The crux of the problem is to transform a matching statement into a decision tree. A matching statement has a value, a sequence of arms, and a trailer. Each arm has a pattern, and code to be executed. When the matching statement is executed, it chooses the first arm whose pattern matches the value, then executes the corresponding code, then executes the trailer. I generate a decision tree to do the job. Each internal node of the decision tree tests a field of a word. It then chooses an edge (child) based on what range constraints can be satisfied by the value of that field, and it continues testing fields until it reaches a leaf, at which time it executes the code associated with that leaf.
The goal of tree generation is not to generate just any tree, but the tree with the fewest nodes. This problem is NP-complete, so I apply a few heuristics. The results, at least for the machine descriptions I use, seem to be as good as what I would come up with by hand. When a pattern is in normal form, it is not obvious what word is tested by a particular range constraint; one needs to know the position of the sequent containing the range constraint. To make the problem simpler, I put the patterns into a new absolute normal form, which is described by the following rules:
absolute_field
in place of a field
.
The absolute_field
gives the bit offset of the word containing the
field (its size is available from the field's class).
<*>= [D->] record adisjunct(aconstraints, name, conditions, length, patlabelbindings) # list of absolute constraints, name, conds record absolute_field(field, offset) # used to make absolute constraints
Definesabsolute_field
,adisjunct
(links are to index).
We have to store the length explicitly in an adisjunct
, because sequents that
constrain no fields are lost. The patlabelbindings
binds label
names to offsets, which are expressed in PC units, not bits.
<*>+= [<-D->] procedure anf(p) return pattern(maplist(anfd, p.disjuncts), p.name) end procedure anfd(d) local offset offset := 0 l := [] t := table() every s := !d.sequents do case type(s) of { "sequent" : { every put(l, aconstraint(!s.constraints, offset)) offset +:= s.class.size } "patlabel" : t[\s.name] := bits_to_pcunits(offset) "latent_patlabel" : &null default : impossible("sequent type") } a := adisjunct(l, d.name, d.conditions, offset, if *t > 0 then t else &null) return gsubst(a, Epatlabel_to_Epc_by_table, t, a) end
Definesanf
,anfd
(links are to index).
During decoding we eliminate the pattern label offsets by using a table of bindings. If the label is already bound, of course, we need do nothing.
<*>+= [<-D->] procedure Epatlabel_to_Epc_by_table(x, t, a) if type(x) == "Epatlabel" then return if /x.l.name then Epatlabel_to_Epc(x) else { write(\mdebug, "====> RESORTED TO TABLE in ", expimage(x)) binop(the_global_pc, "+", \t[\x.l.name]) | impossible("in ", expimage(a), "---Label ", x.l.name, " not used yet, but is not in table:", envimage(t, "pattern_table")) } end
DefinesEpatlabel_to_Epc_by_table
(links are to index).
I don't cache constraints, but I do cache fields. I have absolutely no measurements to justify either decision, but it simplifies the code to make absolute fields unique (as fields are) because they can be inserted into sets.
<*>+= [<-D->] procedure aconstraint(c, offset) return case type(c) of { "constraint" : constraint(afield(c.field, offset), c.lo, c.hi) "fieldbinding" : if x := constant(super_simplify(c.code)) then constraint(afield(c.field, offset), x, x+1) else fieldbinding(afield(c.field, offset), c.code) default : impossible("constraint type") } end
Definesaconstraint
(links are to index).
<*>+= [<-D->] procedure afield(f, offset) static tables initial tables := table() /tables[offset] := table() /tables[offset][f] := absolute_field(f, offset) return tables[offset][f] end
Definesafield
(links are to index).
#line
statements that identify the source of the code.
The original arm gives the arm from which the current arm is derived,
and is useful for many of the heuristics.
<*>+= [<-D->] record matching_stmt(arms,valcode,succptr,trailer) # case arms, code to compute value, id to set to end of p, trailing code record arm(file, line, pattern, eqns, soln, imp_soln, patlen, name, code, original) # line, file, original(pattern) are used for error reporting # These fields are the original contents: # pattern (in absoslute normal form) is pattern to match # eqns are equations given explicitly with arm (or else null) # name is identifier given in square brackets (or else null) # code is the list of code lines on the right hand side of the =>
Definesarm
,matching_stmt
(links are to index).
imp_soln
gives answers and conditions associated with identifiers
that appear as field bindings or constructor operands in the pattern.
These identifiers are the inputs to the equations.
This construct is a little odd, because the meanings of bound
identifiers and the conditions that need to be satisfied are more
naturally associated with disjuncts, not arms.
We ``raise the differences'' by splitting arms until each arm as a
unique such ``implicit solution.''
We further guarantee the uniqueness of the imp_soln
field.
The reason for going to all this trouble is to simplify the task of
dagging the eventual decision tree: we'll be able to unify nodes just
by taking the image()
of the imp_soln
field (along with a few
other goodies, of course).
If succptr
was requested in the corresponding case statment, patlen
gives the length of the pattern in the arm. We split arms as needed to
make lengths unique.
If succptr
wasn't requiested, patlen
is null.
patlen
is assigned by resolve_case_arms
.
Each node of the decision tree is associated with a particular matching
statement.
Internal nodes have children, and a field
and offset
that say
which field of which word
we decided to test on. The edges that point to the children record
the interval of values for the particular child.
Leaf nodes have a name
that records the name of the pattern known
to match at that leaf node.
<*>+= [<-D->] record node(cs, children, field, offset, name, parent) # matching statement, list of edges to children, field chosen, pattern name # (name field used to support name operator, assigned only to leaves) record edge(node, lo, hi) # node pointed to and lo and hi interval of field for this edge
Definesedge
,node
(links are to index).
To create a decision tree, I begin with a node containing the full,
original matching statement. I then use a ``work queue'' approach to check
each node and see if it needs to be split.
If no pattern matches the node, or if the first pattern always matches
(with a unique name), no further splitting needs to be done, and I
assign a name to the leaf.[If the name isn't used, I assign
the name "-unused-"
, because that will make it easier to combine
nodes in the dagging phase.]
Otherwise, I split the node.
<*>+= [<-D->] procedure needs_splitting(n) local name if *n.cs.arms = 0 then fail if not guard_always_satisfied(n.cs.arms[1].imp_soln.constraints) then return # first arm can't always match. p := n.cs.arms[1].pattern name := \p.disjuncts[1].name | p.name every d := !p.disjuncts do { n := \d.name | p.name if n ~=== name then return # needs splitting if names or answers are different else if adalwaysmatches(d) then fail # always matches, needn't split } return # pattern doesn't always match -> split end
Definesneeds_splitting
(links are to index).
I need different procedures to check matching because the patterns are in absolute normal form.
<*>+= [<-D->] procedure aalwaysmatches(p) return adalwaysmatches(!p.disjuncts) end procedure adalwaysmatches(d) if type(!d.aconstraints) == "constraint" then fail else return guard_always_satisfied(d.conditions) end
Definesaalwaysmatches
,adalwaysmatches
(links are to index).
tree
converts a matching statement into a decision tree.
<*>+= [<-D->]
procedure tree(cs)
local armcount, arm, armname, nodename
static heuristics
initial {
heuristics := [leafarms, childarms, nomatch, childdisjuncts, branchfactor]
}
root := node(copy(cs), []) # need empty children in case root not split
work := [edge(root)] # work queue of edges (nodes) to be expanded
while n := get(work).node do
if (needs_splitting(n) & *(afields := mentions(n.cs)) > 0) then {
<split node n
and add children to work queue>
} else {
write(\sdebug, "Not splitting ",
commaseparate(maplist(expimage, n.cs.arms), "\n"))
armcount := *n.cs.arms
trim_impossible_arms(n.cs)
n.name := case *n.cs.arms of {
0 : "-NOMATCH-"
default: get_nodename(n)
}
if \lc_pat_names then n.name := map(\n.name)
if armcount > *n.cs.arms then
write(\sdebug, "Trimmed node is ",
commaseparate(maplist(expimage, n.cs.arms), "\n"))
}
return root
end
Definestree
(links are to index).
We want to assign each leaf node a name, which is derived
from the names of the pattern arms that the node matches.
If all pattern arms in the node have the same name N or
are the null string, i.e., they do not specify a name,
then the node's name is simply N.
This case always holds when the node matches exactly one arm;
one arm and a default (wildcard) arm; or multiple arms that
all match the same constructor (possibly applied to different arguments).
If the names of the pattern arms in the node are not the same,
then the node's name is ambiguous, because no single
name exists for all possible matches.
An ambiguous node name will cause an error in genarm
,
if any of the node's pattern arms attempts to bind a [name]
.
<*>+= [<-D->]
procedure get_nodename(n)
local nodename, armname
nodename := armname := &null
every arm := !n.cs.arms do
if (armname := \(<Get name from pattern arm
>)) then {
write(\sdebug, "[", image(arm.name),"] = ",
image(armname), " for ",expimage(arm.pattern))
if (\nodename ~== armname) then
nodename := <Ambiguous name warning>
else nodename := armname
}
return nodename
end
Definesget_nodename
(links are to index).
<Get name from pattern arm
>= (<-U)
if \arm.name then {
\arm.pattern.disjuncts[1].name | \arm.pattern.name | &null
# "-unnamed-"
} else &null
<Ambiguous name warning>= (<-U) (warning("ambiguous name for pattern arm at ", arm.original.file, ", line ", arm.original.line, ": ", commaseparate(maplist(expimage, n.cs.arms), "\nAre you trying to decode a synthetic instruction?\n")), &null)
Splitting a node involves choosing a field, finding out which intervals of values of that field are interesting, and creating a child node for each such interval of values. The patterns in the matching statement of the child node reflect the knowledge of the value interval of the tested field.
I make the decision by splitting the node on each field mentioned in the matching statement. I then compute some heuristic functions of the children from each splitting and use the best-scoring field.
Some debugging information may be written to hdebug
or sdebug
.
<split node n
and add children to work queue>= (U->)
afields := mentions(n.cs)
*afields > 0 | impossible("internal node mentions no fields")
candidates := table()
every f := !afields do
candidates[f] := split(n, f)
<if debugging, split all and report>
*afields > 1 & write(\hdebug, "Choosing one of ", patimage(afields))
every h := !heuristics do {
if *afields = 1 then break
afields := findmaxima(h, candidates, afields)
write(\hdebug, image(h), " chose ", patimage(afields))
}
*afields > 0 | impossible("no fields")
*afields = 1 | write(\hdebug, "tie among fields", patimage(afields), " near ",
image(n.cs.arms[1].original.file), ", line ",
n.cs.arms[1].original.line)
work |||:= n.children := candidates[n.field := ?afields]
*afields = 1 | write(\hdebug, "arbitrarily chose ", patimage(n.field))
<*>+= [<-D->] procedure parentchoices(n) l := [] n := n.parent while \n do { push(l, n.field); n := n.parent } return l end
Definesparentchoices
(links are to index).
<if debugging, split all and report>= (<-U) if \tryall & \hdebug & *afields > 1 then { write(\hdebug, repl("=",10), " Splitting ", repl("=", 10)) every findmaxima(!heuristics, candidates, afields) do write(\hdebug) write(\hdebug, repl("=", 30), "\n") }
To split a node, I look at each interval of values that might be
interesting. I apply that interval to the matching statement, and if there
can be any match, I create and add a new child node.
f
is an absolute field.
<*>+= [<-D->] procedure split(n, f) local vals,v,d,val,c,p,j,i,newd,cst,child,newp, xxx patterns := [] children := [] every put(patterns, (!n.cs.arms).pattern) r := intervals(patterns, f) <if debugging, write about splitting this node> every i := 1 to *r - 1 do put(children, edge(node(apply(n.cs, f, r[i], r[i+1]),[]), r[i], r[i+1])) write(\sdebug, "Done splitting.\n") every (!children).node.parent := n return children end
Definessplit
(links are to index).
<if debugging, write about splitting this node>= (<-U) writes(\sdebug, "Splitting ") outpattern(\sdebug, patterns[1]) every i := 2 to *patterns do { writes(\sdebug, " | "); outpattern(\sdebug, patterns[i])} write(\sdebug, " on ", f.field.name, " at ", f.offset)
What is the new matching statement that results from applying
lo <=f < hi to cs
?
For each arm, I match the pattern against the interval.
If it succeeds, I create a new arm for the new matching statement,
containing the reduced pattern.
f
is an absolute field.
<*>+= [<-D->] procedure apply(cs, f, lo, hi) local newarm result := copy(cs) result.arms := [] write(\sdebug, " Applying ", stringininterval(patimage(f), lo, hi)) every a := !cs.arms do { newarm := copy(a) put(result.arms, if newarm.pattern := pmatch(a.pattern, f, lo, hi) then newarm) } if *result.arms > 1 & aalwaysmatches(result.arms[1].pattern) & guard_always_satisfied(result.arms[1].imp_soln.constraints) then { # change 21 write(\sdebug, " Trimming results of apply to ", expimage(result.arms[1])) result.arms := [result.arms[1]] } return result end
Definesapply
(links are to index).
pmatch
both tests to see whether lo <=f < hi and, if so, returns
the new p
.
f
is an absolute field.
<*>+= [<-D->]
procedure pmatch(p, f, lo, hi)
result := pattern([], p.name)
every d := !p.disjuncts do
if c := !d.aconstraints & c.field === f & type(c) == "constraint" then
# disjunct mentions f
if c.lo <= lo & hi <= c.hi then { # this constraint is matched
newd := adisjunct([], d.name, d.conditions, d.length,d.patlabelbindings)
every c := !d.aconstraints & c.field ~=== f do
put(newd.aconstraints, c)
put(result.disjuncts, newd)
} else
c.hi <= lo | c.lo >= hi | impossible("bad intervals")
else # disjunct does not mention f
put(result.disjuncts, d)
<if debugging, write about results of pmatch
>
if *result.disjuncts > 0 then return result
end
Definespmatch
(links are to index).
<if debugging, write about results of pmatch
>= (U->)
if *result.disjuncts > 0 then writes(\sdebug, " ===> ") & outpattern(\sdebug, p)
# else writes(\sdebug, " ") & outpattern(\sdebug, p)
if *result.disjuncts > 0 then write(\sdebug, " matches")
# else write(\sdebug, " does not match")
h
, candidate
splittings, and a set of fields, and returns the set of fields with
the largest score on h
.
<*>+= [<-D->] procedure findmaxima(h, candidates, afields) local max S := [] every f := !afields do { score := h(candidates[f], f) write(\hdebug,"Field ", patimage(f), " scores ", score, " on ", image(h)) /max := score - 1 if score > max then { max := score S := [f] } else if score = max then put(S, f) } return set(S) end
Definesfindmaxima
(links are to index).
Here's a big pile of heuristics. I'm not sure I've ever needed more than the first two, but they're amusing and easy enough to write.
<*>+= [<-D->] # leafarms: prefer candidate with most arms that appear at leaf # nodes. Each original arm counted only once. # Not matching is also counted as an arm. procedure leafarms(children, f) arms := set() every n := (!children).node & *n.cs.arms > 0 do if not needs_splitting(n) then insert(arms, n.cs.arms[1].original) return *arms + if *(!children).node.cs.arms = 0 then 1 else 0 end
Definesleafarms
(links are to index).
<*>+= [<-D->] # childarms: prefer the candidate with the fewest arms in children procedure childarms(children, f) sum := 0 every sum -:= *(!children).node.cs.arms return sum end
Defineschildarms
(links are to index).
<*>+= [<-D->] # nomatch: if tied on leafarms and childarms, take candidate # with real leaf in preference to nomatch leaf procedure nomatch(children, f) return if *(!children).node.cs.arms = 0 then -1 else 0 end
Definesnomatch
(links are to index).
<*>+= [<-D->] # childdisjuncts: prefer the candidate with the fewest disjuncts in children procedure childdisjuncts(children, f) sum := 0 every sum -:= *(!(!children).node.cs.arms).pattern.disjuncts return sum end
Defineschilddisjuncts
(links are to index).
<*>+= [<-D->] # branchfactor: prefer the candidate with the fewest children procedure branchfactor(children, f) return - *children end
Definesbranchfactor
(links are to index).
f
is to be used to split patterns,
intervals
returns a sorted list defining the intervals that need to be considered.
<*>+= [<-D->] procedure intervals(patterns, f) cuts := set([0, 2^fwidth(f.field)]) every p := !patterns & d := !p.disjuncts & c := !d.aconstraints & c.field === f & type(c) == "constraint" do every insert(cuts, c.lo | c.hi) return sort(cuts) end
Definesintervals
(links are to index).
mentions
produces the set containing all
absolute fields mentioned in a matching statement.
Mentions in field bindings don't count; this information is for
building decision trees only.
[The original design had no field bindings and omitting them seems to be
the best migration path.]
<*>+= [<-D->] procedure mentions(cs) result := set() every a := !cs.arms & d := !a.pattern.disjuncts & c := !d.aconstraints & type(c) == "constraint" do insert(result, c.field) return result end
Definesmentions
(links are to index).
<*>+= [<-D->] procedure trim_impossible_arms(cs) l := [] every a := !cs.arms do if arm_conditions_always_satisfied(a) then { put(l, a) if *l < *cs.arms then cs.arms := l return cs } else if member(a.imp_soln.constraints, 0) | constant(!(\a.soln).constraints) = 0 then { cs.arms := l return cs } else { put(l, a) } return cs end
Definestrim_impossible_arms
(links are to index).
<*>+= [<-D->] procedure arm_conditions_always_satisfied(a) return guard_always_satisfied(a.imp_soln.constraints) & /a.soln | guard_always_satisfied(a.soln.constraints) end
Definesarm_conditions_always_satisfied
(links are to index).
<*>+= [<-D->] # find_id: tab to and past identifier id, returning its position # ignores quotes, comment brackets procedure find_id(id) static notlnum initial notlnum := ~ (&letters ++ &digits ++ '_') tab(p := find(id)) & p = 1 | (move(-1) & any(notlnum) & move(1)) & =id & pos(0) | any(notlnum) & suspend p end
Definesfind_id
(links are to index).
<*>+= [<-D->] procedure checktree(n, cs) originals := set() every insert(originals, (!cs.arms).original) deletematching(n, originals) every show_unmatched(n, !originals) if hasnomatch(n) then warning("Matching statement at ", image(cs.arms[1].file), ", line ", n.cs.arms[1].line - 1, " doesn't cover all cases") return n end
Defineschecktree
(links are to index).
<*>+= [<-D->] procedure deletematching(n, originals) if *originals = 0 then return else if *n.children > 0 then every deletematching((!n.children).node, originals) else every delete(originals, (!n.cs.arms).original) end
Definesdeletematching
(links are to index).
<*>+= [<-D->] procedure hasnomatch(n) if *n.children > 0 then return hasnomatch((!n.children).node) else if *n.cs.arms = 0 then return # found it end
Defineshasnomatch
(links are to index).
If an arm never matches, I push its pattern through the tree and find out combinations of arms do match that pattern.
<*>+= [<-D->] procedure show_unmatched(n, a) warning("No word matches pattern at ", image(a.file), ", line ", a.line, ".") write(&errout," Covered by patterns at") every find_covering_arms(n, a, !a.pattern.disjuncts) return end procedure find_covering_arms(n, a, ad) if *n.children = 0 then every a := !n.cs.arms do write(&errout, "\t", image(a.file), ", line ", a.line) else { c := find_or_invent_constraint(n.field, ad) every e := !n.children & intervals_intersect(c.lo, c.hi, e.lo, e.hi) do find_covering_arms(e.node, a, ad) } return end
Definesfind_covering_arms
,show_unmatched
(links are to index).
<*>+= [<-D] procedure intervals_intersect(lo1, hi1, lo2, hi2) if hi1 <= lo2 | hi2 <= lo1 then fail else return end # absolute disjuncts! procedure find_or_invent_constraint(f, d) return if type(c := !d.aconstraints) == "constraint" & c.field === f then c else constraint(f, 0, 2^fwidth(f.field)) end
Definesfind_or_invent_constraint
,intervals_intersect
(links are to index).
arm
>: U1, D2
pmatch
>: U1, D2
n
and add children to work queue>: U1, D2