CommonArg
=========

<:CommonArg:> is an optimization pass for the <:SSA:>
<:IntermediateLanguage:>, invoked from <:SSASimplify:>.

== Description ==

It optimizes instances of `Goto` transfers that pass the same
arguments to the same label; e.g.
----
L_1 ()
  ...
  z1 = ?
  ...
  L_3 (x, y, z1)
L_2 ()
  ...
  z2 = ?
  ...
  L_3 (x, y, z2)
L_3 (a, b, c)
  ...
----

This code can be simplified to:
----
L_1 ()
  ...
  z1 = ?
  ...
  L_3 (z1)
L_2 ()
  ...
  z2 = ?
  ...
  L_3 (z2)
L_3 (c)
  a = x
  b = y
----
which saves a number of resources: time of setting up the arguments
for the jump to `L_3`, space (either stack slots or temporaries) for
the arguments of `L_3`, etc.  It may also expose some other
optimizations, if more information is known about `x` or `y`.

== Implementation ==

* <!ViewGitFile(mlton,master,mlton/ssa/common-arg.fun)>

== Details and Notes ==

Three analyses were originally proposed to drive the optimization
transformation.  Only the _Dominator Analysis_ is currently
implemented.  (Implementations of the other analyses are available in
the <:Sources:repository history>.)

=== Syntactic Analysis ===

The simplest analysis I could think of maintains
----
varInfo: Var.t -> Var.t option list ref
----
initialized to `[]`.

* For each variable `v` bound in a `Statement.t` or in the
`Function.t` args, then `List.push(varInfo v, NONE)`.
* For each `L (x1, ..., xn)` transfer where `(a1, ..., an)` are the
formals of `L`, then `List.push(varInfo ai, SOME xi)`.
* For each block argument a used in an unknown context (e.g.,
arguments of blocks used as continuations, handlers, arith success,
runtime return, or case switch labels), then
`List.push(varInfo a, NONE)`.

Now, any block argument `a` such that `varInfo a = xs`, where all of
the elements of `xs` are equal to `SOME x`, can be optimized by
setting `a = x` at the beginning of the block and dropping the
argument from `Goto` transfers.

That takes care of the example above.  We can clearly do slightly
better, by changing the transformation criteria to the following: any
block argument a such that `varInfo a = xs`, where all of the elements
of `xs` are equal to `SOME x` _or_ are equal to `SOME a`, can be
optimized by setting `a = x` at the beginning of the block and
dropping the argument from `Goto` transfers.  This optimizes a case
like:
----
L_1 ()
  ... z1 = ? ...
  L_3 (x, y, z1)
L_2 ()
  ... z2 = ? ...
  L_3(x, y, z2)
L_3 (a, b, c)
  ... w = ? ...
  case w of
    true => L_4 | false => L_5
L_4 ()
   ...
   L_3 (a, b, w)
L_5 ()
   ...
----
where a common argument is passed to a loop (and is invariant through
the loop).  Of course, the <:LoopInvariant:> optimization pass would
normally introduce a local loop and essentially reduce this to the
first example, but I have seen this in practice, which suggests that
some optimizations after <:LoopInvariant:> do enough simplifications
to introduce (new) loop invariant arguments.

=== Fixpoint Analysis ===

However, the above analysis and transformation doesn't cover the cases
where eliminating one common argument exposes the opportunity to
eliminate other common arguments.  For example:
----
L_1 ()
  ...
  L_3 (x)
L_2 ()
  ...
  L_3 (x)
L_3 (a)
  ...
  L_5 (a)
L_4 ()
  ...
  L_5 (x)
L_5 (b)
  ...
----

One pass of analysis and transformation would eliminate the argument
to `L_3` and rewrite the `L_5(a)` transfer to `L_5 (x)`, thereby
exposing the opportunity to eliminate the common argument to `L_5`.

The interdependency the arguments to `L_3` and `L_5` suggest
performing some sort of fixed-point analysis.  This analysis is
relatively simple; maintain
----
varInfo: Var.t -> VarLattice.t
----
{empty}where
----
VarLattice.t ~=~ Bot | Point of Var.t | Top
----
(but is implemented by the <:FlatLattice:> functor with a `lessThan`
list and `value ref` under the hood), initialized to `Bot`.

* For each variable `v` bound in a `Statement.t` or in the
`Function.t` args, then `VarLattice.<= (Point v, varInfo v)`
* For each `L (x1, ..., xn)` transfer where `(a1, ..., an)` are the
formals of `L`}, then `VarLattice.<= (varInfo xi, varInfo ai)`.
* For each block argument a used in an unknown context, then
`VarLattice.<= (Point a, varInfo a)`.

Now, any block argument a such that `varInfo a = Point x` can be
optimized by setting `a = x` at the beginning of the block and
dropping the argument from `Goto` transfers.

Now, with the last example, we introduce the ordering constraints:
----
varInfo x <= varInfo a
varInfo a <= varInfo b
varInfo x <= varInfo b
----

Assuming that `varInfo x = Point x`, then we get `varInfo a = Point x`
and `varInfo b = Point x`, and we optimize the example as desired.

But, that is a rather weak assumption.  It's quite possible for
`varInfo x = Top`.  For example, consider:
----
G_1 ()
  ... n = 1 ...
  L_0 (n)
G_2 ()
  ... m = 2 ...
  L_0 (m)
L_0 (x)
  ...
L_1 ()
  ...
  L_3 (x)
L_2 ()
  ...
  L_3 (x)
L_3 (a)
  ...
  L_5(a)
L_4 ()
  ...
  L_5(x)
L_5 (b)
   ...
----

Now `varInfo x = varInfo a = varInfo b = Top`.  What went wrong here?
When `varInfo x` went to `Top`, it got propagated all the way through
to `a` and `b`, and prevented the elimination of any common arguments.
What we'd like to do instead is when `varInfo x` goes to `Top`,
propagate on `Point x` -- we have no hope of eliminating `x`, but if
we hold `x` constant, then we have a chance of eliminating arguments
for which `x` is passed as an actual.

=== Dominator Analysis ===

Does anyone see where this is going yet?  Pausing for a little
thought, <:MatthewFluet:> realized that he had once before tried
proposing this kind of "fix" to a fixed-point analysis -- when we were
first investigating the <:Contify:> optimization in light of John
Reppy's CWS paper.  Of course, that "fix" failed because it defined a
non-monotonic function and one couldn't take the fixed point.  But,
<:StephenWeeks:> suggested a dominator based approach, and we were
able to show that, indeed, the dominator analysis subsumed both the
previous call based analysis and the cont based analysis.  And, a
moment's reflection reveals further parallels: when
`varInfo: Var.t -> Var.t option list ref`, we have something analogous
to the call analysis, and when `varInfo: Var.t -> VarLattice.t`, we
have something analogous to the cont analysis.  Maybe there is
something analogous to the dominator approach (and therefore superior
to the previous analyses).

And this turns out to be the case.  Construct the graph `G` as follows:
----
nodes(G) = {Root} U Var.t
edges(G) = {Root -> v | v bound in a Statement.t or
                                in the Function.t args} U
           {xi -> ai | L(x1, ..., xn) transfer where (a1, ..., an)
                                      are the formals of L} U
           {Root -> a | a is a block argument used in an unknown context}
----

Let `idom(x)` be the immediate dominator of `x` in `G` with root
`Root`.  Now, any block argument a such that `idom(a) = x <> Root` can
be optimized by setting `a = x` at the beginning of the block and
dropping the argument from `Goto` transfers.

Furthermore, experimental evidence suggests (and we are confident that
a formal presentation could prove) that the dominator analysis
subsumes the "syntactic" and "fixpoint" based analyses in this context
as well and that the dominator analysis gets "everything" in one go.

=== Final Thoughts ===

I must admit, I was rather surprised at this progression and final
result.  At the outset, I never would have thought of a connection
between <:Contify:> and <:CommonArg:> optimizations.  They would seem
to be two completely different optimizations.  Although, this may not
really be the case.  As one of the reviewers of the ICFP paper said:
____
I understand that such a form of CPS might be convenient in some
cases, but when we're talking about analyzing code to detect that some
continuation is constant, I think it makes a lot more sense to make
all the continuation arguments completely explicit.

I believe that making all the continuation arguments explicit will
show that the optimization can be generalized to eliminating constant
arguments, whether continuations or not.
____

What I think the common argument optimization shows is that the
dominator analysis does slightly better than the reviewer puts it: we
find more than just constant continuations, we find common
continuations.  And I think this is further justified by the fact that
I have observed common argument eliminate some `env_X` arguments which
would appear to correspond to determining that while the closure being
executed isn't constant it is at least the same as the closure being
passed elsewhere.

At first, I was curious whether or not we had missed a bigger picture
with the dominator analysis.  When we wrote the contification paper, I
assumed that the dominator analysis was a specialized solution to a
specialized problem; we never suggested that it was a technique suited
to a larger class of analyses.  After initially finding a connection
between <:Contify:> and <:CommonArg:> (and thinking that the only
connection was the technique), I wondered if the dominator technique
really was applicable to a larger class of analyses.  That is still a
question, but after writing up the above, I'm suspecting that the
"real story" is that the dominator analysis is a solution to the
common argument optimization, and that the <:Contify:> optimization is
specializing <:CommonArg:> to the case of continuation arguments (with
a different transformation at the end).  (Note, a whole-program,
inter-procedural common argument analysis doesn't really make sense
(in our <:SSA:> <:IntermediateLanguage:>), because the only way of
passing values between functions is as arguments.  (Unless of course
in the case that the common argument is also a constant argument, in
which case <:ConstantPropagation:> could lift it to a global.)  The
inter-procedural <:Contify:> optimization works out because there we
move the function to the argument.)

Anyways, it's still unclear to me whether or not the dominator based
approach solves other kinds of problems.

=== Phase Ordering ===

On the downside, the optimization doesn't have a huge impact on
runtime, although it does predictably saved some code size.  I stuck
it in the optimization sequence after <:Flatten:> and (the third round
of) <:LocalFlatten:>, since it seems to me that we could have cases
where some components of a tuple used as an argument are common, but
the whole tuple isn't.  I think it makes sense to add it after
<:IntroduceLoops:> and <:LoopInvariant:> (even though <:CommonArg:>
get some things that <:LoopInvariant:> gets, it doesn't get all of
them).  I also think that it makes sense to add it before
<:CommonSubexp:>, since identifying variables could expose more common
subexpressions.  I would think a similar thought applies to
<:RedundantTests:>.
