= x - 1 \end{align*} \caption{An example equation system for which Kleene iteration will not terminate} \label{fig:kleene-infinite} \end{figure} \subsection{W-DFS algorithm} \label{sec:wdfs} 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}[H] \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 \emph{may} 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 \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 $D = \{ x \mapsto v \} \oplus D$ \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} The W-DFS algorithm over-approximates the dependencies for each variable, keeping a map of which variables \emph{may} depend on other variables. The particular variation of W-DFS presented here is designed to return the \emph{greatest} fixpoint of an equation system consisting of only \emph{monotonic} expressions. \section{$\max$-strategy Iteration} The $\max$-strategy iteration can be viewed as an accelerated 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 and our ``values'' to be their subexpressions then we can solve for the best $\max$-strategy using a similar approach and our ``comparison'' Because $\max$-strategy iteration is so similar to a standard fixpoint problem it is possible \subsection{Naive approach} \begin{algorithm}[H] \begin{algorithmic} \Assumptions \begin{tabularx}{0.9\textwidth}{rX} $\sigma $:&$ E_{\max} \to E$, a $\max$ strategy \\ $\varepsilon $:&$ (X \to \D) \to (X \to \D)$, an equation system \\ $\rho $:&$ (X \to D)$, a variable assignment \\ $P_{\max} $:&$ ((E_{\max} \to E_{\max}), (X \to \D)) \to (E_{\max} \to E_{\max})$, a $\max$-strategy improvement operator \end{tabularx} \EndAssumptions \State $n = 0$ \State $\sigma_0 = \lambda x . -\infty$ \State $\rho_0 = \underline{-\infty}$ \Repeat \State $\sigma_{n+1} = P_{\max}(\sigma, \rho)$ \State $\rho_{n+1} = \sigma(\varepsilon)[ \rho_{n} ]$ \State $n = n + 1$ \Until {$\sigma_{n-1} = \sigma_n$} \State \Return $\sigma_n$ \end{algorithmic} \caption{The naive approach to strategy iteration} \label{algo:naive-strategy} \end{algorithm} \subsection{Adapted W-DFS algorithm} \label{sec:adapted-wdfs} 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}[H] \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 $v = bestStrategy(f, \lambda y . \eval(x, y))$ \If {$v \ne \sigma[x]$} \State $\sigma = \{ x \mapsto v\} \oplus \sigma$ \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{Local Demand-driven Strategy Improvement (LDSI)} 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 \emph{Local Demand-driven Strategy Improvement} algorithm, \emph{LDSI}, seeks to do this. By adding an ``invalidate'' function to both W-DFS instances we provide an interface for the two sides of the algorithm to indicate which values have changed and ``destabilise'' that portion of the system. This essentially results in a $\max$-strategy iteration which, at each strategy-improvement step, invalidates a portion of the current fixpoint iteration which may depend on the changed strategy. The fixpoint iteration then re-stabilises itself by evaluating what values have been changed, but only when such values are requested by the $\max$-strategy iteration. The $\max$-strategy can continue as normal, invalidating the fixpoint-iteration as required, and simply assume that the fixpoint-iteration will provide it with the correct values from the greatest fixpoint. This entire approach is demand driven, and so any necessary evaluation is delayed until the point when it is actually required. Additionally, if it is not necessary to evaluate a particular right hand side in order to make a decision then the algorithm will attempt to avoid evaluating it. This algorithm is presented in two parts. Listings \ref{algo:ldsi:fixpoint:globals}, \ref{algo:ldsi:fixpoint:eval}, \ref{algo:ldsi:fixpoint:invalidate}, \ref{algo:ldsi:fixpoint:solve} and \ref{algo:ldsi:fixpoint:stabilise} present 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:ldsi-correctness}. \begin{algorithm}[H] \begin{algorithmic} \Globals \begin{tabularx}{0.9\textwidth}{rX} $D$ & $X \to \D$ - a mapping from variables to values, starting at $\{ x \mapsto \infty \}$ \\ $\sigma$ & $E_{\max} \to E$ - a mapping from $\max$-expressions to their sub-expressions (a $\max$-strategy) \\ $I_{X,X}$ & $X \to X$ - a mapping from a variable to the variables it influences \\ $I_{\max,\max}$ & $E_{\max} \to E_{\max}$ - a mapping from a $\max$-expression to the $\max$-expressions it influences \\ $U_{X}$ & The set of all variables whose values have not stabilised to a final fixpoint value (unstable variables) \\ $S_{\max}$ & The set of all $\max$ expressions whose strategies have stabilised to their final strategy (stable $\max$-expressions) \\ $\varepsilon$ & The equation system, a mapping from a variable to its associated function \\ \end{tabularx} \EndGlobals \end{algorithmic} \caption{Global variables used by the LDSI algorithm} \label{algo:ldsi:fixpoint:globals} \end{algorithm} A few things are of particular note for the global variables. In particular the difference between $U_X$ being an unstable set and $S_{\max}$ being a stable set. In reality these two are entirely equivalent, but because the fixpoint-iteration will be started as being entirely ``stable'' (with values of $-\infty$) it is of more practical benefit to avoid the extra work populating the ``stable'' set by instead storing unstable values. The other variables are just the state from each of the previous algorithms for intelligently performing $\max$-strategy iteration and fixpoint iteration (as were presented in Sections \ref{sec:wdfs} and \ref{sec:adapted-wdfs}). $D$ and $I_{X,X}$ are taken from the W-DFS algorithm, while $\sigma$ and $I_{\max,\max}$ are taken from the Adapted W-DFS algorithm. \begin{algorithm}[H] \begin{algorithmic} \Function {evalfixpoint} {$x \in X$, $y \in X$} \State $\solve \fixpoint(y)$ \State $I_{X,X}[y] = I_{X,X}[y] \cup \{x\}$ \State \Return $D[y]$ \EndFunction \end{algorithmic} \caption{Utility function used to track fixpoint variable dependencies.} \label{algo:ldsi:fixpoint:eval} \end{algorithm} This procedure is exactly the same as the equivalent method in the W-DFS algorithm. It allows us to more easily track dependencies between fixpoint variables by injecting this function as our variable-lookup function. It then both calculates a new value for the variable (the $\solve \fixpoint$ call) and notes the dependency between $x$ and $y$. \begin{algorithm}[H] \begin{algorithmic} \Function {invalidatefixpoint} {$x \in X$} \Comment{Invalidate a fixpoint variable} \If {$x \not \in U_X$} \State $U_X = U_X \cup \{x\}$ \State $D[x] = \infty$ \State $W = I[x]$ \State $I[x] = \emptyset$ \For {$v \in W$} \State $\invalidate \fixpoint(v)$ \EndFor \EndIf \EndFunction \end{algorithmic} \caption{Fixpoint subroutine called from the $\max$-strategy iteration portion to invalidate fixpoint variables} \label{algo:ldsi:fixpoint:invalidate} \end{algorithm} This procedure is not called in the fixpoint iteration process, but is rather the method by which the $\max$-strategy iteration can communicate with the fixpoint-iteration. It allows the $\max$-strategy iteration to inform the fixpoint-iteration which values have changed and will require re-evaluation. This makes it possible to only re-evaluate a partial system (the solving of which is also be delayed until requested by the $\solve \fixpoint$ procedure). \begin{algorithm}[H] \begin{algorithmic} \Function {solvefixpoint} {$x$} \Comment{Solve a specific expression and place its value in $D$} \If {$x \in U_X$} \State $f = \system[x]$ \State $U_X = U_X \backslash \{x\}$ \State $v = \sigma(\system[x])( \lambda y . \eval \fixpoint(x, y) )$ \If {$v \ne D[x]$} \State $D = \{ x \mapsto v \} \oplus D$ \State $W = I_{X,X}[x]$ \State $I_{X,X}[x] = \emptyset$ \State $\invalidate \strategy(x)$ \State $\stable = \stable \backslash W$ \For {$v \in W$} \State $\solve \fixpoint(v)$ \EndFor \EndIf \EndIf \EndFunction \end{algorithmic} \caption{The subroutine of the fixpoint iteration responsible for solving for each variable} \label{algo:ldsi:fixpoint:solve} \end{algorithm} After an evaluation of the $\solve \fixpoint$ procedure, the variable supplied as its argument will have been stabilised within the current $\max$-strategy. This means that it will have taken on the same value as it would take in the greatest fixpoint of this system. Because the $\solve \fixpoint$ calls need not be made immediately it's possible for a variable to be invalidated by several $\max$-strategy improvements without being re-evaluated. \begin{algorithm}[H] \begin{algorithmic} \Function {evalstrategy} {$x \in E_{\max}$, $y \in E_{\max}$} \Comment{Evaluate $y$ for its value and note that when $y$ changes, $x$ must be re-evaluated} \State $\solve \strategy(y)$ \State $I_{\max,\max}[y] = I_{\max,\max}[y] \cup \{x\}$ \State \Return $\sigma[y]$ \EndFunction \Function {evalstrategyfixpoint} {$x \in E_{\max}$, $y \in X$} \Comment{Evaluate $y$ for its value and note that when $y$ changes, $x$ must be re-evaluated} \State $\solve \fixpoint(y)$ \State $I_{\max,\max}[y] = I_{\max,\max}[\system[y]] \cup \{x\}$ \State \Return $D[y]$ \EndFunction \end{algorithmic} \label{algo:ldsi:strategy:eval} \caption{Evaluate a portion of the $\max$-strategy and note a dependency} \end{algorithm} The $\eval \strategy$ function is essentially the same as the $\eval \fixpoint$ function, except that it notes the dependencies in a different variable, $I_{\max,\max}$. This is because the dependency information for the $\max$-strategy iteration is entirely separate to that of the fixpoint iteration. The $\solve \fixpoint$ calls in $\eval \strategy \fixpoint$ are top-level calls, meaning that upon their return we know that $D$ contains the value it would take in the greatest fixpoint of the current strategy $\sigma$. This function, therefore, acts as a simple intermediate layer between the fixpoint-iteration and the $\max$-strategy iteration to allow for dependencies to be tracked. \begin{algorithm}[H] \begin{algorithmic} \Function {invalidatestrategy} {$x \in X$} \Comment{$x$ is a \emph{variable}} \For {$v \in I_{\max,\max}(\system[x])$} \Comment{$v$ is influenced by $x$} \State $\invalidate \strategy (v)$ \EndFor \EndFunction \Function {invalidatestrategy} {$x \in E_{\max}$} \Comment{$x$ is a \emph{$\max$-expression}} \If {$x \in S_{\max}$} \State $S_{\max} = S_{\max} \backslash \{x\}$ \For {$v \in I_{\max,\max}$} \Comment {$v$ is influenced by $x$} \State $\invalidate \strategy (v)$ \EndFor \EndIf \EndFunction \end{algorithmic} \label{algo:ldsi:strategy:invalidate} \caption{Evaluate a portion of the $\max$-strategy and note a dependency} \end{algorithm} Invalidating the $\max$-strategy iteration is slightly more complicated than invalidating the fixpoint iteration stage. As the invalidation interface consists of variables, we must first translate a variable into a $\max$-expression (which is easily done by looking it up in the equation system). We must then invalidate the strategies for each variable which depends on this resultant $\max$-expression. The invalidation for $\max$-expressions consists of transitively invalidating everything which depends on the $\max$-expression, as well as itself. \begin{algorithm}[H] \begin{algorithmic} \Function {solvestrategy} {$x \in E_{\max}$} \Comment{Solve a specific variable and place its value in $\sigma$} \If {$x \not \in S_{\max}$} \State $S_{\max} = S_{\max} \cup \{x\}$ \State $\sigma_{dynamic} = \lambda y . \eval \strategy(x,y)$ \State $e = P_{\max}(\sigma_{dynamic}, \lambda y . \eval \strategy \fixpoint(x, y))(x)$ \If {$e \ne \sigma[x]$} \State $\sigma = \{ x \mapsto e \} \oplus \sigma$ \State $\invalidate \fixpoint(\system^{-1}(x))$ \State $S_{\max} = S_{\max} \backslash I[x]$ \For {$v \in I[x]$} \State $\solve(v)$ \EndFor \EndIf \EndIf \EndFunction \end{algorithmic} \caption{The $\max$-strategy portion of the Combined W-DFS.} \label{algo:ldsi:solve} \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. The fixpoint iteration step invalidates anything that \emph{might} depend on $x$ while it invalidates $x$, thereby ensuring that any further calls to $\solve \fixpoint$ will result in a correct value for the given strategy. The strategy-iteration step invalidates anything that \emph{might} depend on $\system[x]$ while it invalidates $x$, thereby ensuring that any further calls to $\solve \strategy$ will result in a correct value for the given strategy.