\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{\varepsilon} %\mathsf{system}}
\newcommand\parent{\mathsf{parent}}
\newcommand\parents{\mathsf{parents}}
\newcommand\old{\mathsf{old}}

\newcommand\Operators{\mathcal{O}}
\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}

In this chapter the main theoretical contribution of this thesis is
presented: 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{sec:gawlitza-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.

\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}
  \textbf{Variable Assignments:} $X \to \CZ$. 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{Operators:} An operator is a function $o: 2^{\CZ} \to
  \CZ$. They belong to the set $\Operators$.
\end{definition}

\begin{definition}
  \textbf{Expressions:} For the purposes of this discussion 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 are either of the form $e = o\{ e_1, e_2, ..., e_k \}$
  (where $o \in O$ and $e_1, e_2, ... e_k \in E$) or else they are of
  the form $e \in \CZ$ (a constant value) or $e \in X$ (a variable).
  
  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 $E_{\max} \subset E$.

  For an expression $e$, the set of expressions containing it, $\{
  e_\parent \mid e_\parent = o\{ e_1, e_2, ... e_k \}, o \in \O, e_i
  \in E \}$, is denoted $\parents(e)$.
\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).

  An equation system can also be considered 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: $\system(\rho)[x] =
  \system[x](\rho)$. It is only for convenience that we use both
  forms.

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

  An inverse mapping, $E \to X$ also exists for each system, denoted
  by $\system^{-1}$. This is defined as:
  \begin{align*}
    \system^{-1}(e) &= \{ x \mid system(x) = e \} \cup \{
    \system^{-1}(e_\parent) \mid e_\parent \in parents(e) \}
  \end{align*}
  It provides a mapping from each expression to the variables for
  which it is a `subexpression' of the variable's right hand side.
\end{definition}

\begin{definition}
  \textbf{Dependencies:} A variable or expression $x$ is said to
  \emph{depend on} $y$ if $x(\rho) \ne x(\varrho)$, when
  \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}

\begin{definition}
  \textbf{Local:} A solver is said be local if, for some system,
  $\system$, and some $e \in E_\system$, it will for $e_x$ without
  evaluating every element of $E_\system$.
\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 for all fixpoints. An example system is
presented in Figure \ref{fix:kleene-infinite}. In this case our first
iteration will give us $\{ x \mapsto 0, y \mapsto 0 \}$, leading to a
second iteration resulting in $\{ x \mapsto -1, y \mapsto 0\}$. This
will continue without bound resulting 

 In some cases Kleene iteration must iterate
an infinite number of times in order to reach a fixpoint. An example
system is presented as Figure \ref{fig:kleene-infinite}. In this case
$x$ will take the value of $0$ in the first iteration, then $y$ will
evaluate to $-1$. In the next iteration $x$ will also take the value
$-1$, thereby requiring $y$ to take the value $-2$. This will continue
without bound, resulting in the Kleene iteration never reaching the
greatest fixpoint of $\{ x \mapsto -\infty, y \mapsto -\infty \}$.

\begin{figure}[H]
  \begin{align*}
    x &= \min(0, x - 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 $\Theta(|X|)$ 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 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:wdfs}.

\begin{algorithm}
  \begin{algorithmic}
    \Globals
    \begin{tabularx}{0.9\textwidth}{rX}
      $D$ & $\in X \to \CZ$, 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 {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 $\init$ in Listing \ref{algo:wdfs: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 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.

Because each value is only computed when requested, the solver avoids
calculating results which are irrelevant to the final outcome. To
provide a more tangible example, consider the equation system
presented in Figure \ref{fig:wdfs-example}.

\begin{figure}
  \begin{align*}
    x &= \min(0, y) \\
    y &= x \\
    a &= \min(0, b) \\
    b &= c \\
    c &= d \\
    d &= e \\
    e &= a \
  \end{align*}
  \caption{Example equation system for a W-DFS solve}
  \label{fig:wdfs-example}
\end{figure}

It is clear by inspecting this system that it could be split into two
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 sub-systems it is necessary to evaluate the entire sub-system,
but it is not necessary to evaluate the other sub-system. So, for
example, to find the value of $x$ in the greatest fixpoint it is
unnecessary to find the value of the variable $d$.

W-DFS, in this case, would only evaluate the sub-system containing the
requested variable and would leave the other sub-system entirely
unsolved until a variable within it has its value requested.

This means that W-DFS can avoid evaluating a large portion of some
equation systems, thereby making it a \emph{local} solver.





\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
sub-expression. (We will also use $\sigma: \Systems \to \Systems$ as a
shorthand to denote the system obtained by replacing each
$\max$-expression, $x$, in $\system$ with $\sigma(x)$.)

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 \emph{better} in at least one $\max$-expression, and no worse
in the others.

To this end a new function, $P_{\max}: ((E_{\max} \to E), (X \to D))
\to (E_{\max} \to E)$, is used below as a ``strategy improvement
operator''. It 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 $\rho$.

In more precise terms, if have a $\max$-strategy $\sigma$, a variable
assignment $\rho$ and $\varsigma = P_{\max}(\sigma, \rho)$. Then:
\begin{align*}
  \sigma(\system)(\rho) \le \varsigma(\system)(\rho)
\end{align*}

If it is possible for $\sigma$ to be improved then $P_{\max}$ will
return an improvement, otherwise it will return $\sigma$ unchanged.

\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}
        $\sigma$ & $\in E_{\max} \to E$, 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 \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 $\sigma_{k+1} = P_{\max}(\sigma_k, \rho_k)$
        \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}

This approach is quite similar to that of Kleene iteration, but from
the results of Gawlitza et al.\cite{Gawlitza:2007:PFC:1762174.1762203}
it is known that this iteration is guaranteed to terminate. It is
still an inefficient way to solve this problem, however, for many of
the same reasons as Kleene iteration. It may spend significant time
attempting to improve strategies which can not have changed and thus
may perform a significant amount of unnecessary work.

\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}
      $\sigma$ & $\in (E_{\max} \to E)$, 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} \to E), (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 $v = P_{\max}(\lambda y . \eval(x, y), \rho)[x]$
      \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 $P_{\max}$. In Listing \ref{algo:adapted-wdfs} we take the
value of $P_{\max}(\lambda y . \eval(x, y), \rho)[x]$. That is:
improve the strategy, then use only the strategy for $x$. In order for
this algorithm to be correct $P_{\max}$ must always improve the
strategy at $x$ if it is possible to do so.

Additionally, in order to remain efficient, $P_{\max}$ should not
attempt to improve the strategy for any $\max$-expressions until that
expression is requested. This will not affect the correctness of the
algorithm, but will significantly improve the performance of it.

\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
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.

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
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).

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: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.

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.

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}
      $D$ & $\in X \to \CZ$, a mapping from variables to their current
      value \\

      $D_\old$ & $\in X \to \CZ$, a mapping from variables to their
      last stable value \\
      
      $\inflFP$ & $\in X \to 2^X$, 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} \to E), (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.

It is important to note, also, that $D$ and $D_\old$ are both
initialised to \underline{$-\infty$} and $\stableFP$ is initially the
entire set of variables. This is because we are beginning with a known
strategy mapping every max expression to $-\infty$, so we already know
that each fixpoint value must be stable at $-\infty$, so we can save
an entire fixpoint iteration step by taking this into account. This
state corresponds to an already complete fixpoint iteration with the
given initial 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 which
allows the $\max$-strategy iteration to invalidate a portion of the
system to ensure its value is recalculated by the fixpoint iteration
process. The last stable value is also stored in this procedure to
allow \emph{changed} values to be identified, not merely
\emph{invalidated} 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 exactly the same as the equivalent method in the
W-DFS algorithm, except for the fact that $\solve$ has been renamed to
$\solve \fixpoint$. It allows us to 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}
    \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. 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).

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 $\sigma(\system[x])$. The use of $\sigma$ is to ensure that
the fixpoint is solving the variable using the current
$\max$-strategy. This $\max$-strategy 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 $e = P_{\max}(\lambda y . \eval \strategy(x,y),
                          \lambda y . \eval \strategy \fixpoint(x, y))[x]$
      \If {$e \ne \sigma[x]$}
        \State $\sigma = \{ x \mapsto e \} \oplus \sigma$
        \State $\invalidate \fixpoint(\system^{-1}(x))$
        \State $\stableSI = \stableSI \backslash I[x]$
        \For {$v \in \inflSI[x]$}
          \State $\solve(v)$
        \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}

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.
