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diff --git a/tex/thesis/contribution/contribution.tex b/tex/thesis/contribution/contribution.tex index acf9cdd..3e3ed6c 100644 --- a/tex/thesis/contribution/contribution.tex +++ b/tex/thesis/contribution/contribution.tex @@ -57,7 +57,7 @@ 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 +chapter will conclude with our Local Demand-driven Strategy Improvement (LDSI) algorithm to perform efficient strategy iteration and fixpoint iteration. % TODO: fix me, or something @@ -74,6 +74,7 @@ which are already known. \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 $\CZ = \Z \cup \{-\infty, \infty\}$. @@ -86,7 +87,7 @@ variables. We will also define $\CZ = \Z \cup \{-\infty, \infty\}$. 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 in the following way: + 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) \\ @@ -96,9 +97,7 @@ variables. We will also define $\CZ = \Z \cup \{-\infty, \infty\}$. \end{definition} \begin{definition} - \textbf{Expressions:} - - The class of expressions, $E$, is defined inductively: + The class of \textbf{expressions}, $E$, is defined inductively: \begin{itemize} \item If $z \in \CZ$ then $z \in E$ @@ -111,7 +110,7 @@ variables. We will also define $\CZ = \Z \cup \{-\infty, \infty\}$. Nothing else is in E. \end{itemize} - We then define the semantics of these expressions by: + 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 \\ @@ -119,8 +118,8 @@ variables. We will also define $\CZ = \Z \cup \{-\infty, \infty\}$. \lambda \rho . f(\llb e_1 \rrb(\rho),...,\llb e_k \rrb(\rho)) \end{align*} - By abusing our notation we will consider expressions, $e \in E$, as - $e : (X \to \CZ) \to \CZ$ defined by $e = \llb e \rrb$. + 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 @@ -128,8 +127,8 @@ variables. We will also define $\CZ = \Z \cup \{-\infty, \infty\}$. 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 $e_{\parent}$. + 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} @@ -146,15 +145,15 @@ variables. We will also define $\CZ = \Z \cup \{-\infty, \infty\}$. 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 get a new value - for each variable and thus a new variable assignment as a result. + 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. - These two functional forms are equivalent and correspond to changing + The 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. + \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 @@ -171,8 +170,8 @@ variables. We will also define $\CZ = \Z \cup \{-\infty, \infty\}$. \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 + A variable or expression $x$ is said to \textbf{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) @@ -399,7 +398,7 @@ 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 +other system unevaluated. In order to capture this property we will define the notion of a \emph{local} solver. \begin{definition} @@ -408,35 +407,31 @@ define the notion of a \emph{local} solver. 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 $\sigma(e)$. +$\max$-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. This means that each time we improve our strategy we make it -greater in at least one $\max$-expression, and no worse in the -others. That is, we must satisfy $\sigma_{n+1}(\rho) > -\sigma_n(\rho)$, for the $\rho$ in which we are improving. +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 +`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*} @@ -449,8 +444,7 @@ $\sigma$ 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 only done when -requested. +strategy is so the evaluation of each strategy is demand-driven. \begin{algorithm} \begin{algorithmic} @@ -476,9 +470,9 @@ requested. 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}. +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}[H] \begin{algorithmic} @@ -512,24 +506,24 @@ the approach presented in Listing \ref{algo:naive-strategy}. \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 +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 viewed as an accelerated fixpoint +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 $e \in -E_{\max}$. Therefore if we consider our ``variables'' to be +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 allow it to -solve $\max$-strategy iteration problems. +\ref{algo:adapted-wdfs} presents one variation on W-DFS to solve +$\max$-strategy iteration problems. \begin{algorithm}[H] \begin{algorithmic} @@ -643,31 +637,29 @@ computation. \section{Local Demand-driven Strategy Improvement (LDSI)} W-DFS speeds 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. +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{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). +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). Both portions of the LDSI algorithm are \emph{demand driven}, and so any necessary evaluation of strategies or fixpoint values is delayed @@ -948,13 +940,13 @@ 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. +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 also must inspect the +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 @@ -983,10 +975,10 @@ $\invalidate \strategy$. 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. +$\max$-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}[H] |