\floatname{algorithm}{Listing}

\newcommand\inflFP{\mathsf{infl^{FP}}}
\newcommand\stableFP{\mathsf{stable^{FP}}}
\newcommand\touchedFP{\mathsf{touched^{FP}}}
\newcommand\inflSI{\mathsf{infl^{SI}}}
\newcommand\stableSI{\mathsf{stable^{SI}}}

\newcommand\stable{\mathsf{stable}}
\newcommand\system{\mathsf{system}}


\newcommand\Systems{\mathcal{E}}


\newcommand\init{\mathsf{\textsc{init}}}
\newcommand\eval{\mathsf{\textsc{eval}}}
\newcommand\stabilise{\mathsf{\textsc{stabilise}}}
\newcommand\solve{\mathsf{\textsc{solve}}}
\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''. The set of all equation
  systems is denoted by $\Systems$;

  An equation system can be considered as a function $\system : X
  \to ((X \to \D) \to \D)$; $\system[x] = e_x$, mapping from each
  variable to its right-hand-side.

  An equation system can also be considered as a function $\system
  : (X \to \D) \to (X \to \D)$, $\system(\rho)[x] = e_x(\rho)$,
  taking a variable assignment $\rho$ and returning the result of each
  $e_x$'s evaluation in $\rho$.

  These two functional forms are equivalent and correspond to changing
  the argument order in the following way: $\system(\rho)[x] =
  \system[x](\rho)$. It is only for convenience that we use both
  forms.
\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}
    \Function {solvefixpoint} {$\system \in \Systems$}
      \State $k = 0$
      \State $\rho_0 = \underline{\infty}$
      \Repeat
        \State $\rho_{k+1} = \system( \rho_{k} )$
        \State $k = k + 1$
      \Until {$\rho_{k-1} = \rho_k$}
      \State \Return $\rho_k$
    \EndFunction
  \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, y) \\
    y &= x - 1
  \end{align*}
  \caption{An example equation system for which Kleene iteration will
    not terminate}
  \label{fig:kleene-infinite}
\end{figure}

This particular non-termination of Kleene iteration is not a
significant problem in our application, however, as it has been shown
in the work of Gawlitza
et. al. \cite{Gawlitza:2007:PFC:1762174.1762203}, that Kleene
iteration in this specific case is restricted to at most $|X|$
iterations.

\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. The algorithm is presented in
Listing \ref{algo:w-dfs}.

\begin{algorithm}
  \begin{algorithmic}
    \Globals
    \begin{tabularx}{0.9\textwidth}{rX}
      $D$ & $\in (X \to \D)$, a mapping from variables to their current
      values \\
      
      $\inflFP$ & $\in (X \to 2^X)$, a mapping from a variable to the
      variables it \emph{may} influence \\

      $\stableFP$ & $\subseteq X$, a set of ``stable'' variables \\

      $\system$ & $\in \Systems$, an equation system \\
    \end{tabularx}
    \EndGlobals
  \end{algorithmic}

  \begin{algorithmic}
    \Function {solvefixpoint} {$s \in \Systems$}
      \State $D = \underline{\infty}$
      \State $\inflFP = \{x \mapsto \emptyset \mid \forall x \in X\}$
      \State $\stableFP = \emptyset$
      \State $\system = s$
      \State \Return $\lambda x . (\solve(x); D[x])$
    \EndFunction
    \label{algo:w-dfs:init}
  \end{algorithmic}

  \begin{algorithmic}
    \Function {eval} {$x \in X$, $y \in X$}
    \Comment{Evaluate $y$, track $x$ depends on $y$}
      \State $\solve(y)$
      \State $\inflFP[y] = \inflFP[y] \cup \{x\}$
      \State \Return $D[y]$
    \EndFunction 
  \end{algorithmic}

  \begin{algorithmic}
    \Procedure {solve} {$x \in X$}
    \Comment{Solve a $x$ and place its value in $D$}
    \If {$x \not \in \stableFP$}
      \State $e_x = \system[x]$
      \State $\stableFP = \stableFP \cup \{x\}$
      \State $v = e_x( \lambda y . \eval(x, y) )$
      \If {$v \ne D[x]$}
        \State $D = \{ x \mapsto v \} \oplus D$
        \State $W = \inflFP[x]$
        \State $\inflFP(x) = \emptyset$
        \State $\stableFP = \stableFP \backslash W$
        \For {$v \in W$}
          \State $\solve(v)$
        \EndFor
      \EndIf
    \EndIf
    \EndProcedure
  \end{algorithmic}

  \caption{The W-DFS mentioned 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 function $\init$ in Listing \ref{algo:w-dfs:init} is of the type
$\Systems \to (X \to D)$. It takes an equation system and returns a
function to query the greatest fixpoint. Each fixpoint value is solved
on demand and will only evaluate the subset of the equation system
which is required to ensure the correct solution is found.

The algorithm performs no evaluation of any variables unless their
value is requested (whether externally or internally). Once a value is
requested the right-hand-side is evaluated, which may in turn request
the values of other variables (through the $\eval$ function).

In the case of mutually-recursive variables this would result in
infinite recursion, as each time the variable's value must be
calculated it must first calculate its own value. In order to prevent
this a $\stable$ set is maintained of variables whose values are
either currently being computed or whose values have already been
computed. This set provides a ``base'' for the recursion to terminate,
at which point the value is simply looked up in $D$.

Whenever a variable's value changes it will ``destabilise'' and
re-evaluate any variables which \emph{may} depend on it for their
values, as they may change their value if re-evaluated (and hence are
not stable). If these values change then they, too, will destabilise
their dependent variables, and so on.

The W-DFS algorithm provides us with a \emph{local},
\emph{demand-driven} solver for greatest fixpoints of monotonic
fixpoint problems.




\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 to
the W-DFS algorithm presented above.

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}
        $\sigma$ & $\in E_{\max} \to E$, a $\max$ strategy \\

        $\system$ & $\in \Systems$, an equation
        system \\

        $\rho$ & $\in (X \to D)$, a variable assignment \\

        $P_{\max}$ & $ \in ((E_{\max} \to E), (X \to \D)) \to (E_{\max}
        \to E)$, a $\max$-strategy improvement operator \\
      \end{tabularx}
    \EndAssumptions

    \Function {solvestrategy} {$\system \in \Systems$}
      \State $k = 0$
      \State $\sigma_0 = \lambda x . -\infty$
      \State $\rho_0 = \underline{-\infty}$
      \Repeat
        \State $\sigma_{k+1} = P_{\max}(\sigma_k, \rho_k)$
        \State $\rho_{k+1} = \solve \fixpoint(\sigma_{k+1}(\system))$
        \State $k = k + 1$
      \Until {$\sigma_{k-1} = \sigma_k$}
      \State \Return $\sigma_k$
    \EndFunction
  \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}
      $\sigma$ & $\in (E_{\max} \to E)$, a mapping from
      $\max$-expressions to a subexpression \\

      $\inflSI$ & $\in (E_{\max} \to 2^{E_{\max}}$, 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 \\

      $\system$ & $\in \Systems$, an equation system \\
      
      $P_{\max}$ & $\in (E_{\max}, E, (X \to \D)) \to E$, a function
      mapping from a $\max$-expression, the current strategy and a
      variable assignment to a better subexpression than the current
      strategy in that variable assignment \\
    \end{tabularx}
    \EndGlobals
  \end{algorithmic}

  \begin{algorithmic}
    \Function {solvestrategy} {$s \in \Systems$}
      \State $\sigma = \underline{-\infty}$
      \State $\inflSI = \{x \mapsto \emptyset \mid \forall x \in E_{\max}\}$
      \State $\stableSI = \emptyset$
      \State $\system = s$
      \State \Return $\lambda x . (\solve(x); \sigma[x])$
    \EndFunction
    \label{algo:w-dfs:init}
  \end{algorithmic}

  \begin{algorithmic}
    \Function {eval} {$x \in E_{\max}$, $y \in E_{\max}$}
      \State $\solve(y)$
      \State $\inflSI[y] = \inflSI[y] \cup \{x\}$
      \State \Return $\sigma[y]$
    \EndFunction
  \end{algorithmic}

  \begin{algorithmic}
    \Procedure {solve} {$x \in E_{\max}$}
    \If {$x \not \in \stableSI$}
      \State $\stableSI = \stableSI \cup \{x\}$
      \State $\rho = \solve \fixpoint(\sigma(\system))$
      \State $v = P_{\max}(x, \sigma[x], \lambda y . \eval(x, y))$
      \If {$v \ne \sigma[x]$}
        \State $\sigma = \{ x \mapsto v\} \oplus \sigma$
        \State $W = \inflSI[x]$
        \State $\inflSI(x) = \emptyset$
        \State $\stableSI = \stableSI \backslash W$
        \For {$v \in W$}
        \State $\solve(v)$
        \EndFor
      \EndIf
    \EndIf
    \EndProcedure
  \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 time we
seek to improve our strategy we must compute at least a subset of the
fixpoint from a base of no information. Ideally we would be able to
provide some channel for communication between the two W-DFS instances
to reduce the amount of work that must be done at each stage. By
supplying each other with information about which parts of the system
have changed we can jump in partway through the process of finding a
solution, thereby avoiding repeating work that we have already done.

The new \emph{Local Demand-driven Strategy Improvement} algorithm,
\emph{LDSI}, seeks to do this. By adding an ``invalidate'' procedure
to the fixpoint iteration we provide a method for the $\max$-strategy
to invalidate fixpoint variables, and we can modify the fixpoint
iteration algorithm to report which variables it has modified with
each fixpoint iteration step.

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 each variable as it is
requested, tracking which values have changed since the last time they
iterated. The $\max$-strategy iteration then uses this list of changed
variables to determine which portions of the current strategy must be
re-stabilised once more. This process continues until the
$\max$-strategy stabilises (at which point the fixpoint will also
stabilise).

This entire approach is \emph{demand driven}, and so any necessary
evaluation of strategies or fixpoint values is delayed until the point
when it is actually required. This means that the solver will be
\emph{local}.






This algorithm is presented in three parts.

Section \ref{sec:ldsi:top-level} presents the global state and entry
point to the algorithm.

Section \ref{sec:ldsi:max-strategy} presents the fixpoint portion of
the algorithm. This portion bears many similarities to the W-DFS
algorithm presented in Section \ref{sec:wdfs}, with a few
modifications to track changes and allow external invalidations of
parts of the solution.

Section \ref{sec:ldsi:fixpoint} presents the $\max$-strategy portion
of the algorithm. This portion is quite similar to the Adapted W-DFS
algorithm presented in Section \ref{sec:adapted-wdfs}, with some
modifications to allow for communication with the fixpoint portion of
the algorithm.

Section \ref{sec:ldsi:correctness} argues the correctness of this
approach for finding the least fixpoints of monotonic, expansive
equation systems involving $\max$-expressions.

\subsection{Top level} \label{sec:ldsi:top-level}

\begin{algorithm}[H]
  \begin{algorithmic}
    \Globals
    \begin{tabularx}{0.9\textwidth}{rX}
      $D$ & $\in (X \to \D)$, a mapping from variables to their current
      values \\
      
      $\inflFP$ & $\in (X \to 2^X)$, a mapping from a variable to the
      variables it \emph{may} influence \\

      $\stableFP$ & $\subseteq X$, a set of ``stable'' variables \\

      $\touchedFP$ & $\subseteq X$, a set of variables which have been
      ``touched'' by the fixpoint iteration, but not yet acknowledged
      by the $\max$-strategy iteration \\

      \\

      $\sigma$ & $\in (E_{\max} \to E)$, a mapping from
      $\max$-expressions to a subexpression \\

      $\inflSI$ & $\in (E_{\max} \to 2^{E_{\max}}$, 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 \\

      $P_{\max}$ & $\in (E_{\max}, E, (X \to \D)) \to E$, a function
      mapping from a $\max$-expression, the current strategy and a
      variable assignment to a better subexpression than the current
      strategy in that variable assignment \\
      
      \\

      $\system$ & $\in \Systems$, an equation system \\
    \end{tabularx}
    \EndGlobals
  \end{algorithmic}
  \caption{Global variables used by the LDSI algorithm}
  \label{algo:ldsi:fixpoint:globals}
\end{algorithm}

These variables are mostly just a combination of the variables found
in Sections \ref{sec:wdfs} and \ref{sec:adapted-wdfs}. The only
exception is $\touchedFP$, which is a newly introduced variable to
track variables which are touched by the fixpoint iteration steps.



\subsection{$\max$-strategy iteration} \label{sec:ldsi:max-strategy}
\subsection{Fixpoint iteration} \label{sec:ldsi:fixpoint}
\subsection{Correctness} \label{sec:ldsi:correctness}

THIS IS THE END OF WHERE I'M UP TO! The stuff after this still needs
to be fixed up to use the new variables and have the ``touched'' thing
properly put in.

\endinput


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) \\

      $\system$ & 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.
