Why Ssa?

Edit: It was this paper by Brandis and Mössenböck: https://share.google/QNoV9G8yMBWQJqC82

1. Removing that statement (dead code elimination)

2. Deduplicating that statement (available expressions)

3. Reordering that statement with other statements (hoisting; loop-invariant code motion)

4. Duplicating that statement (can be useful to enable other optimizations)

All of the above optimizations are very important in compilers, and they are much, much easier to implement if you don't have to worry about preserving side effects while manipulating the program.

So the point of SSA is to translate a program into an equivalent program whose statements have as few side effects as possible. The result is often something that looks like a functional program. (See: https://www.cs.princeton.edu/~appel/papers/ssafun.pdf, which is famous in the compilers community.) In fact, if you view basic blocks themselves as a function, phi nodes "declare" the arguments of the basic block, and branches correspond to tailcalling the next basic block with corresponding values. This has motivated basic block arguments in MLIR.

The "combinatorial circuit" metaphor is slightly wrong, because most SSA implementations do need to consider state for loads and stores into arbitrary memory, or arbitrary function calls. Also, it's not easy to model a loop of arbitrary length as a (finite) combinatorial circuit. Given that the author works at an AI accelerator company, I can see why he leaned towards that metaphor, though.

Ken zadeck was my office mate for years, so this is my recollection, but it's also been a few decades, so sorry for any errors :)

The reason of why it works is definitely wrong - they weren't even using rewriting forms of SSA, and didn't for a long time. Even the first "fully SSA" compiler (generally considered to be open64) did not rewrite the IR into SSA.

The reason it works so well is because it enables you to perform effective per-variable dataflow. In fact, this is the only problem it solves - the ability to perform unrestricted per-variable dataflow quickly. This was obvious given the historical context at the time, but less obvious now that it is history :)

In the simplest essence, SSA enables you to follow chains of dataflow for a variable very easily and simply, and that's probably what i would have said instead.

It's true that for simple scalar programs, the variable name reuse that breaks these dataflow chains mostly occur at explicit mutation points, but this is not always correct depending on the compiler IR, and definitely not correct you extend it to memory, among other things. It's also not correct at all as you extend the thing SSA enables you to do (per-variable dataflow quickly) to other classes of problems (SSU, SSI, etc).

History wise - this post also sort of implies SSA came out of nowhere and was some revolutionary thing nobody had ever thought about, just sort of developed in the 80's.

In fact, it was a formalization of attempts at per-variable dataflow they had been working on for a while.

I'd probably just say that as a gentle intro, but if you want the rest of history, here you go:

Well before SSA, it was already known that lower bounds on bitvector dataflow (the dominant approach at the time of SSA) were not great. Over the years, it turned out they were much better than initially expected, but in the end, worse than anyone wanted as programs got bigger and bigger. N^2 or N^3 bitvector operations for most problems. Incrementality is basically impossible[2]. They were also hard to understand and debug because of dependencies between related variables, etc.

Attempts at faster/better dataflow existed in two rough forms, neither of which are bound by the same lower bound:

1. Structured dataflow/Interval analysis algorithms/Elimination dataflow - reduce the CFG into various types of regions with a known system of equations, solve the equations, distribute the results to the regions. Only works well on reducible graphs Can be done incrementally with care. This was mostly studied in parallel to bitvectors, and was thought heavily about before it became known that there was a class of rapid dataflow problems (IE before the late 70's). Understanding the region equations well enough to debug them requires a very deep understanding of the basis of dataflow - lattices, etc.

In that sense it was worse than bitvectors to understand for most people. Graph reducibility restrictions were annoying but tractable on forward graphs through node splitting and whatnot (studies showed 90+% of programs at the time had reducible flowgraphs already), but almost all backwards CFG's are not reducible, making backwards dataflow problems quite annoying. In the end, as iterative dataflow became provably faster/etc, and compilers became the province of more than just theory experts, this sort of died[3]. If you ever become a compiler theory nerd, it's actually really interesting to look at IMHO.

2. Per-variable dataflow approaches. This is basically "solve a dataflow problem for single variable or cluster of variables at a time instead of for all variables at once". There were not actually a ton of people who thought this would ever turn into a practical approach:

a. The idea of solving reaching definitions (for example) one variable at a time seemed like it would be much slower than solving it for all variables at once.

b. It was very non-obvious how to be able to effectively partition variables to be able to compute a problem on one variable (or a cluster of variables) at a time without it devolving into either bitvector-like time bounds or structured dataflow like math complexity.

It was fairly obvious at the time that if you culd make it work, you could probably get much faster incremental dataflow solution.

SSA came out of this approach. Kenny's thesis dealt with incremental dataflow and partitioned variable problems, and proved time bounds on various times of partitioned/clustered variable problems. You can even see the basis of SSA in how it thinks about things. Here, i'll quote from a few parts:

" As the density of Def sites per variable increases, the average size of the affected area for each change will decrease...".

His thesis is (amont other things) on a method for doing this that is typical for the time (IE not SSA):

"The mechanism used to enhance performance is raising the density of Def sites for each variable. The most obvious way to increase the Use and Def density is to attempt a reduction on the program flow graph. This reduction would replace groups of nodes by a single new node. This new node would then be labeled with infcrmation that summarized the effects of execution through that group of nodes."

This is because, as i mentioned, figuring out how to do it by partitioning the variables based on dominance relationships was not known - that is literally SSA, and also because Kenny wanted to finish his thesis before the heat death of the sun.

In this sense, they were already aware that if they got "the right number of names" for a variable, the amount of dataflow computation needed for most problems (reaching defs, etc) would become very small, and that changes would be easy to handle. They knew fairly quickly that for the dataflow problems they wanted to solve, they needed each variable to have a single reaching definition (and reaching definitions was well known), etc.

SSA was the incremental culmination of going from there to figuring out a clustering/partitioning of variables that was not based on formally structured control flow regions (which are all-variables things), but instead based on local dataflow for a variable with incorporation of the part of the CFG structure that could actually affect the local result for a given variable. It was approached systematically - understanding the properties they needed, figuring out algorithms that solved them, etc.

Like a lot of things that turn out to be unexpectedly useful, take the world by storm, whatever, etc, history later seems to try to often represent them as a eureka moment.

[1] Bitvectors are assumed to be fixed size, and thus constant cost, but feel free to add another factor of N here if you want.

[2] This is not true in the sense that we knew how to recompute the result incrementally, and end up with a correct result, but doing so provably faster than solving the problem from scratch was not possible.

[3] They actually saw somewhat of a resurgence in the world of GPUs and more general computation graphs because it all becomes heavily applicable again to solving. However, we almost always have eventually developed easier to understand (even if potentially slower theory-wise) global algorithms and use those instead, because we have the compute power to do it, and this tradeoff is IMHO worth it.

Static single assignment (SSA) form provides an efficient intermediate representation for program analysis [Cytron et al., 1989, 1991]. It was introduced as a way of efficiently representing dataflow analyses.

SSA uses a single key idea: that all variables in the program are renamed so that each variable is assigned a value at a single unique statement in the program. From this key idea, SSA is able to provide a number of advantages over other techniques of dataflow analyses:

Factored use-def chain: With a def-use chain, dataflow results may be propagated directly from the assignment of a variable to all of its uses. However, a def-use chain requires an edge from each definition to each use, which may be expensive when a program has many definitions and uses of the same variable. In practice, this occurs in the presence of switch-statements. SSA factors the def-use chain over a φ-node, avoiding this pathological case.

Flow-sensitivity: A flow-insensitive algorithm performed on an SSA form is much more precise than if SSA form were not used. The flow-insensitive problem of multiple definitions to the same variable is solved by the single assignment property. The allows a flow-insensitive algorithm to approach the precision of a flow-sensitive algorithm.

Memory usage: Without SSA form, an analysis must store information for every variable at every program point. SSA form allows a sparse analysis, where an analysis must store information only for every assignment in the program. With a unique version per assignment, the memory usage of storing the results of an analysis can be considerably lower than using bit-vector or set-based approaches.

The llvm tutorials I played with (admittedly a long time ago) made it seem like "just allocate everything and trust mem2reg" basically abstracted SSA pretty completely from a user pov.

The essence of functional languages is that names are created by lambdas, labmdas are first class, and names might not alias themselves (within the same scope, two references to X may be referencing two instances of X that have different values).

The essence of SSA is that names must-alias themselves (X referenced twice in the same scope will definitely give the same value).

There are lots of other interesting differences.

But perhaps the most important difference is just that when folks implement SSA, or CPS, or ANF, they end up with things that look radically different with little opportunity for skills transfer (if you're an SSA compiler hacker then you'll feel like a fish out of water in a CPS compiler).

Folks like to write these "cute" papers that say things that sound nice but aren't really true.

edit: actually even discussed on here

CPS is formally equivalent to SSA, is it not? What are advantages of using CPS o... | Hacker News https://share.google/PkSUW97GIknkag7WY

    while (c < 10) { c *= 3; }

  %2 = alloca i32, align 4
  %3 = alloca i32, align 4
  store i32 %0, ptr %3, align 4
  br label %4, !dbg !18

  4:
  %5 = load i32, ptr %3, align 4, !dbg !19
  %6 = icmp slt i32 %5, 10, !dbg !20
  br i1 %6, label %7, label %10, !dbg !18

  7:
  %8 = load i32, ptr %3, align 4, !dbg !21
  %9 = mul nsw i32 %8, 3, !dbg !21
  store i32 %9, ptr %3, align 4, !dbg !21
  br label %4, !dbg !18

As some evidence to the second point: Haskell is a language that does enforce immutability, but it's compiler, GHC, does not use SSA for main IR -- it uses a "spineless tagless g-machine" graph representation that does, in fact, rely on that immutability. SSA only happens later once it's lowered to a mutating form. If your variables aren't mutated, then you don't even need to transform them to SSA!

Of course, you're welcome to try something else, people certainly have -- take a look at how V8's move to Sea-of-Nodes has gone for them.

As a fan of a functional language, immutability doesn't mean state doesn't exist. You keep state with assignment --- in SSA, every piece of state has a new name.

If you want to keep state beyond the scope of a function, you have to return it, or call another function with it (and hope you have tail call elimination). Or, stash it in a mutable escape hatch.

The functional programmers have decided mutability is evil. The imperative programmers have not.

We functional programmers do crazy stuff and pretend we're not on real hardware - a new variable instead of mutating (how wasteful!) and infinite registers (what real-world machine supports that!?).

Anyway, there's plenty of room for alternatives to SSA/CPS/ANF. It's always possible to come up with something with more mutation.

It's about naming intermediate states so you can refer to them in some way.

Mutation is the result of sloppy thinking about the role of time in computation. Sloppy thinking is a hindrance to efficient and tractable code transformations.

When you "mutate" a value, you're implicitly indexing it on a time offset - the variable had one value at time t_0 and another value at time t_1. SSA simply uses naming to make this explicit. (As do CPS and ANF, which is where that "equivalence" comes from.)

If you don't use SSA, CPS, or ANF for this purpose, you need to do something else to make the time dimension explicit, or you're going to be dealing with some very hairy problems.

"Evil" in this case is shorthand for saying that mutable variables are an unsuitable model for the purpose. That's not a subjective decision - try to achieve similar results without dealing with the time/mutation issue and you'll find out why.

Sure, you still need to keep those algorithms in place for being able to reason about memory loads and stores. But if you put effort into kicking memory operations into virtual register operations (where you get SSA for free), then you can also make the compiler faster since you're not constantly rerunning these analyses, but only on demand for the handful of passes that specifically care about eliminating or moving loads and stores.

SSA isn't (primarily) concerned with memory, it's concerned with local variables. It completely virtualizes the storage of local variables--in fact, all intermediate computations. By connecting computations through dataflow edges and not storage, it removes ordering (except that induced by dependence edges) from consideration.

It is, after all, what CPUs do under the hood with register renaming! They are doing dynamic SSA, a trick they stole from us compiler people!

And SICP is still relevant, even more so today with concurrency problems all over. If in Intel Chips, threads with locking solutions, or multi-processing. Shared state should not exist, and functional languages did win there.

SSA makes dataflow between operations explicit; it completely eliminates the original (incidental) names from programs. Because of that, all dataflow problems (particularly forward dataflow problems) get vastly simpler.

With SSA you can throw basically all forward dataflow problems (particularly with monotonic transformations) into a single pass and they all benefit each other. Without SSA, you have every single transformation tripping over itself to deal with names from the source program and introducing transformations that might confuse other analyses.

I know we teach different compiler optimizations at different stages, but it's really important to realize that all of them need to work together and that having each as a separate pass is a good way to fail at the phase ordering problem.

And going further with the sea-of-nodes representation just makes them all more powerful; I really do recommend reading Cliff Click's thesis.

The transition from one function to another is to copy the arguments to some common location, jump the instruction counter, then copy the values our of that location to give the initial values of the parameters.

These are the same thing, except that the branch in the first case is always at the end of the block. In the tail position.

Your preferred looping construct of branching between basic blocks is _identically_ tail calls between functions, except that some of the block arguments are represented as phi nodes. This is because the blocks are functions.