Rotom: Layout Representation updates for tiled layouts

[edwjchen]

Three attribute-layer changes preparing the layout search for split and tiled packings:

Part 1 of #2980.

• preprocessLayoutData produces a LayoutPiece list and a deduped axes table that maps one ISL domain variable per logical tensor dim. A traversal piece is a mixed-radix digit (i / divBy) mod modBy of its axis's index, so an axis split into several pieces (tiled or ct/slot-straddling layouts) shares one ISL domain variable.
• The axes table canonicalizes to ascending dim id. Previously row- and column-major materialized identical relations, making conversions between them silently free when lowered to the ISL domain.
• The dim assembly separates ciphertext dims from slot dims with | (e.g., [ct pieces | slot pieces]). Unused slots is written as an explicit gap piece.
• Replication and gap dims print as R and G ([R:4:1], [G:4:1]); the numeric ids -1/-2 are still accepted on input and round-trip to the letter forms.

[j2kun]

I guess one thing I don't quite understand about the design here is explained with this example:

#rotom.layout<n = 4, dims = [[0:8:1]]> 

Here you'd split the single dim into two pieces because it has size 8 and there are only 4 slots, right?

What is the benefit of doing that instead of, say, explicitly representing what piece is slots and what piece is ciphertext?

#rotom.layout<n = 4, ct_dims = [[0:2:4]], slot_dims = [[0:4:1]]> 

I guess what I'm saying is: if you explicitly represented this, would you be able to avoid most of the work in this PR?

[edwjchen]

The layout representation is now updated to have a clear separation between ct and slot dimensions:

#rotom.layout<n = 4, dims = [[0:2:4]  | [0:4:1]]>

where LayoutPieces before the | correspond to ct dims and after the | correspond to slot dims.

The remaining changes in this PR are for adding robustness checks to constructing a Rotom layout (in preprocessLayoutData) and facilitating lowering a Rotom layout into an ISL relation (by canonicalizing the relationship between LayoutPieces and the original tensor axes.