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I guess that depends on what you mean by equivalence between CPS and SSA.

My prefered form is basic blocks with arguments. Where the final jump has parameters it passes to it's destination blocks.

That form has a 1-1 equivalence with SSA and I find it much easier to reason about. Every SSA algorithm I have looked at so far works just fine on the representation. Plus there is none of the annoying transforming into and out of SSA.

If you omit the mucking about with return closures in CPS what is left is basic blocks with arguments.

To see basic blocks with arguments as a form of SSA just see every basic block argument as a phi function, and the callers parameters that feed into that argument as the other end of the sources of the phi function.

Just from your description I think your pizlo special SSA is actually a form of basic blocks with arguments. What I don't see from your description is why Upsilon and phi aren't combined into a single notion (What I would call an incoming block parameter).

Pizlo form is not basic block with arguments, because basic blocks don't have arguments in Pizlo form.

I don't think basic block with arguments is CPS because CPS, in practice, involves return closures.

What is Upsilon? It sounds like something that figures out the value coming into a block, which I would call a basic block argument.

How does Upsilon know which values from other basic blocks it could receive? That information is critical for transformations like constant propagation, and register allocation.

Upsilon doesn't know anything about basic blocks. It's just an assignment.

Upsilon uses an SSA value, and assigns to a Phi. The use of an SSA value is just a normal SSA use.

The assignment to the Phi is special, and follows Static Single Use law.

In practice, you'll want to either have all Phis know about the Upsilons that refer to them, or you'll want to have a handy analysis that tells you about the Upsilons that refer to Phis. This part is no different from how your SSA form will either have user tracking (like LLVM IR) or an analysis that tells you the users.

Nit: the correspondence between phi-SSA and basic block arguments is not one to one. With block arguments, you can jump to the same block with different arguments depending on a condition. You can’t do that in SSA without adding new blocks.

> My prefered form is basic blocks with arguments. Where the final jump has parameters it passes to it's destination blocks.

This was my idea too; I also think this is better.

Isn't it supposed to be ANF ("A-normal form") rather than full CPS? That looks a bit closer to what you're talking about re: basic blocks w/ arguments.

Have you had experience implementing SSA via sea-of-nodes representation? Could it be that in this case dominance frontier is no longer important and one could use the simpler SSA construction algorithms that do not require dominance frontiers?

The Pizlo special approach sounds a bit like converting out of SSA form via compensating `alloca`s in LLVM. E.g. one `alloca` per SSA variable, with a `store` into the `alloca` in the source block, and the `phi` replaced by a `load`.

If this is the case, this is an approach I've taken in the past to unify how LLVM-based taint tracking instrumentation of "normal" `alloca`s and phi nodes works, e.g.: https://github.com/lifting-bits/vmill/blob/master/tools/Tain...

It's not, because the Phi's in Pizlo form introduce a Static Single Use variable, which makes them easy to analyze.

Allocas have no requirement that there's a static single use. They are quite hard to analyze.

>Phi just loads the value out of its shadow variable and assigns it to its implicit variable.

Isn't this similar to block arguments? These shadow variables would have been block parameters, just unordered and maybe not explicitly collected in a set.

Please write up your "Pizlo special" on Phi nodes

Sure.

Let's first define a syntax for SSA and some terminology. Here's an example SSA node:

    A = Add(B, C)

In reality, this will be a single object in your in-memory representation, and the names are really addresses of those objects. So, this node has an "implicit variable" called A; it's the variable that is implicitly assigned to when you execute the node. If you then do:

    X = Sub(A, 1)

Then "A" is just a pointer to the Add node, and we're using the implicit variable "A".

Here's an example function:

    int foo(int a, int b)
    {
        int x;
        if (a)
            x = b + 1
        else
            x = b * 2
        return x + 42;
    }

Here's an SSA program with a Phi in Pizlo form:

    root:
        A = GetArgument(0)
        B = GetArgument(1)
        Branch(A, then, else)
    then:
        X1 = Add(B, 1)
        Upsilon(X1, ^X)
        Jump(return)
    else:
        X2 = Mul(B, 2)
        Upsilon(X2, ^X)
        Jump(return)
    return:
        X = Phi()
        R = Add(X, 42)
        Return(R)

In Pizlo form:

- Every SSA node has an implicit variable, as mentioned above.

- Every Phi node has a shadow variable in addition to the implicit variable.

Let's say that given a Phi like "X = Phi()", the implicit variable is called "X", and the shadow variable is called "^X".

Therefore, the semantics of an upsilon like "Upsilon(X1, ^X)" is just "set ^X = X1". And the semantics of a Phi like "X = Phi()" is just "set X = ^X".

In other words, you can think of Upsilon as being a side effect (a store to a shadow variable). And you can think of Phi as being a side effect (a load from a shadow variable). You can model them that way in your effect analysis to block reordering Upsilons and Phis.

But also, the shadow variables of Phis in Pizlo form are "Static Single Use" (SSU) variables. This falls out naturally from the fact that the only syntax for loading a shadow variable is the Phi itself. So you can think of Pizlo form as "SSA-SSU form".

The main benefit of this form is that basic blocks - and all CFG data structures - have zero knowledge about SSA. There are no basic block arguments. There's no requirement that Phis appear at the tops of blocks. In fact, this is a valid program in Pizlo form (albeit suboptimal):

    M = Stuff(...)
    Upsilon(M, ^N)
    N = Phi()
    MoreStuff(N)

Here, there's a Phi in them middle of a basic block, and there's an Upsilon just before it. That's fine. This is important, because it means that you can do CFG transforms that blow away control flow edges without worrying about fixing your Phis.

In any Pizlo-form compiler, you'll want to have a Phi simplification pass, which you can implement either by running Cytron or by running any other SSA converter. The simplest is just to just fixpoint the rule that if you have a Phi that has Upsilons that only use the Phi or exactly one other value, then replace the Phi with that other value.

Mind if I quote you on this in the post?

Don't mind at all!

Worth calling out that this is the form used by two forms in JavaScriptCore:

- DFG-SSA (used for high level optimizations in the FTL JIT)

- B3 (used for low level optimizations in the FTL JIT)

B3 is somewhat well documented: https://www.webkit.org/docs/b3/intermediate-representation.h...

Thank you!

I posted this to a gist, so it's a bit easier to find/cite: https://gist.github.com/pizlonator/79b0aa601912ff1a0eb1cb925...

> CPS isn’t really anything like SSA.

Though with CPS you can do fun stuff like copy-and-patch compilation and the fancified musttail interpreter thing (which I'm not sure even has a proper name). Admittedly the two are so similar that the second one could (probably) be converted into the first one without too much trouble. From what I understand, IANAL &etc.