> If your static analysis is over SSA, then generally the static analysis is easier and (potentially) storing facts is easier. This is due to this property called sparseness. Where a static analysis over non-SSA programs has to store facts about all variables at all program points, an analysis over SSA need only store facts about all variables, independent of context.
Not sure what this has to do with SSA or not? Maybe you're saying if you don't have SSA then value numbering requires extra book-keeping to keep track of the names? that's true but the book-keeping only kicks when a name is updated, not at every program point.
> If we want to keep our information sparse, though, we have to add a new definition to the IR.
I mean this isn't like a law of nature, it's just a design point - a question of whether the facts have to be serializable for some reason. If they don't (and in general, I can't think of a reason they would have to be), you can just keep things in-memory in general and do either sparse or dense analysis on an as-needed basis. To wit MLIR, an SSA IR, handily supports both sparse and dense dataflow analysis:
https://github.com/llvm/llvm-project/blob/main/mlir/include/...
https://github.com/llvm/llvm-project/blob/main/mlir/include/...
Note, both analyses propagate lattices which support meet and join.
> Anyway, SSA only stores type information about instructions and does not encode information that we might later learn in the IR. With basic SSA, there’s not a good way to encode refinements…
This is true of course but I struggle to think of a reasonable semantic that would enable you to refine/change/alter the type of a value that wasn't an operation. The conditional example is specious but could be rectified if it were modeled correctly: the conditional is an operation which has operands and two regions, each with a basic block with basic block args and it's the types of the bb args that can show as "refined". MLIR's scf.if isn't "isolated from above" and its bbs don't have args but it could have both such things and such an operation would model what you want precisely and in the IR/types themselves.