\floatname{algorithm}{Listing}

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

\newcommand\stable{\mathsf{stable}}
\newcommand\system{\mathcal{E}} %\mathsf{system}}
\newcommand\parent{\mathsf{parent}}
\newcommand\parents{\mathsf{parents}}
\newcommand\val{\mathsf{val}}
\newcommand\old{\mathsf{old}}
\newcommand\best{\mathsf{best}}
\newcommand\all{\mathsf{all}}

\newcommand\fix{\mathsf{fix}}
\newcommand\solnFP{\mathsf{soln^{FP}}}
\newcommand\RFP{\mathsf{R^{FP}}}
\newcommand\solnSI{\mathsf{soln^{SI}}}
\newcommand\RSI{\mathsf{R^{SI}}}

\newcommand\post{\mathsf{post}}
\newcommand\pre{\mathsf{pre}}
\newcommand\last{\mathsf{last}}

\newcommand\Operators{\mathcal{O}}
\newcommand\Systems{\mathbb{E}}

\newcommand\improve{\mathsf{\textsc{improve}}}
\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:\\}}{}
\algblockdefx[ReturnLambdaFn]{ReturnLambdaFn}{EndReturnLambdaFn}[1]{\textbf{return function} (#1)}{\textbf{end function}}


\chapter{Demand-driven strategy improvement} \label{chap:contribution}

In this chapter the main theoretical contribution of this thesis is
presented: an extension on a $\max$-strategy iteration algorithm for
solving fixpoint equations over the integers with monotonic
operators~\cite{Gawlitza:2007:PFC:1762174.1762203}. We devise an
algorithm which performs considerably better for practical equation
systems.

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 discuss a modification
of the algorithm to allow it to perform $\max$-strategy iteration. The
chapter will conclude with our Demand-driven Strategy
Improvement (DSI) algorithm to perform efficient strategy iteration
and fixpoint iteration. % TODO: fix me, or something

The existing 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 results in no change. Gawlitza et al.~proved that
these iterations converge in a finite number of
steps\cite{Gawlitza:2007:PFC:1762174.1762203} , but in practice this
naive approach performs a large number of re-calculations of results
which are already known.
% TODO: add something here about variable dependencies

\section{Preliminaries}

% TODO: something about equation systems, variables and domains (very generally)
In order to aid our explanation of these algorithms we will now define
a few terms and notations. We will assume a fixed, finite set $X$ of
variables. We will also define $\CZ = \Z \cup \{-\infty, \infty\}$.

\begin{definition}
  A \textbf{variable assignment} is a partial function, $\rho:
  X \leadsto \CZ$ that maps variables to values. The domain of a
  variable assignment, $\rho$, is represented by
  $\mathsf{domain}(\rho) \subseteq X$. An underlined value,
  $\underline{a}$, indicates a variable assignment $\{ x \mapsto
  a \mid \forall x \in X \}$.

  Variable assignments $\rho : X \to \CZ$ and $\varrho : X \to \CZ$
  may be composed with the $\oplus$ operator as follows:
  \begin{align*}
    (\rho \oplus \varrho)(x) = \left\{\begin{array}{lc}
        \varrho(x) & x \in \mathsf{domain}(\varrho) \\
        \rho(x) & \mbox{otherwise}
      \end{array}\right., \forall x \in X
  \end{align*} 
\end{definition}

\begin{definition}
  The class of \textbf{expressions}, $E$, is defined inductively:
  \begin{itemize}
    \item
      If $z \in \CZ$ then $z \in E$
    \item
      If $x \in X$ then $x \in E$
    \item
      If $f : \CZ^k \to \CZ$ and $e_1,...,e_k \in E$ then
      $f(e_1,...,e_k) \in E$
    \item
      Nothing else is in E.
  \end{itemize}

  We define the semantics of these expressions by:
  \begin{align*}
    \llb z \rrb &= \lambda \rho . z  & \forall z \in \CZ \\
    \llb x \rrb &= \lambda \rho . \rho(x) & \forall x \in X \\
    \llb f(e_1,...,e_k) \rrb &=
      \lambda \rho . f(\llb e_1 \rrb(\rho),...,\llb e_k \rrb(\rho))
  \end{align*}

  By abusing our notation we consider expressions, $e \in E$, as $e :
  (X \to \CZ) \to \CZ$ defined by $e = \llb e \rrb$.
  
  The subset of expressions of the form $\max\{ e_1, e_2, ... e_k \}$,
  with $e_1, e_2, ..., e_k \in E$ are referred to
  as \emph{$\max$-expressions}, denoted by $E_{\max} \subset E$.

  We define a function $\parents: E \to 2^E$, taking an expression,
  $e$, to the set of expressions containing $e$, $\{ e_\parent \mid
  e_\parent = f\{ ..., e, ... \}, f: \CZ^k \to \CZ \}$. The expression
  $e$ is said to be a \emph{subexpression} of $e_{\parent}$.
\end{definition}

\begin{definition}
  \textbf{Equation System:} An equation system is a mapping from
  variables to expressions. $\system : X \to E$. We define
  $E_\system \subset E$ to be the preimage of $\system$, that is
  $E_\system = \{ e \in E \mid system(x) = e, \forall x \in X \}$. As
  each element of $E$ is also a function $(X \to \CZ) \to \CZ$, so an
  equation system is considered as a function $\system : X \to
  ((X \to \CZ) \to \CZ)$ (which is equivalent to the above).

  Alternatively an equation system can be represented as a function
  $\system : (X \to \CZ) \to (X \to \CZ)$, $\system(\rho)[x] =
  e_x(\rho)$, taking a variable assignment $\rho$ and returning the
  result of each $e_x$'s evaluation in $\rho$. We use this second form
  as a convenience for when we are evaluating the entire equation
  system in the context of a variable assignment: we obtain a new
  value for each variable and thus a new variable assignment as a
  result.

  The two functional forms are equivalent and correspond to changing
  the argument order in the following way: $\system(\rho)[x] =
  \system[x](\rho)$. The representations are differentiated by context
  (as the arguments are of differing types) as well as by whether
  parentheses or brackets are used.

  The expressions in $E_\system$ are referred to as ``right hand
  sides'' of the equation system $\system$. We denote the set of all
  equation systems by $\Systems$.

  We define an inverse mapping, $\system^{-1} : E \to 2^X$, taking an
  expression, $e$, to the set of variables, $x$ for which it is a
  transitive subexpression of $\system[x]$. This is captured in the
  following definition:
  \begin{align*}
    \system^{-1}(e) &= \{ x \mid system(x) = e \} \cup \{
    \system^{-1}(e_\parent) \mid e_\parent \in parents(e) \}
  \end{align*}
\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}[tbphH]
  \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, resulting in a
considerable inefficiency in practice, requiring the repeated
evaluation of many right-hand-sides for the same value. Thus an
approach which evaluates 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 for all fixpoints. An example system is
presented in Figure \ref{fig:kleene-infinite}, where our first
iteration will give $\{ x \mapsto 0, y \mapsto 0 \}$, leading to a
second iteration resulting in $\{ x \mapsto -1, y \mapsto 0\}$. The
iteration will continue without bound, tending towards $\{ x \mapsto
-\infty, y \mapsto 0 \}$ rather than the greatest fixpoint of $\{
x \mapsto -\infty, y \mapsto -\infty \}$.

\begin{figure}[tbphH]
  \begin{align*}
    x &= \min(0, x - 1) \\
    y &= \min(0, \infty \oplus x) &
    \text{where } \infty \oplus -\infty &= -\infty
  \end{align*}
  \caption{An example equation system for which Kleene iteration will
    not terminate}
  \label{fig:kleene-infinite}
\end{figure}

It has been shown by Gawlitza et
al.\cite{Gawlitza:2007:PFC:1762174.1762203} that Kleene iteration is
at most restricted to $|X|$ iterations in the particular case of the
$\max$-strategy iteration algorithm considered in this thesis. Kleene
iteration is therefore restricted to $\Theta(|X|^3)$ right hand side
evaluations at most in our application. We seek to improve on this
performance in the next section by taking into account dependencies
between expressions.

\begin{definition}
  An expression $x$ is said to \textbf{depend} on $y$ if and only if
  $x(\rho) \ne x(\varrho)$ and
  \begin{align*}
    \rho(z) &= \varrho(z) & \forall z \in X \backslash \{y\} \\
    \rho(y) &\ne \varrho(y)
  \end{align*}

  Intuitively, $x$ depends on $y$ if and only if a change in 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}


\subsection{W-DFS algorithm} \label{sec:wdfs}

Several algorithms have been presented in literature for finding
solutions to general fixpoint equation systems. In particular has been
the \emph{topdown solver} of Charlier and Van
Hentenryck\cite{TD-fixpoint} and the \emph{worklist} solver of
Kildall\cite{Kildall:1973:UAG:512927.512945} which has been further
extended by others\cite{Jorgensen94findingfixpoints,FP-efficient}. The
W-DFS algorithm of Seidl et al.\cite{DBLP:tr/trier/MI96-11} is an
attempt to combine the best aspects of these general fixpoint
algorithms.

The W-DFS algorithm takes into account dependencies between variables
as it solves the system. By taking into account dependencies we can
leave portions of the system unevaluated when we are certain that
those values have not changed. The algorithm is presented in Listing
\ref{algo:wdfs}.

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

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

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

  \begin{algorithmic}
    \Function {init} {$s \in \Systems$}
      \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:wdfs: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:wdfs}
\end{algorithm}

The function $\init: \Systems \to (X \to D)$ in
Listing \ref{algo:wdfs:init} takes an equation system as its argument
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 is calculated we
first calculate the variable's 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 certainly stable). If these values change then they, too, will
destabilise their dependent variables, and so on.

\begin{figure}[tbphH]
  \begin{tikzpicture}
    \node[draw] (full) at (0,-2) {
      \begin{minipage}{0.3\textwidth}
        \begin{align*}
          x &= \min(0, y) \\
          y &= x \\
          a &= \min(0, b) \\
          b &= c \\
          c &= d \\
          d &= e \\
          e &= a
        \end{align*}
      \end{minipage}
    };

    \node[draw] (first) at (10, 1) {
      \begin{minipage}{0.3\textwidth}
        \begin{align*}
          x &= \min(0, y) \\
          y &= x
        \end{align*}
      \end{minipage}
    };
    \node[draw] (second) at (10,-4) {
      \begin{minipage}{0.3\textwidth}
        \begin{align*}
          a &= \min(0, b) \\
          b &= c \\
          c &= d \\
          d &= e \\
          e &= a
        \end{align*}
      \end{minipage}
    };

    \draw (full) -- (first.west);
    \draw (full) -- (second.west);
  \end{tikzpicture}
  \caption{Example of an equation system which can be separated into
    two independent sub-systems}
  \label{fig:wdfs-example}
\end{figure}

Because each value is only computed when requested, the solver avoids
calculating results which are irrelevant to the final outcome. To
provide a more concrete example, consider the equation system
presented in Figure \ref{fig:wdfs-example}. This equation system can
be separated into two independent equation systems, one consisting of
$\{x, y\}$ and the other of $\{a, b, c, d, e\}$. In order to find the
value of a variable in either of these subsystems it is necessary to
evaluate the entire subsystem, but it is not necessary to evaluate the
other subsystem. To compute the value of $x$ in the greatest fixpoint
it is unnecessary to compute the value of the variable $d$, but it is
necessary to compute the value of $y$. The W-DFS algorithm, will only
evaluate the necessary subsystem for a requested variable, leaving the
other subsystem unevaluated.



\section{$\max$-strategy Iteration}

$\max$-strategy iteration finds a mapping $\sigma: E_{\max} \to E$
from each $\max$-expression to its greatest subexpression. We use
$\sigma: \Systems \to \Systems$ as a shorthand to denote the system
obtained by replacing each $\max$-expression, $x$, in $\system$ with
$\sigma(x)$. Similarly $\sigma_{\all}: E \to E$ denotes the expression
obtained by replacing each $\max$-subexpression in $e$ with
$\sigma(e)$.

In the particular case of \emph{monotonic}, \emph{expansive} operators
it has been shown\cite{Gawlitza:2007:PFC:1762174.1762203} that this
process will take a finite number of iterations if we ensure at each
iteration that we make a \emph{strict} improvement on our
strategy. Each time we improve our strategy we make it greater in at
least one $\max$-expression, and no worse in the others. We must
satisfy $\sigma_{n+1}(\rho) > \sigma_n(\rho)$, for the $\rho$ in which
we are improving.

A new function, $\improve: ((E_{\max} \to E) \times (X \to D)) \to
(E_{\max} \to E)$, is used below as a \emph{strategy improvement
  operator}. $\improve$ takes a $\max$-strategy and a variable
assignment and returns a new $\max$-strategy which constitutes an
`improvement' of the strategy in the variable assignment $\rho$. If we
have a $\max$-strategy $\sigma$, a variable assignment $\rho$ and
$\varsigma = \improve(\sigma, \rho)$. Then
\begin{align*}
  \sigma(\system)(\rho) \le \varsigma(\system)(\rho)
\end{align*}

If there is a possible improvement to be made on $\sigma$ then
$\improve$ will return an improvement, otherwise it will return
$\sigma$ unchanged. One possible implementation of $\improve$ is
presented in Listing \ref{algo:pmax}. This implementation will try
every option of subexpressions for this $\max$-expression and will
then return the best. The use of an anonymous function as the returned
strategy is so the evaluation of each strategy is demand-driven.

\begin{algorithm}
  \begin{algorithmic}
    \Function {improve} {$\sigma: E_{\max} \to E$, $\rho: X \to D$}
      \ReturnLambdaFn {$e \in E_{\max}$}
        \State $e_{\best} = \sigma(e)$
        \State $\val_{\best} = \sigma_{\all}(e_{\old})(\rho)$
        \For {$e_i \in e$}
          \If {$\sigma_{\all}(e_i)(\rho) > \val_{\best}$}
            \State $e_{\best} = e_i$
            \State $\val_{\best} = \sigma(e_i)(\rho)$
          \EndIf
        \EndFor
        \State \Return $e_{\best}$
      \EndReturnLambdaFn
    \EndFunction
  \end{algorithmic}
  \caption{The variation of $\improve$ used in our algorithm}
  \label{algo:pmax}
\end{algorithm}

\subsection{Naive approach}

The problem of $\max$-strategy iteration is one of constantly
improving until no further improvement can be made, so a simple
approach is to perform a loop, improving the strategy at each step,
until an equilibrium point is reached. The approach is presented in
Listing \ref{algo:naive-strategy}.

\begin{algorithm}[tbphH]
  \begin{algorithmic}
    \Assumptions
      \begin{tabularx}{0.9\textwidth}{rX}
        $\sigma$ & $: E_{\max} \to E$, a $\max$ strategy \\

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

        $\rho$ & $: X \to D$, a variable assignment \\

        $\improve$ & $: ((E_{\max} \to E) \times (X \to \CZ)) \to (E_{\max}
        \to E)$, a $\max$-strategy improvement operator \\
      \end{tabularx}
    \EndAssumptions

    \Function {solvestrategy} {$\system \in \Systems$}
      \State $k = 0$
      \State $\sigma_0 = \{ x \mapsto -\infty \mid \forall x \in X \}$
      \State $\rho_0 = \underline{-\infty}$
      \Repeat
        \State $\sigma_{k+1} = \improve(\sigma_k, \rho_k)$
        \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}

This approach is similar to that of Kleene iteration, and from the
results of Gawlitza et al.\cite{Gawlitza:2007:PFC:1762174.1762203} it
is known that this iteration is guaranteed to terminate. This naive
approach is inefficient, however, in the same way as Kleene
iteration. A large amount of time will be spent attempting to improve
portions of the strategy for which no improvement can be made.

\subsection{Adapted W-DFS algorithm} \label{sec:adapted-wdfs}

The $\max$-strategy iteration can be seen 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 $\max$-expression
$e \in E_{\max}$. 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 in Section \ref{sec:wdfs}. Listing
\ref{algo:adapted-wdfs} presents one variation on W-DFS to solve
$\max$-strategy iteration problems.

\begin{algorithm}[tbphH]
  \begin{algorithmic}
    \Globals
    \begin{tabularx}{0.9\textwidth}{rX}
      $\sigma$ & $: (E_{\max} \to E)$, a mapping from
      $\max$-expressions to a subexpression \\

      $\inflSI$ & $: (E_{\max} \to 2^{E_{\max}}$, a mapping from a
      $\max$-expression to the subexpressions it influences \\

      $\stableSI$ & $\subseteq E_{\max}$, the set of all
      $\max$-expressions whose strategies are stable \\

      $\rho$ & $: X \to \CZ$, a variable assignment \\

      $\system$ & $\in \Systems$, an equation system \\
      
      $\improve$ & $: ((E_{\max} \to E) \times (X \to \CZ)) \to (E_{\max}
      \to E)$, a $\max$-strategy improvement operator \\
    \end{tabularx}
    \EndGlobals
  \end{algorithmic}

  \begin{algorithmic}
    \Function {init} {$s \in \Systems$}
      \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:adapted-wdfs: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 $v = \improve(\lambda y . \eval(x, y), \rho)[x]$
      \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
      \State $\rho_{\last} = \rho$
      \State $\rho = \solve \fixpoint(\sigma(\system))$
      \For {$y \in X$}
        \If {$\rho_{\last}[y] = \rho[y]$}
          \State $\stableSI = \stableSI \backslash \{\system[y]\}$
          \State $\solve(\system[y])$
        \EndIf
      \EndFor
    \EndIf
    \EndProcedure
  \end{algorithmic}

  \caption{W-DFS, this time modified to find the best $\max$-strategy.}
  \label{algo:adapted-wdfs}
\end{algorithm}

This approach retains the benefits of the W-DFS algorithm for solving
fixpoints, but applies them to $\max$-strategy iteration
problems. Dependencies between $\max$-expressions are considered and
an improvement on the strategy at an expression $e$ is only attempted
if some of the $\max$-expressions which influence $e$ themselves
improve strategy. We also destabilise $\max$-expressions which have
their values changed in the fixpoint iteration step.

For this particular algorithm to work, however, we must assume another
property of $\improve$. In Listing \ref{algo:adapted-wdfs} we take the
value of $\improve(\lambda y . \eval(x, y), \rho)[x]$; we improve the
strategy, then use only the strategy for $x$. In order for this
algorithm to be correct $\improve$ must always improve the strategy at
$x$ if it is possible to do so. If $\improve$ did not improve the
strategy at $x$, while a permissible improvement existed, then we
would consider $x$ to be ``stable'' and would not attempt to improve
our strategy at $x$ until the improvement of another portion of our
strategy invalidated $x$. It is not guaranteed that another strategy
improvement will invalidate the strategy at $x$, however, so the
strategy at $x$ may still be able to be improved when the algorithm
terminates. If the strategy at $x$ can still be improved then we have
not reached the solution of our $\max$-strategy iteration. The
particular $\improve$ that we are using (see Listing \ref{algo:pmax})
satisfies this assumption, as it attempts to improve the strategy at
the point of individual $\max$-expressions.

Additionally, in order to remain efficient, $\improve$ should not
attempt to improve the strategy for any $\max$-expressions until that
expression is requested. Whether or not $\improve$ is lazy in this way
will not affect the correctness of the algorithm, as it will only
result in the $\max$-strategy improvement at each step being more
expensive. The extra work to improve other portions of the strategy
will be discarded with only $\improve(...)[x]$ being used. The
particular $\improve$ that we are using (see Listing \ref{algo:pmax})
satisfies this property using an anonymous function to delay the
computation.

\section{Demand-driven Strategy Improvement (DSI)}

W-DFS speeds up both the $\max$-strategy iteration and the fixpoint
iteration as two independent processes, but each time we improve our
strategy we compute at least a subset of the greatest fixpoint from a
base of no information. We provide a 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 fixpoint, thereby avoiding repeating
calculations.

The new \emph{Demand-driven Strategy Improvement} algorithm,
\emph{DSI}, 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 modify the fixpoint iteration
algorithm to report which variables it has modified with each fixpoint
iteration step. We then have 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
were stabilised. The $\max$-strategy iteration then uses this list of
changed variables to determine which portions of the current strategy
are now unstable. This process continues until the $\max$-strategy
stabilises (at which point the fixpoint will also stabilise).






This algorithm is presented in three parts.

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

\item
  Section \ref{sec:ldsi:fixpoint} 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.

\item
  Section \ref{sec:ldsi:max-strategy} 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.
\end{itemize}

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

\begin{algorithm}[tbphH]
  \begin{algorithmic}
    \Globals
    \begin{tabularx}{0.9\textwidth}{rX}
      $D$ & $: X \to \CZ$, a mapping from variables to their current
      value \\

      $D_\old$ & $: X \to \CZ$, a mapping from variables to their
      last stable value \\
      
      $\inflFP$ & $: X \to 2^X$, a mapping from a variable to the
      variables it \emph{may} influence \\

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

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

      \\

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

      $\inflSI$ & $: E_{\max} \to 2^{E_{\max}}$, a mapping from a
      $\max$-expression to the subexpressions it influences \\

      $\stableSI$ & $\subseteq E_{\max}$, the set of all
      $\max$-expressions whose strategies are stable \\

      $\improve$ & $: ((E_{\max} \to E) \times (X \to \CZ)) \to
      (E_{\max} \to E)$, a $\max$-strategy improvement operator \\
      
      \\

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

\begin{algorithm}[tbphH]
  \begin{algorithmic}
    \Function {init} {$s \in \Systems$}
      \State $D = \underline{-\infty}$
      \State $D_\old = \underline{-\infty}$
      \State $\inflFP = \{x \mapsto \emptyset \mid \forall x \in X\}$
      \State $\stableFP = X$
      \State $\touchedFP = \emptyset$

      \State $\sigma = \underline{-\infty}$
      \State $\inflSI = \{x \mapsto \emptyset \mid \forall x \in E_{\max}\}$
      \State $\stableSI = \emptyset$

      \State $\system = s$

      \State \Return $\lambda x . (\solve \strategy (x); \solve
      \fixpoint (x) ; D[x])$
    \EndFunction
  \end{algorithmic}
  \caption{LSDI init function and entry point}
  \label{algo:ldsi:init}
\end{algorithm}

The $\init$ function is, similarly, just a combination of the $\init$
functions presented in Sections \ref{sec:wdfs} and
\ref{sec:adapted-wdfs}. The $\touchedFP$ and $D_\old$ variables are
also initialised, as they was not present in either of the previous
$\init$ functions.

$D$ and $D_\old$ are both initialised to \underline{$-\infty$} and
$\stableFP$ is initially the entire set of variables. The values of
$D$, $D_\old$ and $\stableFP$ are set to their result after a complete
fixpoint iteration over each variable with the initial
$\max$-strategy.

The type of this function is different to either of the original
$\init$ functions. $\init : \Systems \to (X \to \CZ)$. The function
returned by $\init$ performs the entire process of $\max$-strategy
iteration and fixpoint iteration and calculates the final value of
each variable for the least fixpoint, returning a value in $\CZ$.



\subsection{Fixpoint iteration} \label{sec:ldsi:fixpoint}

The process of fixpoint iteration is similar to the fixpoint iteration
algorithm presented in Section \ref{sec:wdfs}. The only difference is
that we have an $\invalidate \fixpoint$ procedure for the
$\max$-strategy iteration to invalidate a portion of the system. By
invalidating only a portion of the system we reuse the values which
have already been computed to avoid performing additional work. The
last stable value is stored in this procedure to identify
\emph{changed} values, rather than only \emph{unstable} ones.

\begin{algorithm}[tbphH]
  \begin{algorithmic}
    \Function {evalfixpoint} {$x \in X$, $y \in X$}
      \State $\solve \fixpoint(y)$
      \State $\inflFP[y] = \inflFP[y] \cup \{x\}$
      \State \Return $D[y]$
    \EndFunction
  \end{algorithmic}
  \caption{Utility function used to track fixpoint variable
    dependencies. This function is very similar to the $\eval$
    function presented in Listing \ref{algo:wdfs}}
  \label{algo:ldsi:fixpoint:eval}
\end{algorithm}

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 the same function as
the $\eval$ function in Listing \ref{algo:wdfs}.

\begin{algorithm}[tbphH]
  \begin{algorithmic}
    \Procedure {invalidatefixpoint} {$x \in X$}
    \Comment{Invalidate a fixpoint variable}
    \If {$x \in \stableFP$}
      \State $\stableFP = \stableFP \backslash \{x\}$
      \State $\D_\old[x] = D[x]$
      \State $D[x] = \infty$
      \State $W = \inflFP[x]$
      \State $\inflFP[x] = \emptyset$
      \State $\touchedFP = \touchedFP \cup \{ x \}$
      \For {$v \in W$}
        \State $\invalidate \fixpoint(v)$
      \EndFor
    \EndIf
    \EndProcedure
  \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. A call to $\invalidate
\fixpoint$ indicates to the fixpoint-iteration that a variable's value
\emph{may} have changed as a result of a $\max$-strategy improvement,
and that it will require re-calculation. The invalidation provided by
this procedure permits us to only re-evaluate a partial system (the
solving of which is delayed until requested by the $\solve \fixpoint$
procedure).

While invalidating variables we store their $\old$ value, and mark
them as ``touched''. As the $\old$ value is only set in this
procedure, and only when $x$ is stable, it gives us the ability to see
whether a variable has stabilised to a new value, or whether it has
merely re-stabilised to the same value as last time. This refines the
set of changed variables for the $\max$-strategy iteration phase.



\begin{algorithm}[tbphH]
  \begin{algorithmic}
    \Procedure {solvefixpoint} {$x \in X$}
    \If {$x \not \in \stableFP$}
      \State $\stableFP = \stableFP \cup \{ x \}$
      \State $v = \sigma(\system[x])( \lambda y . \eval \fixpoint(x, y) )$
      \If {$v \ne D[x]$}
        \State $D = \{ x \mapsto v \} \oplus D$
        \State $W = \inflFP[x]$
        \State $\inflFP[x] = \emptyset$
        \State $\stableFP = \stableFP \backslash W$
        \For {$v \in W$}
          \State $\solve \fixpoint(v)$
        \EndFor
      \EndIf
    \EndIf
    \EndProcedure
  \end{algorithmic}
  \caption{The subroutine of the fixpoint iteration responsible for
    solving for each variable. This is very similar to the $\solve$
    procedure in Listing \ref{algo:wdfs}}
  \label{algo:ldsi:fixpoint:solve}
\end{algorithm}

The $\solve \fixpoint$ procedure is similar to the $\solve$ procedure
presented in Listing \ref{algo:wdfs}. There are two main differences:
the self-recursive call has been renamed to reflect the change in the
function's name, and instead of using $\system[x]$ we must use
$\sigma(\system[x])$. We use $\sigma$ to provide the
fixpoint-iteration with the current $\max$-strategy. $\sigma$ is
stable for the duration of the $\solve \fixpoint$ execution, so it
will result in fixpoint iteration being performed over a stable
system.

After an evaluation of the $\solve \fixpoint$ procedure, the variable
$x$ will have its value in the current $\max$-strategy's greatest
fixpoint stabilised, and that value will be stored in
$D[x]$. Stabilising this may result in other related variables also
being stabilised.



The fixpoint iteration portion of the algorithm is concerned with
finding the greatest fixpoint, $\solnFP(s) : X \to D$ for the system $s
= \sigma(\system)$. If we consider all valid fixpoints, $\fix(s) = \{
\rho \mid \rho = s(\rho) \}$, then we find that $\solnFP(s) \in \fix(s)$
and $\solnFP(s) \ge \rho ~ \forall \rho \in \fix(s)$. Our general
approach will be to consider our variable assignment $D \ge \solnFP(s)$
and show that it decreases until $D \in \fix(s)$ and thus $D =
\solnFP(s)$.

We have three invariants for the fixpoint portion of the DSI
algorithm. We use $\depFP^{+}(x)$ to be the set of all variables which
$x$ may depend on. $\depFP^{+}$ is defined as the transitive closure
of $\depFP$, $\depFP[x] = \{ y \in X \mid
x \in \inflFP[y] \}$. $\RFP \subseteq X$ is defined as the variables
currently on our recursion stack.
\begin{align}
  D &\ge \solnFP(s) \label{eqn:fp-inv1} \\
%
  x &\in \inflFP[y] & \forall x \in \stableFP, y \in \depFP[x]
    \label{eqn:fp-inv3} \\
%
  D &= \solnFP(s)[x] & \mbox{if }
      x \not \in \RFP,
      \depFP^{+}[x] \cap \RFP = \emptyset,
      x \in \stableFP
    \label{eqn:fp-inv2}
\end{align}

(\ref{eqn:fp-inv1}) follows from monotonicity. We begin our iteration
at some variable assignment $D \ge \solnFP(s)$ and monotonically descend
from there. We know that $s(D)[x] \ge soln(s)[x] ~\forall D \ge
\solnFP(s), x \in X$, thus each evaluation of $s[x]$ in the context of
$D$ will result in a new value for $D[x]$ which is either closer to or
equal to $\solnFP(s)[x]$. As we only ever evaluate in the context of
variables taken from $D$, and $D[x]$ is set to $\infty$ when $x$ is
invalidated, we know that $D \ge \solnFP(s)$ holds.

(\ref{eqn:fp-inv3}) follows from the use of the $\eval \fixpoint$
function. Each variable lookup results in a direct insertion into
$\inflFP$, so during the evaluation of $x$ all relevant $\inflFP$ sets
are updated to reflect the current dependencies.

For (\ref{eqn:fp-inv2}) we will examine what has happened when each of
the conditions is false:
\begin{itemize}
\item
  If $x \in \RFP$ then we are currently in the context of an earlier
  evaluation of $x$, so the value of $D[x]$ will be changed again by
  that earlier call. In addition, we conclude that $x \in
  \depFP^{+}[x]$ and thus, from the following condition, that $x$ will
  be re-evaluated at some point after this one.

\item
  If $\depFP^{+}[x] \cap \RFP \ne \emptyset$ then a variable which $x$
  depends on is in $\RFP$ at the moment. This will result in the
  re-evaluation of $x$ if the value of any of the variables in
  $\depFP^{+}[x] \cap \RFP$ have changed.

\item
  If $x \not \in \stableFP$ then a re-evaluation of $x$ will occur
  with the next call to $\solve \fixpoint(x)$. This may result in a
  change to the value of $D[x]$, or it may leave $D[x]$ stable for the
  moment. If we denote the variable assignment before the evaluation
  of $x$ by $D_{\pre}$, and the variable assignment after the
  evaluation of $x$ by $D_{\post}$ then we find that $D_{\post} \le
  D_{\pre}$, due to the monotonicity of $s$. After each change of
  $D[x]$, everything which \emph{may} have depended on $D[x]$ is
  re-evaluated (see the previous two items), leading to a final point
  of stability when $D[x] = \solnFP(s)[x]$, by the definition of
  $\solnFP(s)[x]$ and (\ref{eqn:fp-inv1}).
\end{itemize}

After an evaluation of $\solve \fixpoint(x)$ we know that $x \in
\stableFP$. If it is also the case that $\RFP = \emptyset$, as is the
case for top-level calls to $\solve \fixpoint$, then we know that
$D[x] = \solnFP(s)[x]$. This means that the function $\lambda x
. (\solve \fixpoint(x); D[x])$ will act as a variable assignment
solving for the greatest fixpoint of $s = \sigma(\system)$, as is
required by our $\max$-strategy iteration. As the $\max$-strategy
iteration changes $\sigma$ it also induces a change in $s =
\sigma(\system)$, so in order to maintain the correctness of this
algorithm we must show that our above invariants are maintained by
this process.

When the $\max$-strategy iteration changes the current $\max$-strategy
$\sigma$ at the variable $x$ it changes the equation system $s =
\sigma(\system)$ that the fixpoint iteration uses. The $\max$-strategy
iteration then invalidates the affected portion of the fixpoint
iteration by doing two things: it removes the variable $x$ from the
$\stableFP$ set, and it sets $D[x] = \infty$. The invalidation of
variables is then propagated to each variable which transitively
depends on $x$: $\{ y \in X \mid x \in \depFP^{+}[y] \} = \{ y \in X
\mid y \in \inflFP^{+}[x] \}$, where $\inflFP^{+}$ is the transitive
closure of $\inflFP$. We know from (\ref{eqn:fp-inv3}) that this
transitive closure of $\inflFP$ will identify the entire subsystem
which may depend on the value of $x$. The invalidation of transitive
dependencies ensures that (\ref{eqn:fp-inv1}) holds for the changed
subsystem, as $\infty \ge z ~ \forall z \in \CZ$. From
(\ref{eqn:fp-inv1}) we can then conclude that (\ref{eqn:fp-inv2})
holds as well, as the removal of $x$ from $\stableFP$ combined with
(\ref{eqn:fp-inv1}) leads to (\ref{eqn:fp-inv2}). These invariants now
hold for the affected sub-system and are still true for the unaffected
sub-system. Thus our invariants hold for our entire system and our
fixpoint iteration will continue to be correct in the presence of
invalidation by the $\max$-strategy iteration.



\subsection{$\max$-strategy iteration} \label{sec:ldsi:max-strategy}



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

    \Function {evalstrategyfixpoint} {$x \in E_{\max}$, $y \in X$}
      \State $\solve \fixpoint(y)$
      \State $\inflSI[y] = \inflSI[\system[y]] \cup \{ x \}$ 
      \For {$v \in \touchedFP$}
        \If {$v \in \stableFP$ and $D[x] \ne D_\old[x]$}
          \For {$w \in \inflSI[\strategy(\system[v])]$}
            \State $\invalidate \strategy(\system[v])$
          \EndFor
          \State $\touchedFP = \touchedFP \backslash \{v\}$
        \EndIf
      \EndFor
      \State \Return $D[y]$
    \EndFunction
  \end{algorithmic}
  \label{algo:ldsi:strategy:eval}
  \caption{Functions which evaluate a portion of the
    $\max$-strategy/fixpoint and note a dependency}
\end{algorithm}

The $\eval \strategy$ function is essentially the same as the $\eval$
function in Listing \ref{algo:adapted-wdfs}, except that it calls the
$\solve \strategy$ procedure.

The $\solve \fixpoint$ calls in $\eval \strategy \fixpoint$ are
top-level fixpoint-iteration calls, upon their return we know that the
value of $D[x]$ is the value that $x$ takes in the greatest fixpoint
of the current strategy $sigma$. This function, therefore, acts as a
intermediate layer between the fixpoint-iteration and the
$\max$-strategy iteration to allow for dependencies to be tracked.

Upon the conclusion of the $\solve \fixpoint$ we inspect the
$\touchedFP$ set to determine which variables have values which may
have changed. If their values have stabilised since being invalidated,
and if they stabilised to a different value to their previous
stabilisation, then we will invalidate all strategies which depend on
them. We do not have to invalidate the variable's right hand side
directly, but if it is dependent on its own value (as in $\{ x = x + 1
\}$, for example), then it will be destabilised by the transitivity of
$\invalidate \strategy$.

\begin{algorithm}[tbphH]
  \begin{algorithmic} 
    \Procedure {invalidatestrategy} {$x \in E_{\max}$}
      \If {$x \in \stableSI$}
        \State $\stableSI = \stableSI \backslash \{x\}$
        \For {$v \in \inflSI$}
          \State $\invalidate \strategy (v)$
        \EndFor
      \EndIf
    \EndProcedure
  \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 essentially the same as
the invalidation in the fixpoint-iteration stage, except that we do
not need to keep track of the last stable values, nor which
$\max$-expressions we have invalidated. We remove them from the
$\stableSI$ set, thereby indicating that they need to be re-stabilised
by a $\solve \strategy$ call. All $\max$-expressions which
(transitively) depend on $x$ are destabilised recursively.


\begin{algorithm}[tbphH]
  \begin{algorithmic}    
    \Procedure {solvestrategy} {$x \in E_{\max}$}
    \If {$x \not \in \stableSI$}
      \State $\stableSI = \stableSI_{\max} \cup \{x\}$
      \State $e = \improve(\lambda y . \eval \strategy(x,y),
                           \lambda z . \eval \strategy \fixpoint(x, z))[x]$
      \If {$e \ne \sigma[x]$}
        \State $\sigma = \{ x \mapsto e \} \oplus \sigma$
        \For {$v \in \system^{-1}(x)$}
          \State $\invalidate \fixpoint(\system^{-1}(x))$
        \EndFor
        \State $\stableSI = \stableSI \backslash I[x]$
        \For {$v \in \inflSI[x]$}
          \State $\solve \strategy(v)$
        \EndFor
      \EndIf
    \EndIf
    \EndProcedure
  \end{algorithmic}
  \caption{The $\max$-strategy portion of the Combined W-DFS.}
  \label{algo:ldsi:solve}
\end{algorithm}

$\solve \strategy$ is very similar to the other $\solve$ calls we have
already seen. The procedure attempts a strategy improvement for the
$\max$-expression $x$ and, if any improvement is possible, it improves
the strategy $\sigma$ before invalidating and re-evaluating any other
$\max$-expressions which may depend on $x$. We here assume that
$\improve$ has the same properties as those which are assumed in
Section \ref{sec:adapted-wdfs}.



In the following brief analysis we use $\rho: X \to \CZ$ as $\rho
= \lambda x . (\solve \fixpoint(x); D[x])$ to denote a variable
assignment which will always calculate the greatest fixpoint of
$\sigma(\system)$ for the current strategy $\sigma$. Each time $\rho$
is queried for a variable's value it will also record which variables
have had their values changed, whether or not those changed values are
final, in the set $\touchedFP$. This represents the effect of the
fixpoint iteration portion of the DSI algorithm.

We can establish similar invariants for the $\max$-strategy iteration
as we had for the fixpoint iteration. We denote the optimal strategy
by $\solnSI(\system)$ and we define $\RSI \subseteq E_{\max}$ to be
the set of $\max$-expressions on our recursion stack.
\begin{align}
  e &\in \inflSI[y] & \forall e \in \stableSI, y \in \depSI[e]
    \label{eqn:si-inv1} \\
%
  \sigma &= \solnSI(\system)[e] & \mbox{if }
      e \not \in \RSI,
      \depSI^{+}[e] \cap \RSI = \emptyset,
      e \in \stableSI
    \label{eqn:si-inv2}
\end{align}

(\ref{eqn:si-inv1}) follows from the same logic as (\ref{eqn:fp-inv3})
for the fixpoint iteration. The $\eval \strategy$ directly updates the
$\inflSI$ sets whenever they are used. This ensures our dependency
information is correct.

(\ref{eqn:si-inv2}) follows a similar argument to (\ref{eqn:fp-inv2})
for the fixpoint iteration. At each step we must make a strategy
improvement, such that $\sigma_{n+1}(\system)(\rho) >
\sigma_n(\system)(\rho)$. Every step must be an improvement and, by the
assumptions made in Section \ref{sec:adapted-wdfs}, we know that we
will make an improvement for each individual variable if it is
possible to do so. The invalidation for a changed strategy is
identical to that which is performed for variables in the fixpoint
iteration, so it will not be re-considered here. An additional
complexity is that $\rho$ is not a constant mapping during
$\max$-strategy iteration. In order to maintain (\ref{eqn:si-inv2}) we
ensure that on changes in $\rho$ we invalidate some of our
$\max$-expressions. The set $\touchedFP$ over-approximates the set of
changed variables, so we are certain that all changes will be found by
enumerating $\touchedFP$. We only invalidate those $\max$-expressions
which are the right-hand side of a stable, changed variable. If a
variable has not stabilised then it's value is has not been used
directly (see (\ref{eqn:fp-inv2})) and thus cannot directly influence
the $\max$-strategy. If the variable has stabilised to the same value
as it had previously then it has not changed and thus has no effect on
the $\max$-strategy.

