\section{Converting trees to dags}
The problem with the trees generated in the previous section is that
there's a different edge, and therefore a different child, for each
possible interval of the field tested, even if those children both
execute exactly the same ``original'' arm of the case statement.
The code in this section converts the trees to dags, and as part of
the process it combines edges pointing to the same node.
This can reduce the size of the tree by huge factors.

To make the transformation work, I have to represent a {\em set of
intervals} on each edge, not just a single interval.  Because no two intervals
overlap, I can use a wonderful dirty trick, detailed below.
I also {\em may} convert a node's name string to a [[namearray]] mapping
field values to strings.  The goal is for children of the same
parent to share a single name array; that way the edges can be merged and
the name operator can be implemented with an array reference.
If I don't convert a node's name, the only penalty is that the tree
might be bigger.
(Code generation will be different for the two cases.)
@
Now, the dirty representation trick: 
I can represent a set of numbers $S$ (a union of intervals) as two
sets, $lo$ and $hi$, such that
\begin{itemize}
% l2h substitution cap intersect#
% l2h substitution cup union#
% l2h substitution emptyset empty#set
\item[] $lo \cap hi = \emptyset$
\item[] if ${\tt sort}(lo \cup hi) = a, b, c, d, \ldots$, then
      $S = [a,b-1] \cup [c,d-1] \cup \ldots$.
\end{itemize}
The procedure [[addinterval]] adds a new interval to such a set $S$,
relying on the fact that no two intervals overlap.
<<*>>=
procedure addinterval(loset, hiset, lonum, hinum)
    if member(loset, hinum) then delete(loset, hinum) else insert(hiset, hinum)
    if member(hiset, lonum) then delete(hiset, lonum) else insert(loset, lonum)
    return
end
@
To convert trees to dags I need to be able to compare two nodes
for structural identity, and the easiest way is to compute a canonical
representation as a string:\par\noindent [[
   node : [fname:patimage(list of edges)]
        | <NOMATCH>
        | (image(node.name):image(node.cs.arms[1].original))
   edge : patimage(list of sort(loset ++ hiset)):node
]]
<<*>>=
procedure nodetostring(n, depth)
    static cache 
    initial cache := table()
    /depth := 0
    if /cache[n] then
        if *n.children > 0 then {
            result := "[" || n.field.name || ":"
	    every result ||:= edgetostring(!n.children, depth+2)
            cache[n] := result || "]"
        } else if *n.cs.arms = 0 then
            cache[n] := "<NOMATCH>"
        else 
            cache[n] := "(" || image(n.name) || ":" || image(n.cs.arms[1].original) ||")"
    return \cache[n]
end

procedure edgetostring(e,depth)
    return left("\n", depth) || 
          "{" || patimage(sort(e.lo ++ e.hi)) || ":" || nodetostring(e.node,depth) || "}"
end
@
Conversion to dag is the usual bottom-up hashing; here I compute the
string and then use the string to index into a table.
The real work of merging edges is done by [[combinechildren]].
If edge merging results in a single each, the node is replaced by
its child, provided the edge really covers all possible values
of the field.
<<*>>=
procedure tree2dag(n, nodetable, depth)
    /nodetable := table()
    /depth := 0
    if *n.children > 0 then
        combinechildren(n, nodetable, depth+2)	# converts edges to set form
    if *n.children = 1 then {
        e := n.children[1]
        if covers(n.children[1], n.field.hi - n.field.lo) then
            n := n.children[1].node     # all roads to child: hoist it
        else
            warning("node with one child doesn't match all cases")    
    }
    s := nodetostring(n, depth)
    /nodetable[s] := n
    return nodetable[s]
end
@
Here's where I check coverage.
Only success or failure of [[covers]] is meaningful, not
the value returned.
<<*>>=
procedure covers(e, width)
    l := sort(e.lo ++ e.hi)
    return *l = 2 & l[1] = 0 & l[2] = 2^width
end
@
The complicated stuff here is identifying a name array.
At each node, either all edges go in an exiting name array 
or a new name array is used.
If not, I create a new one.
<<*>>=
record namearray(field, tbl, hi, codename)
	# field used as index, table[integer] of name, bound on table, name of this array
global natable
<<*>>=
procedure arraycandidates(n)
    initial MAXRANGE := 32
    suspend e := !n.children & type(e.node.name) == "string" & 
            e.hi - e.lo <= MAXRANGE & e
end

procedure combinechildren(n, nodetable, depth)
    initial natable := table()

    if arraycandidates(n).node.name ~== arraycandidates(n).node.name then {
        <<change names of children from strings to namearrays when possible>>
    }

    lotable := table()
    hitable := table()
    every e := !n.children & child := tree2dag(e.node, nodetable, depth) do {
        /lotable[child] := set()
        /hitable[child] := set()
        addinterval(lotable[child], hitable[child], e.lo, e.hi)
    }
    n.children := []
    every child := key(lotable) do
        put(n.children, edge(child, lotable[child], hitable[child]))
    return
end
<<change names of children from strings to namearrays when possible>>=
mightuse := set()           # name arrays we might use must have right field
every na := !\natable[n.field] do
    insert(mightuse, na)
every e := arraycandidates(n) & na := !mightuse do
    if \na.tbl[e.lo to e.hi - 1] ~== e.node.name then  # slot used with wrong name
        delete(mightuse, na)
if *mightuse > 0 then
    willuse := ?mightuse
else {
    /natable[n.field] := set()
    insert(natable[n.field], willuse := namearray(n.field, table(), 0))
}
every e := arraycandidates(n) &
      e.lo - willuse.hi <= MAXRANGE do {
          every willuse.tbl[e.lo to e.hi - 1] := e.node.name;
          e.node.name := willuse
          willuse.hi <:= e.hi
      }
<<*>>=
procedure namesused(n, result)
    /result := set()
    if type(n.name) == "namearray" then insert(result, n.name)
    every namesused((!n.children).node, result)
    return result
end
