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1 # MLIR: The case for a simplified polyhedral form
4 representing and transforming dense numerical kernels, but it uses a form that
10 that explored the tradeoffs of using this simplified form vs the traditional
11 polyhedral schedule list form. At some point, this document could be dusted off
13 it in this crafty form than not to. Beware that this document uses archaic
26 represented in multidimensional loop form. The idea is that a CFG function is
54 exactly what form is best to perform the analyses and transformations we want?
80 to represent this with a classical form like (syntax details are not important
169 This form has great power, and the polyhedral code generator (which lowers from
178 form, we would have a representation where all instructions inside of a given
180 the simplified form, we also have an "if" instruction that takes an affine
225 density. The 'traditional' form introduces an extra level of abstraction
228 out after code generation. With the simplified form, transformations have to do
254 one form are representable in the other form in all cases.
273 its loops. The simplified form pays for this by eliminating schedules and
276 simplified form supports instruction duplication.
278 It is important to point out that the traditional form wins on compactness in
286 required as part of lowering to the target machine. The simplified form has a
290 In contrast, the traditional form has significant complexity in the lowering
297 An advantage for the traditional form is that it is easier to perform certain
302 In practice, the simplified form requires moving the complexity of code
316 The simplified form is much easier for analyses and transformations to build
319 what target code will be generated. With the traditional form, these analyses
336 The simplified form defines this away: the concepts in the IR remain simple, and
344 they are defined based on traditional dominance. In the simplified form, this is
349 In the traditional form though, this is not the case: it seems that a lot of
351 form is correct or not. For example, this is invalid code:
367 produce SSA form will need to be aware of this and be able to model what codegen
375 The traditional form has multiple encodings for the same sorts of behavior: you
389 compactness of the traditional form is a negative for this purpose: reasoning
394 In the simplified form, the effect of "code generation" is always obvious from
405 1. Early performance transformations could be done on the traditional form.
406 1. Partial code generation lowers to the simplified form
408 transforms that don't benefit much from the traditional form could be run.