\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{\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\Operators{\mathcal{O}}
\newcommand\Systems{\varepsilon}

\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{Contribution} \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} (see Section
\ref{sec:gawlitza-algorithm}). We devise an algorithm which runs in
linear time in the best case.

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 then conclude with our Local Demand-driven Strategy
Improvement (LDSI) algorithm to perform efficient strategy iteration
and fixpoint iteration. % TODO: fix me, or something

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 results in no change. Gawlitza et
al.~proved 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 re-calculating results which are already known.
% TODO: add something here about variable dependencies

%By considering which variables depend on, and influence, each other it
%is possible to reduce the amount of work that is to be done quite
%considerably.

\section{Preliminaries}

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, $\rho: X \to
  \CZ$, from a variable to a value. 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 $\rho \in X \to \CZ$ and $\varrho \in X \to
  \CZ$ may be composed with the $\oplus$ operator 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:} We will consider expressions, $e \in E$, as $e
  : (X \to \CZ) \to \CZ$, a mapping from a variable assignment to the
  expression's value in that assignment.

  Expressions have three possible forms:
  \begin{itemize}
    \item
      $e = f( e_1, e_2, ..., e_k )$, \\
      with $f: \CZ^k \to \CZ$ and
      $e_1, e_2, ... e_k \in E$ and $e(\rho) = f(e_1(\rho),
      e_2(\rho), ..., e_k(\rho))$
    \item $e = x \in X$, with $e(\rho) = \rho(x)$
    \item $e = z \in \CZ$, with $e(\rho) = z$
  \end{itemize}
  
  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 $e$, $\{ e_\parent \mid
  e_\parent = f\{ ..., e, ... \}, f: \CZ^k \to \CZ \}$. $e$ is then
  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. $\system : X \to E_\system$, where
  $E_\system \subset E$. As each element of $E$ is also a function $(X
  \to \CZ) \to \CZ$, so an equation system can be 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 $\rho$.

  These two functional forms are equivalent and correspond to changing
  the argument order in the following way: $\system(\rho)[x] =
  \system[x](\rho)$. We use both forms as a convenience for when we
  only wish to use one argument. 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}

\begin{definition}
  \textbf{Dependencies:} A variable or expression $x$ is said to
  \emph{depend on} $y$ 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*}

  If $x$ depends on $y$ then $y$ is said to \emph{influence} $x$.
\end{definition}

\section{Fixpoint Iteration}
\subsection{Kleene Iteration}

A simple approach to fixpoint iteration over monotonic equations is to
simply iterate over the system repeatedly until a reevaluation results
in no change to the values of any variables. This approach will always
reach the least/greatest solution if there is one to be found, but it
will often perform many more evaluations than are necessary. This
algorithm is presented in Listing \ref{algo:kleene}.

\begin{algorithm}[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, resulting in a
considerable inefficiency in practice, requiring the repeated
evaluation of many right-hand-sides 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 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}[H]
  \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}

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. It does,
however, require $\Theta(|X|^3)$ right hand side evaluations, which is
very inefficient.

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

The W-DFS algorithm of Seidl et al.~takes into account dependencies
between variables as it solves the system. By taking into account
dependencies it can leave portions of the system unevaluated when it
is 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
must 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.

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

\begin{figure}[H]
  \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);
    \draw[->] (full) -- (second);
  \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 system untouched. In order to capture this property we will
define the notion of a \emph{local} solver.

\begin{definition}
  \textbf{Local:} A solver is said be \emph{local} if, for some
  equation system, $\system$, and some $e \in E_\system$, it will
  solve for $e_x$ without evaluating every element of $E_\system$.
\end{definition}

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



\section{$\max$-strategy Iteration}

The problem of $\max$-strategy iteration is trying to find a mapping
$\sigma: E_{\max} \to E$ from each $\max$-expression to its greatest
subexpression. We will 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
$\sigma_{\all}: E \to E$ will denote 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. This means that each time we improve our strategy we must
make it greater in at least one $\max$-expression, and no worse in the
others.

To this end a new function, $\improve: ((E_{\max} \to E) \times
(X \to D)) \to (E_{\max} \to E)$, 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 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 it is possible for $\sigma$ to be improved then $\improve$ will
return an improvement, otherwise it will return $\sigma$
unchanged. One possible implementation of $\improve$ is presented in
Listing \ref{algo:pmax}. It 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 only done when requested.

\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 simply to perform a straightforward loop, improving the
strategy at each step, until an equilibrium point is reached. This is
the approach presented in Listing \ref{algo:naive-strategy}.

\begin{algorithm}[H]
  \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 quite 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 an inefficient way to solve this problem, however, for
many of the same reasons 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 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 in Section \ref{sec:wdfs}. Listing
\ref{algo:adapted-wdfs} presents one variation on W-DFS to allow it to
solve $\max$-strategy iteration problems.

\begin{algorithm}[H]
  \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 \\

      $\system$ & $\in \Systems$, 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 $\rho = \solve \fixpoint(\sigma(\system))$
      \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
    \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. It
will take into account dependencies between the $\max$-expressions and
only attempt to improve the strategy for a given $\max$-expression if
some of the $\max$-expressions which may influence it are changed.

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 by way of its anonymous function.

\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 compute at least a subset of the
greatest fixpoint from a base of no information. We would like to 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 fixpoint, thereby avoiding repeating work that we have
already done when it is unnecessary to do so.

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. 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 must be re-stabilised once more. This process
continues until the $\max$-strategy stabilises (at which point the
fixpoint will also stabilise).

Both portions of the LDSI algorithm are \emph{demand driven}, and so
any necessary evaluation of strategies or fixpoint values is delayed
until the point when it is required. This means that the solver will
be \emph{local} (see Section \ref{sec:wdfs}).






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.

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

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

\begin{algorithm}[H]
  \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 LDSI 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}[H]
  \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 also 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 extremely 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 also stored in this procedure to
identify \emph{changed} values, rather than only \emph{unstable} ones.

\begin{algorithm}[H]
  \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 exactly the same
function as the $\eval$ function in Listing \ref{algo:wdfs}.

\begin{algorithm}[H]
  \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 also be delayed until requested by the $\solve
\fixpoint$ procedure).

While invalidating variables we also note 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 can prevent
us from doing extra work in the $\max$-strategy iteration phase.



\begin{algorithm}[H]
  \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 extremely 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.



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



\begin{algorithm}[H]
  \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
        \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, 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.

Upon the conclusion of the $\solve \fixpoint$ we also must 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}[H]
  \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. All we need to do is 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
themselves destabilised as well.


\begin{algorithm}[H]
  \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}





\subsection{Correctness} \label{sec:ldsi:correctness}

To argue the correctness of our LDSI algorithm we will first establish
some invariants, after which it should be much easier to see why our
algorithm is correct.

%\begin{invariant}
First of all $\fixpoint \solve$. 
%\end{invariant}

In order to argue that LDSI is correct we must first assume that the
W-DFS algorithm of Seidl et al.~is correct and will solve the equation
system for its least/greatest fixpoint.




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.

\subsection{Implementation} \label{sec:ldsi:implementation}
