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Canonicalization and canonical forms are one dimension of organizing the work of an optimizing compiler.


A lot of code constructs can be written in multiple ways. For example:

   x + 4
   4 + x
   (x + 2) + 2

Canonicalization means picking one of these forms to be the canonical form, and then going through the program and rewriting all constructs which are equivalent to the canonical form into the canonical form.

In the case of add, it’s common to pick the form that has a constant on the right side, so we’d rewrite all these constructs to x + 4.

Why is canonicalization useful?

The goal of canonicalization is to make subsequent optimizations more effective. This is a key point, and we’ll get into some of the subtleties below. But there are a lot of cases where it’s just obviously a good thing to do. It means that subsequent optimizations that look for specific patterns of code only have to look for the canonical forms, rather than all forms.

Another way of saying this is, having a canonicalization pass is a way of factoring out the parts in a compiler that know all the different forms x + 4 could take, so that most optimization passes don’t have to worry about this. They don’t have to look for 4 + x, because they can assume that looking for x + 4 covers that. Handy!

How do we choose a canonical form?

Sometimes it’s easy. The canonical form for 2 + 3 is 5, because that’s clearly simpler in every possible way.

Sometimes it’s an arbitrary choice. It often doesn’t matter than much whether one picks 4 + x over x + 4, but it is helpful to pick one or the other, so sometimes it’s just human aesthetics.

It’s tempting to pick whatever form would be fastest, or optimal, on the target machine. And indeed, sometimes what’s fastest aligns with what’s simplest. 2 + 3 canonicalizing 5 is typically such a case. But sometimes it doesn’t.

Yeah so what about x * 2

x * 2 is equivalent to x + x; which of these should be the canonical form? It might seem like we might want to say: pick whatever’s optimal for the target architecture. Addition is generally faster than multiplication, so that would suggest we pick x + x as the canonical form.

But, x + x can actually make things harder for subsequent optimizations, because it means that now x has multiple uses. Having multiple uses makes some optimizations more complex – in terms of the dependence graph, this is a DAG rather than a tree, and trees are generally simpler to work with. So maybe x * 2 is actually a better canonical form, even if it’s a worse optimal form.

That said, we might consider canonicalizing this to x << 1, which has only one use of x, and has the nice property of making it as obvious as possible that the least significant bit of the result is zero. That way, we don’t have to have as much random knowledge of multiplication by powers of two strewn throughout the compiler.

Efficiency on the target machine still matters, but we can defer thinking about that until codegen, where it’s no longer as important to enable subsequent optimizations. At that point, we’re going to start caring about picking between +, *, and << based on which one executes fastest.

The basic philosophy of canonicalization says that canonical forms should be translated into optimal forms toward the back of the compiler, after all mid-level optimizations which benefit from canonical form are done. It’s also worth noting that codegen itself benefits from having the code coming into it be in canonical form, so that it doesn’t have to recognize all the ways to write x << 1, and can just recognize one pattern for that and emit the optimal code for it.

This is often a source of confusion: Is canonicalization the same as optimization? It’s often done as part of the “optimizer”, and many of the things it does produce more optimal code directly. But ultimately, in its purest form, canonicalization just focuses on removing unnecessary variation so that subsequent optimizations can be simpler.

Canonical form, canonical form
Canonical form hates optimal form
They have a fight, canonical wins
Canonical form…

(sung to the tune of “Particle Man” by They Might Be Giants)

Sometimes it’s ambiguous: redundancy elimination

Is redundancy elimination a canonicalization or an optimization?

It’s certainly simpler to compute a given value once and reuse the value, rather than compute it twice. But does that aid subsequent optimizations? It depends.

A case where it does aid subsequent optimizations is when it eliminates redundant memory accesses. That way, it’s essentially saying that no later passes have to even ask what the dependencies are for a given memory access, because the memory access has been eliminated.

A case where it doesn’t is where it can take expression trees where everything has a single use and give some values multiple uses. Some kinds of optimization passes are harder to do on DAGs than on trees, so this can result in pessimizations in some cases, depending on what kinds of things the rest of the compiler is doing.

However, we typically do think of redundancy elimination as being a canonicalization. It’s trivial to convert multiple-use values into single-use values by duplicating code, while going the other direction requires some analysis.

Even more ambiguous: inlining

Inlining can act like canonicalization, especially in cases where the inlined function body can be optimized away. However, thinking of inlining in terms of canonicalization doesn’t lead to a natural threshold. Should we maximally inline everything as far as possible? Or should we do the reverse and maximally outline and deduplicate outlined functions? Neither extreme is particularly practical, so most compilers use heuristics in practice rather than having a rigid definition of canonical form relative to calls.

That said, even though there’s no clear boundary, we can imagine a rough guideline. Think of old-school C code, written at a time when “C is a portable assembly language” was more true than it is today, where calls that were important to inline were written as macros. In many ways, one of the jobs of compilers for higher-level languages is to compile them down to roughly this level, so that they can be optimized using optimization techniques which work well at this level. In theory, we could define the task of a canonicalizing inliner to just be to inline code down to what it would have looked like if that same code had been written in old-school C (at least with respect to inlining). That’s not very pure, and it’s difficult to precisely describe, but it’s an intuitive and relatively practical compromise between extremes.

Canonical form isn’t just for arithmetic expressions!

Everything in an IR may be subjected to canonicalization.

For example, a control flow canonicalization might involve sorting the basic blocks of a program into Reverse Post-Order (RPO). Doing this is a simple way to ensure that the code is optimized the same regardless of how the user organizes the code inside their functions.

Dead code elimination is also a kind of canonicalization. The canonical form for dead code is no code.

In aggressive loop-transforming compilers, another form of canonicalization is to maximally fission loops into as many parts as possible, and then assume subsequent passes will fuse them back together into optimal loops that effectively utilize available registers and cache space.

Canonicalization as compression

It’s often the case that smaller forms are preferred over longer forms, so canonicalization tends to make code smaller, making it a form of compression. Also, since it effectively reduces non-essential entropy, it can make subsequent general-purpose compression more effective as well.

If we could conceptually perform all theoretically possible canonicalizations on a program, we’d end up with something related to its Kolmogorov Complexity (it may not be identical, since canonicalization puts the needs of subsequent optimizations first, rather than absolute compression). Maximal canonicalization is frequently impossible in practice, because of the halting problem, but also because even in cases where it’s theoretically possible, it can require impractical amounts of computation.

That said, it is pretty fun to think that for any given program, there is a theoretical “maximally canonical form” for that program, that all equivalent ways of writing that program could be reduced to. It is tempting to think of this as a kind of pure essence of the program.

Excessive canonicalization

Canonicalization discards inessential information. However, sometimes that information can be useful to preserve.

An example arises in instruction scheduling: Say a user writes code like

  x = a + b;
  y = c * d;

Assuming there’s no aliasing going on here, there’s no reason why one of these statements has to be ordered before the other. Their order in the user’s source code is inessential information. “Sea of nodes” style compilers may canonicalize to the point where there is no inherent ordering between these two statements.

The compiler backend ultimately has to produce machine code, which on conventional architectures requires the compiler to pick some ordering. Compilers can be pretty smart, and can take into consideration many things, such as available execution resources before and after these statements to know what order a CPU would prefer to see them in. However, as smart as they can be, compilers can’t always find the optimal answers. Optimal instruction scheduling is NP complete, but also, it may come down to runtime factors that ahead-of-time compilers don’t have. And CPU hardware performance characteristics aren’t always fully documented.

“Do No Harm”?

Most software has no idea how it’ll be mapped on to CPU pipelines. But some does. And it’s these cases where getting the mapping right is most important.

On such software, optimizations that make actual improvements are fine, however it can be important that compilers not make anything worse than if they had translated the code naively. Aggressively canonicalizing compilers have a risk that they will throw away information and at the end reconstruct a form which is worse than if they had just simply translated the code as it was written.

This, along with the ambiguous cases above, suggests that canonicalization be used in practical rather than rigid ways.

The theoretical shape of optimization.

A compiler typically starts with human-written source code.

Typical human-written code follows various human-oriented sensibilities. Different humans may have different aesthetic sensibilities, and this can lead to writing the same code in different ways. But of course this isn’t interesting to optimizers.

So the first then we typically do when the code hits the optimizers is to start canonicalizing. Throw away useless fluff that humans imbue their code with.

Canonicalizing optimizations are cascading; often doing one canonicalization will enable more. Folding an expression to a constant may allow other expressions to be folded to constants. Replacing a load by forwarding a value through a prior store may create a direct edge between two expression trees and introduce opportunities to simplify further.

So ignoring the practical concerns we mentioned above, we can imagine a theoretical compiler that does all the canonicalization it knows how to do up front. And while we don’t realistically approach Kolmogorov Complexity levels, we do lift the program closer to what we might think of as its essence, the simplest form that does what it needs to do.

And then, the compiler can begin to optimize, rewriting canonical forms into optimal forms, as it lowers the code all the way down to assembly code.

For practical reasons, it isn’t always possible to build compilers in terms of a purely canonicalizing phase and a purely optimizing phase, however this can be a useful reference point for understanding compilers in practice.