*I’ll be defending on Friday, December 13th; that will likely
mark the end of my copious free time. I hope to write a couple more
braindumps on SBCLy topics in the near future, before I forget
everything.*

SBCL is known for its flow-sensitive type propagation pass… or, rather, for its effects. That is probably why it can be so shocking when SBCL fails to infer obvious invariants. I feel like I can now describe the root causes of these surprising weaknesses and propose a series of fixes. Coding that up would likely take months, but the result should be more precise type propagation with improved asymptotic efficiency, and even a tunable knob between precision and compile times.

The compiler in SBCL, Python, doesn’t work on an IR in SSA form, nor does it convert internally to CPS à la Rabbit/Orbit. However, a clever – I don’t recall seeing it described anywhere else – design decision achieves similar effects. Operations receive their arguments as linear vars (LVARs) and write their results to an LVAR. Each LVAR has exactly one destination (one reader node), and, on any execution path, can only be used (written to) by one node. LVARs may nevertheless have multiple uses: these correspond to the value of conditionals and of conditional non-local exits. Assignment and access to lexical variables are represented as SET and REF nodes that receive and produce values through LVARs. This means that nearly everything can work in terms of LVARs when manipulating code and reasoning about dataflow. The conversion of read-once lexical variables to LVARs further simplifies a lot of mostly functional code.

Python takes this split to the extreme: only the constraint (type) propagation pass really handles full-blown lexical variables. Everything else depends on the flow-sensitive type information summarised in CFG nodes and joined in LVARs. Want to check whether a given argument is always positive? Simply test for subtyping on the LVAR’s derived type to implicitly leverage flow-sensitive type information. Want to insert some computation (e.g. a type assertion or an explicit modulo for overflowing arithmetic) between a value and its destination? Insert a new node in the CFG and substitute around nothing but opaque LVARs. These operations are so natural in Python that I only recently realised the design decision that makes them so easy.

However, I believe that Python went too far in relegating lexical
variables to only the constraint pass. That pass *only* handles
lexical variables (LAMBDA-VARs) and flow-sensitive information about
them, and only propagates three kinds of constraints (derived
information):

- lexical variable V is (is not) of type T at this program point (because of a type test or assertion);
- lexical variable V is (is not) < or > than a value of type T at this program point (because of a comparison);
- lexical variable V is (is not) EQL to an LVAR, to another lexical variable U, or to a constant at this program point (because of an assignment/reference to a variable or a comparison).

The importance of EQL constraints is subtle. These constraints represent the way that, e.g., information about a lexical variable is also true of the result of a lookup for that variable’s value (and vice versa), even if that information is learned after the variable is accessed. Similarly, when two lexical variables are EQL, information about one is true of the other. Finally, equality with a constant is always useful for constant propagation.

All three constraint types are centered on lexical variables. The reason is that no flow-sensitive information needs to be computed for LVARs: they are only used once. For example, if a type test is performed on an LVAR, that is the LVAR’s only appearance and there is no information to propagate… unless that information is also true of some lexical variable.

There’s some more clever engineering and the analysis is simplified by disregarding variables that may be assigned from closures, but that’s the gist of it. A vanilla dataflow analysis worklist loop propagates information around, and constraints sets for each basic block shrink until the least (in terms of types, greatest when measuring information) fixed point is reached.

The first weakness is caused by the aforementioned cleverness: knowledge about the state of the world as each basic block is entered and exited is represented with sets of constraints (bitsets indexed with consecutive indices assigned to constraints on demand). In the interest of speed, join and meet are implemented as (bit-)set union/intersection. The problem with this approach is that it completely discards information about a lexical variable when it is not identically present in both sets. For example, when one predecessor says that V is a positive integer and another that V is an integer, their successor discards all information about the type of V.

In practice, this is a lesser problem than it first appears to be: LVARs merge information about their multiple uses (writes) without any loss. The kind of code that suffers is code that does things like:

```
(lambda (x)
(ecase x
(1 ...)
(2 ...))
[work on X, knowing that it is either 1 or 2])
```

The two branches of the ecase that eventually continue execution respectively derive that X is EQL to 1 and to 2. However, their shared successor combines that information by forgetting about both equalities rather than weakening/downgrading the two constraints into “X is of type (INTEGER 1 2).”

Another interesting case is

```
(lambda (x)
(let ((y nil)) ; IF expression translated to an effectful IR
(if x
(setf y 1)
(setf y 2))
y))
```

for which Python derives that the result is either 1, 2, or NIL (the
union of all the values ever assigned to Y). The equivalent code
`(lambda (x) (if x 1 2))`

compiles everything to LVARs and the result is
known to be either 1 or 2 even before constraint propagation.

This can be addressed with a redundant representation that adds, to each constraint set, a dictionary from lexical variables to the (downgradable) information known about them: their types and the types they’re known to be < or > than. When intersecting constraint sets, these dictionaries make it easy to weaken information and insert the corresponding constraints.

The second weakness is deeply embedded in the structure of the compiler: the constraint propagation pass only pushes around pre-computed flow-insensitive information. That information is valid because the flow-insensitive pass (that computes things like the type of the values produced by each CFG node) works from an older (over) approximation of the flow-sensitive information. Once a fixed point is reached in the constraint propagation pass, the flow-insensitive pass is re-executed and information refined. In the end, there is a back-and-forth between the simple flow-sensitive constraint propagation pass and the flow-insensitive “optimization” pass that leverages a huge knowledge base about Common Lisp operators and functions.

That exchange of information not only takes many (hefty) iterations to converge, but is also lossy: the constraint propagation pass is flow-sensitive, but manipulates types based on an earlier pass or on type declarations. In effect, constraint propagation computes the least fixed point of a function that is defined by an earlier coarse upper approximation… a process that ends up preserving a lot of the initial weakness. This strange choice presents an interesting characteristic: every iteration the outer communication loop between flow -sensitive and -insensitive analysis produces valid (but overly conservative) information. So convergence is slow, but at least intermediate results can guide rewrites safely.

The obvious fix is to derive node types within the constraint propagation pass, even as the final result of the flow-sensitive pass is approximated from below. For example, upon assignment to a lexical variable, the current pass adds a new constraint: the type of that variable is that of the LVAR for the assignment’s value, and the LVAR’s type is computed based on the information from the previous constraint propagation pass. Instead, it should be based on the current (overly agressive) flow-sensitive information. That would eventually converge to a smaller fixed point. This alternative design would also obviate the need for a questionable hack that improves precision for some lucky iteration variables.

A weaker form of this change would be to use preliminary (perhaps overly aggressive) information only to tentatively detect impossible conditional branches. This would enable Python to derive that the return value of

```
(lambda ()
(loop with x = 0
while (foo)
do (case x
(1 (setf x 2))
(2 (setf x 1)))
finally (return x)))
```

is always 0 (and never 1 or 2), or that the expression `(if (eql x y) (eql x y) t)`

is always true.

With suitable handling of calls to local functions, useful types might even be derived for recursive functions. The current type propagation pass derives insanely stupid types for recursive functions and for assignment to variables in loops (unless the iteration variable hack suffices). For example

```
(lambda ()
(labels ((foo (x)
(if (bar)
(foo (1+ x))
x)))
(foo 42)))
```

is derived to return an arbitrary NUMBER. Instead, the return type of local functions can be initialised to the empty type (in which case propagation must be stopped for the caller block) and extended as argument types grow and as recursion triggers more propagation.

The problem is that this more precise flow-sensitive pass loses the finite chain condition: there is no guarantee that the least fixed point will always be reached in finite time. The current constraint propagation pass operates on a finite lattice: only variables and types that exist in the input IR1 graph can appear in constraints, so the fixpointed function is defined over the powerset of this finite set of constraints for each basic block. In contrast, the full CL type universe is most definitely not finite (type tests aren’t even decidable in general).

There is a simple solution to this problem: before extending the constraint universe, widen types to satisfy the finite chain condition, i.e., make sure that types eventually stop growing. For example, when taking a non-trivial union or intersection of types, the result could be quantised to a coarse set of types (singleton types, named class types, a few key numeric ranges, etc.). Appropriate widening can also accelerate finite but slow convergence; the resulting fixed point isn’t guaranteed to be minimal, but is always valid. In fact, widening is likely to be necessary for the first improvement (with redundant dictionaries) as well.

Another issue in the constraint propagation pass is an instance of the phase ordering problem. Some source-to-source transforms help type propagation (e.g., turning complicated functions into simpler ones for which type propagation is feasible), and others hinder it (e.g., converting integer division by a constant to multiply and shift or MAPCAR to an inline loop). Worse, there’s a dependency cycle as transforms rely on type propagation to determine whether they are applicable. There’s been some academic work that exposes multiple equivalent code sequences to analyses, and it seems possible to do something similar in Python as well: source-to-source transforms generate functions, so we’d “only” have to analyse the function without splicing it in. However, that’s likely to be impractically slow (even termination is non-obvious)… perhaps it will be doable for select transforms that expand into pure expressions.

The only suggestion to accelerate constraint propagation so far is to widen types, and that mostly compensates for slowdowns (or infinite convergence) introduced by other changes. There’s a classic trick that ought to accelerate even the current pass: guide the order in which basic blocks are analysed by partitioning them in strongly connected components. Python exploits a simple ordering technique, a depth-first ordering that minimises backward edges in reducible control-flow graphs. Basic blocks could instead be partitioned into strongly connected components: contracting each strongly connected component into a single vertex leaves a directed acyclic graph. It then suffices to fixpoint on each strong component and traverse the condensed DAG so that each component is only visited after all its predecessors. Python already computes loop nests, so that’s an easy change; it’s likely even better to look at nested loops and recursively order analysis within each component. A few other passes perform some form of dataflow analysis and would benefit from that improved ordering… perhaps compile times can be really improved (Amdahl says that’s unlikely until splicing in source-to-source transformations is a lot quicker).

Recap :

- Add a dictionary for TYPEP and </> constraints on LAMBDA-VARs in consets; use it to merge and weaken information in consets.
- Derive flow-insensitive types within the flow-sensitive pass, for quicker final convergence to a tighter solution.
- Simplify conditional branches with flow-sensitive information, even when that information is still optimistic.
- Widen types for quicker/finite convergence.
- See what source-to-source transformations are applicable with the current information, and use that to derive stronger types.
- Analyse basic blocks based on loop nests: fixpoint a loop nest before visiting its successors, and wait until all predecessors have reached a fixed point before analysing a loop’s blocks.

In terms of implementation difficulty, these changes probably go 1-6-3-4-2-5. This also happens to be the order in which I’d try to implement them.