X-Git-Url: http://www.chiark.greenend.org.uk/ucgi/~ian/git?p=topbloke-formulae.git;a=blobdiff_plain;f=strategy.tex;h=e2bcf46fc50dfa768b2108b930b9728a0d39cae7;hp=9bed7cf094dd3f0a8940ff8d9a0338cb982e6b6a;hb=be08dfe44764573fcf4d903dbe74576e3762ee04;hpb=64a3ef281cdbfa63f1c306222d6e3e2b507d21a0

diff --git a/strategy.tex b/strategy.tex
index 9bed7cf..e2bcf46 100644
--- a/strategy.tex
+++ b/strategy.tex
@@ -6,7 +6,7 @@ remove dependencies from patches.
 
 Broadly speaking the update proceeds as follows: during the Ranking
 phase we construct the intended graph of dependencies between patches
-(which involves select a merge order for the base branch of each
+(and incidentally select a merge order for the base branch of each
 patch).  Then during the Traversal phase we walk that graph from the
 bottom up, constructing for each patch by a series of merges and other
 operations first a new base branch head commit and then a new tip
@@ -50,13 +50,6 @@ the $\le$-maximal elements of $\bigcup_{J \in \set J} \pendsof{J}{\p}$
 Convenience notation for
 $\bigforall_{E \in \pendsof{\set X}{\p}} E \le T$
 
-\item[ $\Gamma_{\pc}$ ]
-The desired direct dependencies of $\pc$, a set of patches.
-
-\item[ $\allpatches$ ]
-The set of all the patches we are dealing with (constructed
-during the update algorithm).
-
 %\item[ $\set E_{\pc}$ ]
 %$ \bigcup_i \pendsof{S_{\pc,i}}{\pc} $.
 %All the ends of $\pc$ in the sources.
@@ -92,6 +85,21 @@ dependencies to use.  This allows the specification of any desired
 
 \end{basedescript}
 
+\stdsection{Important variables and values in the update algorithm}
+
+\begin{basedescript}{
+\desclabelwidth{5em}
+\desclabelstyle{\nextlinelabel}
+}
+\item[ $\Gamma_{\pc}$ ]
+The desired direct dependencies of $\pc$, a set of patches.
+
+\item[ $\allpatches$ ]
+The set of all the patches we are dealing with (constructed
+during the update algorithm).
+
+\end{basedescript}
+
 \section{Ranking phase}
 
 We run the following algorithm:
@@ -100,29 +108,29 @@ We run the following algorithm:
 \item Repeatedly:
 \begin{enumerate}
 \item Clear out the graph $\hasdirdep$ so it has no edges.
-\item Execute {\bf Rank-Recurse}($\pc_0$)
+\item Execute $\alg{Rank-Recurse}(\pc_0)$
 \item Until $\allpatches$ remains unchanged.
 \end{enumerate}
 \end{enumerate}
 
-{\bf Rank-Recurse}($\pc$) is:
+$\alg{Rank-Recurse}(\pc)$ is:
 \begin{enumerate}
 
-\item If we have already done {\bf Rank-Recurse}($\pc$) in this
+\item If we have already done $\alg{Rank-Recurse}(\pc)$ in this
 ranking iteration, do nothing.  Otherwise:
 
 \item Add $\pc$ to $\allpatches$ if it is not there already.
 
-\item Let
+\item Set
 $$
-  \set S = h(\pcn)
+  \set S \iassign h(\pcn)
      \cup 
         \bigcup_{\p \in \allpatches}
         \bigcup_{H \in h(\pn) \lor H \in h(\py)}
          \{ \baseof{E} \; | \; E \in \pendsof{H}{\pcy} \}
 $$
 
-and $W = w(h(\pcn))$
+and $W \iassign w(h(\pcn))$
 
 \item While $\exists_{S \in \set S} S \ge W$,
 update $W \assign S$ and $\set S \assign \set S \, \backslash \{ S \}$
@@ -130,16 +138,16 @@ update $W \assign S$ and $\set S \assign \set S \, \backslash \{ S \}$
 (This will often remove $W$ from $\set S$.  Afterwards, $\set S$
 is a collection of heads to be merged into $W$.)
 
-\item Choose an order of $\set S$, $S_i$ for $i=1 \ldots n$.
+\item Choose an ordering of $\set S$, $S_i$ for $i=1 \ldots n$.
 
 \item For each $S_i$ in turn, choose a corresponding $M_i$
 such that $$
    M_i \le S_i \land \left[
-   M_i \le W \lor \bigexists_{S_i, j<i} M_i \le S_i
+   M_i \le W \lor \bigexists_{j<i} M_i \le S_j
    \right]
 $$
 
-\item Set $\Gamma = \depsreqof{W}$.
+\item Set $\Gamma \iassign \depsreqof{W}$.
 
 If there are multiple candidates we prefer $M_i \in \pcn$
 if available.
@@ -147,16 +155,19 @@ if available.
 \item For each $i \ldots 1..n$, update our putative direct
 dependencies:
 $$
-\Gamma \assign \text{\bf set-merge}\left[\Gamma, 
- \left( \begin{cases} 
-  M_i \in \pcn :     & \depsreqof{M_i} \\
-  M_i \not\in \pcn : & \{ \}
- \end{cases} \right),
- \depsreqof{S_i}
- \right]
+\Gamma \assign \setmergeof{
+    \Gamma
+  }{
+    \begin{cases}
+     M_i \in \pcn :     & \depsreqof{M_i} \\
+     M_i \not\in \pcn : & \{ \}
+    \end{cases}
+  }{
+    \depsreqof{S_i}
+  }
 $$
 
-TODO define {\bf set-merge}
+TODO define $\setmerge$
 
 \item Finalise our putative direct dependencies
 $
@@ -169,9 +180,9 @@ $
 \item Add an edge $\pc \hasdirdep \pd$ to the digraph (adding nodes
 as necessary).
 If this results in a cycle, abort entirely (as the function $g$ is
-inappropriate; a different $g$ could work.)
+inappropriate; a different $g$ could work).
+\item Run $\alg{Rank-Recurse}(\pd)$.
 \end{enumerate}
-\item Run ${\text{\bf Rank-Recurse}}(\pd)$.
 
 \end{enumerate}
 
@@ -186,24 +197,43 @@ $ \allpatches, \hasdirdep $ and hence the completion of $\hasdirdep$
 into the partial order $\hasdep$.
 
 \item
-For each $\pc \in \allpatches$, the base branch starting point commit $W_{\pcn} = W$.
+For each $\pc \in \allpatches$,
+the base branch starting point commit $W^{\pcn} = W$.
 
 \item
 For each $\pc$,
-the direct dependencies $\Gamma_{\pc}$.
+the direct dependencies $\Gamma^{\pc} = \Gamma$.
 
 \item
 For each $\pc$,
-the ordered set of base branch sources $\set S_{\pcn} = \set S,
-S_{\pcn,i} = S_i$
-and corresponding merge bases $M_{\pcn,i} = M_i$.
+the ordered set of base branch sources $\set S^{\pcn} = \set S,
+S^{\pcn}_i = S_i$
+and corresponding merge bases $M^{\pcn}_i = M_i$.
 
 \end{itemize}
 
+\subsection{Proof of termination}
+
+$\alg{Rank-Recurse}(\pc)$ recurses but only downwards through the
+finite graph $\hasdirdep$, so it must terminate.  
+
+The whole ranking algorithm iterates but each iteration involves
+adding one or more patches to $\allpatches$.  Since there are
+finitely many patches and we never remove anything from $\allpatches$
+this must complete eventually.
+
+$\qed$
+
 \section{Traversal phase}
 
+For each patch $C \in \allpatches$ in topological order by $\hasdep$,
+lowest first:
 
+\begin{enumerate}
+
+\item Optionally, attempt $\alg{Merge-Base}(\pc)$.
 
+\end{enumerate}
 
 
 \section{Planning phase}